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THE CONTRIBUTIONS OF
JOSEPH SAUVEUR (1653-1716)
TO ACOUSrpICS
VOLTlME I
Presented by
Robert E Maxham
To fulfill tne disserta tion requirement for the degree of
Doctor of Phtlosophy
Department of Musicology and Music History
Thesis Director Erich P Schwandt
Eastman School of Music
of the
University of Rochester
March 1976
iA i 2
i
~ ) ~ )
vrrA
Robert 11nxham was born in Erie Pennsyl vnnla on
March 1 1947 and attended pnrochial schools there radushy
sting from Cathedral preparatory School with honors 1n
ItJGb liaving obtained his bachelors deproe with
distinction at the Eastman School of fIIusic in 1969 he
studied philosophy and theology at st Marks Seminary
in Erie bull Charles Horromeo Seminary in Philadelphia
where he won the Hanna Cusick Ryan Prize for General
Excellence in Bunrlamental Theology and St Mary 1 s
Seminary and University in Baltimore tie returned to the
stlJdy of musiC receiving the Master of Arts degree in
Musicology from the strnan School in 1975 ~he next
year was spent in preparation in absentia of the
present study
ii
PREFACE
Although Joseph Sauveur (1653-1716) has with some
justification been named as the founder of the modern
science of acoustics a science to which he contributed
not only clarificatory terminology ingenious scales and
systems of measurement brilliant insights and wellshy
reasoned principles but the very name itself his work
has been neglected in recent times The low estate to
which his fortune has fallen is grimly illustrated by the
fact that Groves Dictionary not containing an article
devoted exclusively to him includes an article on
acoustics which does not mention his name even in passing
That a more thorough investigation of Sauveurs
works may provide a basis for further exploration of the
performance practices of the period during which he lived
is suggested by Erich Schwandt 1 s study of the tempos of
dances of the French court as they are indicated by
-Michel LAffilard Schwandt contends that LAffilard
misapplied Sauveurs scale for the measl~ement of temporal
duration and thus speci fied tempos which are twice too
fast
Sauveurs division of the octave into 43 a~d
further into 301 logarithmic degrees is mentioned in the
various works on the theory and practice of temperament
iii
written since his time A more tho~ou~h inveot1vntinn of
Sauveurs works should make possible a more just assessment
of his position in the history of that sctence or art-shy
temperinp the 1ust scale--to which he is I1811a] 1y
acknowled~ed to have h~en an i111nortant contrihlltor
rhe relationship of Srntvenr to tho the()rl~~t T~nnshy
Philippe Rameau ~hould also he illuminate~ by a closer
scrutiny of the works of Sauvcllr
It shall he the program of this study to trace
ttroughout Sauveurs five oub1ished Mfmo5res the developshy
ment (providing demonstrations where they are lacking or
unclear) of four of his most influential ideas the
chronometer or scale upon which teMporal ~urntions cnu16
be measured within a third (or a sixtieth of a second) of
time the division of the octave into 43 and further
301 equal p(lrtfl and the vnr10u8 henefi ts wrich nC(~-rlH~ fr0~~
snch a division the establishment of a tone with a 1Ptrgtl_
mined number of vibrations peT second as a fixed Ditch to
which all others could be related and which cou]n thus
serve as a standard for comparing the VqriOl~S standaTds
of pitch in use throughout the world an~ the ~rmon1c
series recognized by Sauveur as arisin~ frnm the vib~ation
of a string in aliquot parts The vRrious c 1aims which
have been mane concerning Sauveurs theories themselves
and thei r influence on th e works of at hels shall tr en be
more closely examined in the l1ght of the p-receding
exposition The exposition and analysis shall he
1v
accompanied by c ete trans tions of Sauveu~ls five
71Aemoires treating of acoustics which will make his works
available for the fipst time in English
Thanks are due to Dr Erich Schwandt whose dedishy
cation to the work of clarifying desi~nRtions of tempo of
donees of the French court inspiled the p-resent study to
Dr Joel Pasternack of the Department of Mathematics of
the University of Roc ster who pointed the way to the
solution of the mathematical problems posed by Sauveurs
exposition and to the Cornell University Libraries who
promptly and graciously provided the scientific writings
upon which the study is partly based
v
ABSTHACT
Joseph Sauveur was born at La Flampche on March 24
1653 Displayin~ an early interest in mechanics he was
sent to the Tesuit Collere at La Pleche and lA-ter
abandoning hoth the relipious and the medical professions
he devoted himsel f to the stl1dy of Mathematics in Paris
He became a hi~hly admired geometer and was admitted to
the lcad~mie of Paris in 1696 after which he turned to
the science of sound which he hoped to establish on an
equal basis with Optics To that end he published four
trea tises in the ires de lAc~d~mie in 1701 1702
1707 and 1711 (a fifth completed in 1713 was published
posthu~ously in 1716) in the first of which he presented
a corrprehensive system of notation of intervaJs sounds
Lonporal duratIon and harrnonlcs to which he propo-1od
adrlltions and developments in his later papers
The chronometer a se e upon which teMporal
r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth
of a second) of time represented an advance in conception
he~Tond the popLllar se e of Etienne Loulie divided slmnly
into inches which are for the most part incomrrensurable
with seco~ds Sauveurs scale is graduated in accordance
wit~1 the lavl that the period of a pendulum is proportional
to the square root of the length and was taken over by
vi
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
iVORKS CITED
Backus tJohn rrhe Acollstical Founnatlons of Music New York W W Norton amp Company 1969
I~ull i w f~ouse A ~hort Acc011nt of the lIiltory of ~athemrltics New York Dovor Publications l~)GO
Barbour tTames Murray Equal rPemperament Its History from Ramis (1482) to Hameau (1737)ff PhD dissertashytion Cornell University 1932
Tuning and Temnerament ERst Lansing Michigan State College Press 1951
Bosanquet H H M 1f11 emperament or the division of the octave iiusical Association Froceedings 1874-75 pp 4-1
Cajori 1lorian ttHistory of the Iogarithmic Slide Rule and Allied In[-)trllm8nts II in Strinr Fipures rtnd Other F[onorrnphs pre 1-121 Edited by lrtf N OUSG I~all
5rd ed iew York Ch01 sea Publ i shinp Company 1 ~)G9
Culver Charles A Nusical Acoustics 4th edt NeJI York McGraw-hill Book Comnany Inc 1956
Des-Cartes Hene COr1pendium Musicae Rhenum 1650
Dtctionaryof Physics Edited hy II bullbullT Gray New York John lJiley and Sons Inc 1958 Sv Pendulum 1t
Dl0 MllSlk In Gcnchichte lJnd Goponwnrt Sv Snuvellr J os e ph II by llri t z Iiinc ke] bull
Ellis Alexander J liOn an Improved Bimodular Method of compnting Natural and Tahular Loparlthms and AntishyLogarithms to Twelve or Sixteen Places vi th very brief Tables II ProceedinFs of the Royal Society of London XXXI (February 188ry-391-39S
~~tis F-J Uiographie universelle des musiciens at b i bl i ographi e generale de la musi que deuxi erne ed it -j on bull Paris Culture et Civilisation 185 Sv IISauveur Joseph II
154
155
Fontenelle Bernard Ie Bovier de Elove de M Sallveur
Histoire de J Acad~mie des Sciences Ann~e 1716 Amsterdam Chez Pierre Mortier 1736 pre 97-107
bull Sur la Dete-rmination d un Son Fixe T11stoire ---d~e--1~1 Academia Hoyale des Sciences Annee 1700
Arns terdnm Chez Pierre fvlortler 17)6 pp Id~--lDb
bull Sur l Applica tion des Sons liarmonique s alJX II cux ---(1--O---rgues Histol ro de l Academia Roya1e des Sc ienc os
Ann~e 1702 Amsterdam Chez Pierre Mortier l7~6 pp 118-122
bull Sur un Nouveau Systeme de Musique Histoi 10 ---(~l(-)~~llclo(rnlo lioynle des ~)clonces AJneo 1701
Amsterdam Chez Pierre Nlortier 1706 pp 158-180
Galilei Galileo Dialogo sopra i dlJe Tv1assimiSistemi del Mondo n in A Treasury of 1Vorld Science fJP 349shy350 Edited by Dagobe-rt Hunes London Peter Owen 1962
Glr(l1 o~~tone Cuthbert tTenn-Philippe Ram(~lll His Life and Work London Cassell and Company Ltd 1957
Groves Dictionary of Music and usiciA1s 5th ed S v If Sound by Ll S Lloyd
Harding Hosamond The Oririns of li1usical Time and Expression London Oxford l~iversity Press 1938
Harvard Dictionary of Music 2nd ed Sv AcoustiCS Consonance Dissonance If uGreece U flIntervals iftusic rrheory Savart1f Solmization
Helmholtz Hermann On the Sensations of Tone as a Physiologic amp1 Basis for the Theory of hriusic Translnted by Alexander J Ellis 6th ed New York Patnl ~)nllth 191JB
Henflinrr Konrad Specimen de novo suo systemnte musieo fI
1iseel1anea Rerolinensla 1710 XXVIII
Huygens Christian Oeuvres completes The Ha[ue Nyhoff 1940 Le Cycle lIarmoniauetI Vol 20 pp 141-173
Novus Cyelns Tlarmonicus fI Onera I
varia Leyden 1724 pp 747-754
Jeans Sir tTames Science and Music Cambridge at the University Press 1953
156
L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b
Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945
Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______
1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7
flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696
Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II
Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
iA i 2
i
~ ) ~ )
vrrA
Robert 11nxham was born in Erie Pennsyl vnnla on
March 1 1947 and attended pnrochial schools there radushy
sting from Cathedral preparatory School with honors 1n
ItJGb liaving obtained his bachelors deproe with
distinction at the Eastman School of fIIusic in 1969 he
studied philosophy and theology at st Marks Seminary
in Erie bull Charles Horromeo Seminary in Philadelphia
where he won the Hanna Cusick Ryan Prize for General
Excellence in Bunrlamental Theology and St Mary 1 s
Seminary and University in Baltimore tie returned to the
stlJdy of musiC receiving the Master of Arts degree in
Musicology from the strnan School in 1975 ~he next
year was spent in preparation in absentia of the
present study
ii
PREFACE
Although Joseph Sauveur (1653-1716) has with some
justification been named as the founder of the modern
science of acoustics a science to which he contributed
not only clarificatory terminology ingenious scales and
systems of measurement brilliant insights and wellshy
reasoned principles but the very name itself his work
has been neglected in recent times The low estate to
which his fortune has fallen is grimly illustrated by the
fact that Groves Dictionary not containing an article
devoted exclusively to him includes an article on
acoustics which does not mention his name even in passing
That a more thorough investigation of Sauveurs
works may provide a basis for further exploration of the
performance practices of the period during which he lived
is suggested by Erich Schwandt 1 s study of the tempos of
dances of the French court as they are indicated by
-Michel LAffilard Schwandt contends that LAffilard
misapplied Sauveurs scale for the measl~ement of temporal
duration and thus speci fied tempos which are twice too
fast
Sauveurs division of the octave into 43 a~d
further into 301 logarithmic degrees is mentioned in the
various works on the theory and practice of temperament
iii
written since his time A more tho~ou~h inveot1vntinn of
Sauveurs works should make possible a more just assessment
of his position in the history of that sctence or art-shy
temperinp the 1ust scale--to which he is I1811a] 1y
acknowled~ed to have h~en an i111nortant contrihlltor
rhe relationship of Srntvenr to tho the()rl~~t T~nnshy
Philippe Rameau ~hould also he illuminate~ by a closer
scrutiny of the works of Sauvcllr
It shall he the program of this study to trace
ttroughout Sauveurs five oub1ished Mfmo5res the developshy
ment (providing demonstrations where they are lacking or
unclear) of four of his most influential ideas the
chronometer or scale upon which teMporal ~urntions cnu16
be measured within a third (or a sixtieth of a second) of
time the division of the octave into 43 and further
301 equal p(lrtfl and the vnr10u8 henefi ts wrich nC(~-rlH~ fr0~~
snch a division the establishment of a tone with a 1Ptrgtl_
mined number of vibrations peT second as a fixed Ditch to
which all others could be related and which cou]n thus
serve as a standard for comparing the VqriOl~S standaTds
of pitch in use throughout the world an~ the ~rmon1c
series recognized by Sauveur as arisin~ frnm the vib~ation
of a string in aliquot parts The vRrious c 1aims which
have been mane concerning Sauveurs theories themselves
and thei r influence on th e works of at hels shall tr en be
more closely examined in the l1ght of the p-receding
exposition The exposition and analysis shall he
1v
accompanied by c ete trans tions of Sauveu~ls five
71Aemoires treating of acoustics which will make his works
available for the fipst time in English
Thanks are due to Dr Erich Schwandt whose dedishy
cation to the work of clarifying desi~nRtions of tempo of
donees of the French court inspiled the p-resent study to
Dr Joel Pasternack of the Department of Mathematics of
the University of Roc ster who pointed the way to the
solution of the mathematical problems posed by Sauveurs
exposition and to the Cornell University Libraries who
promptly and graciously provided the scientific writings
upon which the study is partly based
v
ABSTHACT
Joseph Sauveur was born at La Flampche on March 24
1653 Displayin~ an early interest in mechanics he was
sent to the Tesuit Collere at La Pleche and lA-ter
abandoning hoth the relipious and the medical professions
he devoted himsel f to the stl1dy of Mathematics in Paris
He became a hi~hly admired geometer and was admitted to
the lcad~mie of Paris in 1696 after which he turned to
the science of sound which he hoped to establish on an
equal basis with Optics To that end he published four
trea tises in the ires de lAc~d~mie in 1701 1702
1707 and 1711 (a fifth completed in 1713 was published
posthu~ously in 1716) in the first of which he presented
a corrprehensive system of notation of intervaJs sounds
Lonporal duratIon and harrnonlcs to which he propo-1od
adrlltions and developments in his later papers
The chronometer a se e upon which teMporal
r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth
of a second) of time represented an advance in conception
he~Tond the popLllar se e of Etienne Loulie divided slmnly
into inches which are for the most part incomrrensurable
with seco~ds Sauveurs scale is graduated in accordance
wit~1 the lavl that the period of a pendulum is proportional
to the square root of the length and was taken over by
vi
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
iVORKS CITED
Backus tJohn rrhe Acollstical Founnatlons of Music New York W W Norton amp Company 1969
I~ull i w f~ouse A ~hort Acc011nt of the lIiltory of ~athemrltics New York Dovor Publications l~)GO
Barbour tTames Murray Equal rPemperament Its History from Ramis (1482) to Hameau (1737)ff PhD dissertashytion Cornell University 1932
Tuning and Temnerament ERst Lansing Michigan State College Press 1951
Bosanquet H H M 1f11 emperament or the division of the octave iiusical Association Froceedings 1874-75 pp 4-1
Cajori 1lorian ttHistory of the Iogarithmic Slide Rule and Allied In[-)trllm8nts II in Strinr Fipures rtnd Other F[onorrnphs pre 1-121 Edited by lrtf N OUSG I~all
5rd ed iew York Ch01 sea Publ i shinp Company 1 ~)G9
Culver Charles A Nusical Acoustics 4th edt NeJI York McGraw-hill Book Comnany Inc 1956
Des-Cartes Hene COr1pendium Musicae Rhenum 1650
Dtctionaryof Physics Edited hy II bullbullT Gray New York John lJiley and Sons Inc 1958 Sv Pendulum 1t
Dl0 MllSlk In Gcnchichte lJnd Goponwnrt Sv Snuvellr J os e ph II by llri t z Iiinc ke] bull
Ellis Alexander J liOn an Improved Bimodular Method of compnting Natural and Tahular Loparlthms and AntishyLogarithms to Twelve or Sixteen Places vi th very brief Tables II ProceedinFs of the Royal Society of London XXXI (February 188ry-391-39S
~~tis F-J Uiographie universelle des musiciens at b i bl i ographi e generale de la musi que deuxi erne ed it -j on bull Paris Culture et Civilisation 185 Sv IISauveur Joseph II
154
155
Fontenelle Bernard Ie Bovier de Elove de M Sallveur
Histoire de J Acad~mie des Sciences Ann~e 1716 Amsterdam Chez Pierre Mortier 1736 pre 97-107
bull Sur la Dete-rmination d un Son Fixe T11stoire ---d~e--1~1 Academia Hoyale des Sciences Annee 1700
Arns terdnm Chez Pierre fvlortler 17)6 pp Id~--lDb
bull Sur l Applica tion des Sons liarmonique s alJX II cux ---(1--O---rgues Histol ro de l Academia Roya1e des Sc ienc os
Ann~e 1702 Amsterdam Chez Pierre Mortier l7~6 pp 118-122
bull Sur un Nouveau Systeme de Musique Histoi 10 ---(~l(-)~~llclo(rnlo lioynle des ~)clonces AJneo 1701
Amsterdam Chez Pierre Nlortier 1706 pp 158-180
Galilei Galileo Dialogo sopra i dlJe Tv1assimiSistemi del Mondo n in A Treasury of 1Vorld Science fJP 349shy350 Edited by Dagobe-rt Hunes London Peter Owen 1962
Glr(l1 o~~tone Cuthbert tTenn-Philippe Ram(~lll His Life and Work London Cassell and Company Ltd 1957
Groves Dictionary of Music and usiciA1s 5th ed S v If Sound by Ll S Lloyd
Harding Hosamond The Oririns of li1usical Time and Expression London Oxford l~iversity Press 1938
Harvard Dictionary of Music 2nd ed Sv AcoustiCS Consonance Dissonance If uGreece U flIntervals iftusic rrheory Savart1f Solmization
Helmholtz Hermann On the Sensations of Tone as a Physiologic amp1 Basis for the Theory of hriusic Translnted by Alexander J Ellis 6th ed New York Patnl ~)nllth 191JB
Henflinrr Konrad Specimen de novo suo systemnte musieo fI
1iseel1anea Rerolinensla 1710 XXVIII
Huygens Christian Oeuvres completes The Ha[ue Nyhoff 1940 Le Cycle lIarmoniauetI Vol 20 pp 141-173
Novus Cyelns Tlarmonicus fI Onera I
varia Leyden 1724 pp 747-754
Jeans Sir tTames Science and Music Cambridge at the University Press 1953
156
L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b
Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945
Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______
1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7
flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696
Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II
Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
PREFACE
Although Joseph Sauveur (1653-1716) has with some
justification been named as the founder of the modern
science of acoustics a science to which he contributed
not only clarificatory terminology ingenious scales and
systems of measurement brilliant insights and wellshy
reasoned principles but the very name itself his work
has been neglected in recent times The low estate to
which his fortune has fallen is grimly illustrated by the
fact that Groves Dictionary not containing an article
devoted exclusively to him includes an article on
acoustics which does not mention his name even in passing
That a more thorough investigation of Sauveurs
works may provide a basis for further exploration of the
performance practices of the period during which he lived
is suggested by Erich Schwandt 1 s study of the tempos of
dances of the French court as they are indicated by
-Michel LAffilard Schwandt contends that LAffilard
misapplied Sauveurs scale for the measl~ement of temporal
duration and thus speci fied tempos which are twice too
fast
Sauveurs division of the octave into 43 a~d
further into 301 logarithmic degrees is mentioned in the
various works on the theory and practice of temperament
iii
written since his time A more tho~ou~h inveot1vntinn of
Sauveurs works should make possible a more just assessment
of his position in the history of that sctence or art-shy
temperinp the 1ust scale--to which he is I1811a] 1y
acknowled~ed to have h~en an i111nortant contrihlltor
rhe relationship of Srntvenr to tho the()rl~~t T~nnshy
Philippe Rameau ~hould also he illuminate~ by a closer
scrutiny of the works of Sauvcllr
It shall he the program of this study to trace
ttroughout Sauveurs five oub1ished Mfmo5res the developshy
ment (providing demonstrations where they are lacking or
unclear) of four of his most influential ideas the
chronometer or scale upon which teMporal ~urntions cnu16
be measured within a third (or a sixtieth of a second) of
time the division of the octave into 43 and further
301 equal p(lrtfl and the vnr10u8 henefi ts wrich nC(~-rlH~ fr0~~
snch a division the establishment of a tone with a 1Ptrgtl_
mined number of vibrations peT second as a fixed Ditch to
which all others could be related and which cou]n thus
serve as a standard for comparing the VqriOl~S standaTds
of pitch in use throughout the world an~ the ~rmon1c
series recognized by Sauveur as arisin~ frnm the vib~ation
of a string in aliquot parts The vRrious c 1aims which
have been mane concerning Sauveurs theories themselves
and thei r influence on th e works of at hels shall tr en be
more closely examined in the l1ght of the p-receding
exposition The exposition and analysis shall he
1v
accompanied by c ete trans tions of Sauveu~ls five
71Aemoires treating of acoustics which will make his works
available for the fipst time in English
Thanks are due to Dr Erich Schwandt whose dedishy
cation to the work of clarifying desi~nRtions of tempo of
donees of the French court inspiled the p-resent study to
Dr Joel Pasternack of the Department of Mathematics of
the University of Roc ster who pointed the way to the
solution of the mathematical problems posed by Sauveurs
exposition and to the Cornell University Libraries who
promptly and graciously provided the scientific writings
upon which the study is partly based
v
ABSTHACT
Joseph Sauveur was born at La Flampche on March 24
1653 Displayin~ an early interest in mechanics he was
sent to the Tesuit Collere at La Pleche and lA-ter
abandoning hoth the relipious and the medical professions
he devoted himsel f to the stl1dy of Mathematics in Paris
He became a hi~hly admired geometer and was admitted to
the lcad~mie of Paris in 1696 after which he turned to
the science of sound which he hoped to establish on an
equal basis with Optics To that end he published four
trea tises in the ires de lAc~d~mie in 1701 1702
1707 and 1711 (a fifth completed in 1713 was published
posthu~ously in 1716) in the first of which he presented
a corrprehensive system of notation of intervaJs sounds
Lonporal duratIon and harrnonlcs to which he propo-1od
adrlltions and developments in his later papers
The chronometer a se e upon which teMporal
r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth
of a second) of time represented an advance in conception
he~Tond the popLllar se e of Etienne Loulie divided slmnly
into inches which are for the most part incomrrensurable
with seco~ds Sauveurs scale is graduated in accordance
wit~1 the lavl that the period of a pendulum is proportional
to the square root of the length and was taken over by
vi
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
iVORKS CITED
Backus tJohn rrhe Acollstical Founnatlons of Music New York W W Norton amp Company 1969
I~ull i w f~ouse A ~hort Acc011nt of the lIiltory of ~athemrltics New York Dovor Publications l~)GO
Barbour tTames Murray Equal rPemperament Its History from Ramis (1482) to Hameau (1737)ff PhD dissertashytion Cornell University 1932
Tuning and Temnerament ERst Lansing Michigan State College Press 1951
Bosanquet H H M 1f11 emperament or the division of the octave iiusical Association Froceedings 1874-75 pp 4-1
Cajori 1lorian ttHistory of the Iogarithmic Slide Rule and Allied In[-)trllm8nts II in Strinr Fipures rtnd Other F[onorrnphs pre 1-121 Edited by lrtf N OUSG I~all
5rd ed iew York Ch01 sea Publ i shinp Company 1 ~)G9
Culver Charles A Nusical Acoustics 4th edt NeJI York McGraw-hill Book Comnany Inc 1956
Des-Cartes Hene COr1pendium Musicae Rhenum 1650
Dtctionaryof Physics Edited hy II bullbullT Gray New York John lJiley and Sons Inc 1958 Sv Pendulum 1t
Dl0 MllSlk In Gcnchichte lJnd Goponwnrt Sv Snuvellr J os e ph II by llri t z Iiinc ke] bull
Ellis Alexander J liOn an Improved Bimodular Method of compnting Natural and Tahular Loparlthms and AntishyLogarithms to Twelve or Sixteen Places vi th very brief Tables II ProceedinFs of the Royal Society of London XXXI (February 188ry-391-39S
~~tis F-J Uiographie universelle des musiciens at b i bl i ographi e generale de la musi que deuxi erne ed it -j on bull Paris Culture et Civilisation 185 Sv IISauveur Joseph II
154
155
Fontenelle Bernard Ie Bovier de Elove de M Sallveur
Histoire de J Acad~mie des Sciences Ann~e 1716 Amsterdam Chez Pierre Mortier 1736 pre 97-107
bull Sur la Dete-rmination d un Son Fixe T11stoire ---d~e--1~1 Academia Hoyale des Sciences Annee 1700
Arns terdnm Chez Pierre fvlortler 17)6 pp Id~--lDb
bull Sur l Applica tion des Sons liarmonique s alJX II cux ---(1--O---rgues Histol ro de l Academia Roya1e des Sc ienc os
Ann~e 1702 Amsterdam Chez Pierre Mortier l7~6 pp 118-122
bull Sur un Nouveau Systeme de Musique Histoi 10 ---(~l(-)~~llclo(rnlo lioynle des ~)clonces AJneo 1701
Amsterdam Chez Pierre Nlortier 1706 pp 158-180
Galilei Galileo Dialogo sopra i dlJe Tv1assimiSistemi del Mondo n in A Treasury of 1Vorld Science fJP 349shy350 Edited by Dagobe-rt Hunes London Peter Owen 1962
Glr(l1 o~~tone Cuthbert tTenn-Philippe Ram(~lll His Life and Work London Cassell and Company Ltd 1957
Groves Dictionary of Music and usiciA1s 5th ed S v If Sound by Ll S Lloyd
Harding Hosamond The Oririns of li1usical Time and Expression London Oxford l~iversity Press 1938
Harvard Dictionary of Music 2nd ed Sv AcoustiCS Consonance Dissonance If uGreece U flIntervals iftusic rrheory Savart1f Solmization
Helmholtz Hermann On the Sensations of Tone as a Physiologic amp1 Basis for the Theory of hriusic Translnted by Alexander J Ellis 6th ed New York Patnl ~)nllth 191JB
Henflinrr Konrad Specimen de novo suo systemnte musieo fI
1iseel1anea Rerolinensla 1710 XXVIII
Huygens Christian Oeuvres completes The Ha[ue Nyhoff 1940 Le Cycle lIarmoniauetI Vol 20 pp 141-173
Novus Cyelns Tlarmonicus fI Onera I
varia Leyden 1724 pp 747-754
Jeans Sir tTames Science and Music Cambridge at the University Press 1953
156
L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b
Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945
Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______
1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7
flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696
Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II
Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
written since his time A more tho~ou~h inveot1vntinn of
Sauveurs works should make possible a more just assessment
of his position in the history of that sctence or art-shy
temperinp the 1ust scale--to which he is I1811a] 1y
acknowled~ed to have h~en an i111nortant contrihlltor
rhe relationship of Srntvenr to tho the()rl~~t T~nnshy
Philippe Rameau ~hould also he illuminate~ by a closer
scrutiny of the works of Sauvcllr
It shall he the program of this study to trace
ttroughout Sauveurs five oub1ished Mfmo5res the developshy
ment (providing demonstrations where they are lacking or
unclear) of four of his most influential ideas the
chronometer or scale upon which teMporal ~urntions cnu16
be measured within a third (or a sixtieth of a second) of
time the division of the octave into 43 and further
301 equal p(lrtfl and the vnr10u8 henefi ts wrich nC(~-rlH~ fr0~~
snch a division the establishment of a tone with a 1Ptrgtl_
mined number of vibrations peT second as a fixed Ditch to
which all others could be related and which cou]n thus
serve as a standard for comparing the VqriOl~S standaTds
of pitch in use throughout the world an~ the ~rmon1c
series recognized by Sauveur as arisin~ frnm the vib~ation
of a string in aliquot parts The vRrious c 1aims which
have been mane concerning Sauveurs theories themselves
and thei r influence on th e works of at hels shall tr en be
more closely examined in the l1ght of the p-receding
exposition The exposition and analysis shall he
1v
accompanied by c ete trans tions of Sauveu~ls five
71Aemoires treating of acoustics which will make his works
available for the fipst time in English
Thanks are due to Dr Erich Schwandt whose dedishy
cation to the work of clarifying desi~nRtions of tempo of
donees of the French court inspiled the p-resent study to
Dr Joel Pasternack of the Department of Mathematics of
the University of Roc ster who pointed the way to the
solution of the mathematical problems posed by Sauveurs
exposition and to the Cornell University Libraries who
promptly and graciously provided the scientific writings
upon which the study is partly based
v
ABSTHACT
Joseph Sauveur was born at La Flampche on March 24
1653 Displayin~ an early interest in mechanics he was
sent to the Tesuit Collere at La Pleche and lA-ter
abandoning hoth the relipious and the medical professions
he devoted himsel f to the stl1dy of Mathematics in Paris
He became a hi~hly admired geometer and was admitted to
the lcad~mie of Paris in 1696 after which he turned to
the science of sound which he hoped to establish on an
equal basis with Optics To that end he published four
trea tises in the ires de lAc~d~mie in 1701 1702
1707 and 1711 (a fifth completed in 1713 was published
posthu~ously in 1716) in the first of which he presented
a corrprehensive system of notation of intervaJs sounds
Lonporal duratIon and harrnonlcs to which he propo-1od
adrlltions and developments in his later papers
The chronometer a se e upon which teMporal
r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth
of a second) of time represented an advance in conception
he~Tond the popLllar se e of Etienne Loulie divided slmnly
into inches which are for the most part incomrrensurable
with seco~ds Sauveurs scale is graduated in accordance
wit~1 the lavl that the period of a pendulum is proportional
to the square root of the length and was taken over by
vi
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
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Tuning and Temnerament ERst Lansing Michigan State College Press 1951
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Des-Cartes Hene COr1pendium Musicae Rhenum 1650
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154
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156
L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b
Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945
Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______
1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7
flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696
Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II
Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
accompanied by c ete trans tions of Sauveu~ls five
71Aemoires treating of acoustics which will make his works
available for the fipst time in English
Thanks are due to Dr Erich Schwandt whose dedishy
cation to the work of clarifying desi~nRtions of tempo of
donees of the French court inspiled the p-resent study to
Dr Joel Pasternack of the Department of Mathematics of
the University of Roc ster who pointed the way to the
solution of the mathematical problems posed by Sauveurs
exposition and to the Cornell University Libraries who
promptly and graciously provided the scientific writings
upon which the study is partly based
v
ABSTHACT
Joseph Sauveur was born at La Flampche on March 24
1653 Displayin~ an early interest in mechanics he was
sent to the Tesuit Collere at La Pleche and lA-ter
abandoning hoth the relipious and the medical professions
he devoted himsel f to the stl1dy of Mathematics in Paris
He became a hi~hly admired geometer and was admitted to
the lcad~mie of Paris in 1696 after which he turned to
the science of sound which he hoped to establish on an
equal basis with Optics To that end he published four
trea tises in the ires de lAc~d~mie in 1701 1702
1707 and 1711 (a fifth completed in 1713 was published
posthu~ously in 1716) in the first of which he presented
a corrprehensive system of notation of intervaJs sounds
Lonporal duratIon and harrnonlcs to which he propo-1od
adrlltions and developments in his later papers
The chronometer a se e upon which teMporal
r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth
of a second) of time represented an advance in conception
he~Tond the popLllar se e of Etienne Loulie divided slmnly
into inches which are for the most part incomrrensurable
with seco~ds Sauveurs scale is graduated in accordance
wit~1 the lavl that the period of a pendulum is proportional
to the square root of the length and was taken over by
vi
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
iVORKS CITED
Backus tJohn rrhe Acollstical Founnatlons of Music New York W W Norton amp Company 1969
I~ull i w f~ouse A ~hort Acc011nt of the lIiltory of ~athemrltics New York Dovor Publications l~)GO
Barbour tTames Murray Equal rPemperament Its History from Ramis (1482) to Hameau (1737)ff PhD dissertashytion Cornell University 1932
Tuning and Temnerament ERst Lansing Michigan State College Press 1951
Bosanquet H H M 1f11 emperament or the division of the octave iiusical Association Froceedings 1874-75 pp 4-1
Cajori 1lorian ttHistory of the Iogarithmic Slide Rule and Allied In[-)trllm8nts II in Strinr Fipures rtnd Other F[onorrnphs pre 1-121 Edited by lrtf N OUSG I~all
5rd ed iew York Ch01 sea Publ i shinp Company 1 ~)G9
Culver Charles A Nusical Acoustics 4th edt NeJI York McGraw-hill Book Comnany Inc 1956
Des-Cartes Hene COr1pendium Musicae Rhenum 1650
Dtctionaryof Physics Edited hy II bullbullT Gray New York John lJiley and Sons Inc 1958 Sv Pendulum 1t
Dl0 MllSlk In Gcnchichte lJnd Goponwnrt Sv Snuvellr J os e ph II by llri t z Iiinc ke] bull
Ellis Alexander J liOn an Improved Bimodular Method of compnting Natural and Tahular Loparlthms and AntishyLogarithms to Twelve or Sixteen Places vi th very brief Tables II ProceedinFs of the Royal Society of London XXXI (February 188ry-391-39S
~~tis F-J Uiographie universelle des musiciens at b i bl i ographi e generale de la musi que deuxi erne ed it -j on bull Paris Culture et Civilisation 185 Sv IISauveur Joseph II
154
155
Fontenelle Bernard Ie Bovier de Elove de M Sallveur
Histoire de J Acad~mie des Sciences Ann~e 1716 Amsterdam Chez Pierre Mortier 1736 pre 97-107
bull Sur la Dete-rmination d un Son Fixe T11stoire ---d~e--1~1 Academia Hoyale des Sciences Annee 1700
Arns terdnm Chez Pierre fvlortler 17)6 pp Id~--lDb
bull Sur l Applica tion des Sons liarmonique s alJX II cux ---(1--O---rgues Histol ro de l Academia Roya1e des Sc ienc os
Ann~e 1702 Amsterdam Chez Pierre Mortier l7~6 pp 118-122
bull Sur un Nouveau Systeme de Musique Histoi 10 ---(~l(-)~~llclo(rnlo lioynle des ~)clonces AJneo 1701
Amsterdam Chez Pierre Nlortier 1706 pp 158-180
Galilei Galileo Dialogo sopra i dlJe Tv1assimiSistemi del Mondo n in A Treasury of 1Vorld Science fJP 349shy350 Edited by Dagobe-rt Hunes London Peter Owen 1962
Glr(l1 o~~tone Cuthbert tTenn-Philippe Ram(~lll His Life and Work London Cassell and Company Ltd 1957
Groves Dictionary of Music and usiciA1s 5th ed S v If Sound by Ll S Lloyd
Harding Hosamond The Oririns of li1usical Time and Expression London Oxford l~iversity Press 1938
Harvard Dictionary of Music 2nd ed Sv AcoustiCS Consonance Dissonance If uGreece U flIntervals iftusic rrheory Savart1f Solmization
Helmholtz Hermann On the Sensations of Tone as a Physiologic amp1 Basis for the Theory of hriusic Translnted by Alexander J Ellis 6th ed New York Patnl ~)nllth 191JB
Henflinrr Konrad Specimen de novo suo systemnte musieo fI
1iseel1anea Rerolinensla 1710 XXVIII
Huygens Christian Oeuvres completes The Ha[ue Nyhoff 1940 Le Cycle lIarmoniauetI Vol 20 pp 141-173
Novus Cyelns Tlarmonicus fI Onera I
varia Leyden 1724 pp 747-754
Jeans Sir tTames Science and Music Cambridge at the University Press 1953
156
L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b
Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945
Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______
1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7
flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696
Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II
Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
ABSTHACT
Joseph Sauveur was born at La Flampche on March 24
1653 Displayin~ an early interest in mechanics he was
sent to the Tesuit Collere at La Pleche and lA-ter
abandoning hoth the relipious and the medical professions
he devoted himsel f to the stl1dy of Mathematics in Paris
He became a hi~hly admired geometer and was admitted to
the lcad~mie of Paris in 1696 after which he turned to
the science of sound which he hoped to establish on an
equal basis with Optics To that end he published four
trea tises in the ires de lAc~d~mie in 1701 1702
1707 and 1711 (a fifth completed in 1713 was published
posthu~ously in 1716) in the first of which he presented
a corrprehensive system of notation of intervaJs sounds
Lonporal duratIon and harrnonlcs to which he propo-1od
adrlltions and developments in his later papers
The chronometer a se e upon which teMporal
r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth
of a second) of time represented an advance in conception
he~Tond the popLllar se e of Etienne Loulie divided slmnly
into inches which are for the most part incomrrensurable
with seco~ds Sauveurs scale is graduated in accordance
wit~1 the lavl that the period of a pendulum is proportional
to the square root of the length and was taken over by
vi
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
iVORKS CITED
Backus tJohn rrhe Acollstical Founnatlons of Music New York W W Norton amp Company 1969
I~ull i w f~ouse A ~hort Acc011nt of the lIiltory of ~athemrltics New York Dovor Publications l~)GO
Barbour tTames Murray Equal rPemperament Its History from Ramis (1482) to Hameau (1737)ff PhD dissertashytion Cornell University 1932
Tuning and Temnerament ERst Lansing Michigan State College Press 1951
Bosanquet H H M 1f11 emperament or the division of the octave iiusical Association Froceedings 1874-75 pp 4-1
Cajori 1lorian ttHistory of the Iogarithmic Slide Rule and Allied In[-)trllm8nts II in Strinr Fipures rtnd Other F[onorrnphs pre 1-121 Edited by lrtf N OUSG I~all
5rd ed iew York Ch01 sea Publ i shinp Company 1 ~)G9
Culver Charles A Nusical Acoustics 4th edt NeJI York McGraw-hill Book Comnany Inc 1956
Des-Cartes Hene COr1pendium Musicae Rhenum 1650
Dtctionaryof Physics Edited hy II bullbullT Gray New York John lJiley and Sons Inc 1958 Sv Pendulum 1t
Dl0 MllSlk In Gcnchichte lJnd Goponwnrt Sv Snuvellr J os e ph II by llri t z Iiinc ke] bull
Ellis Alexander J liOn an Improved Bimodular Method of compnting Natural and Tahular Loparlthms and AntishyLogarithms to Twelve or Sixteen Places vi th very brief Tables II ProceedinFs of the Royal Society of London XXXI (February 188ry-391-39S
~~tis F-J Uiographie universelle des musiciens at b i bl i ographi e generale de la musi que deuxi erne ed it -j on bull Paris Culture et Civilisation 185 Sv IISauveur Joseph II
154
155
Fontenelle Bernard Ie Bovier de Elove de M Sallveur
Histoire de J Acad~mie des Sciences Ann~e 1716 Amsterdam Chez Pierre Mortier 1736 pre 97-107
bull Sur la Dete-rmination d un Son Fixe T11stoire ---d~e--1~1 Academia Hoyale des Sciences Annee 1700
Arns terdnm Chez Pierre fvlortler 17)6 pp Id~--lDb
bull Sur l Applica tion des Sons liarmonique s alJX II cux ---(1--O---rgues Histol ro de l Academia Roya1e des Sc ienc os
Ann~e 1702 Amsterdam Chez Pierre Mortier l7~6 pp 118-122
bull Sur un Nouveau Systeme de Musique Histoi 10 ---(~l(-)~~llclo(rnlo lioynle des ~)clonces AJneo 1701
Amsterdam Chez Pierre Nlortier 1706 pp 158-180
Galilei Galileo Dialogo sopra i dlJe Tv1assimiSistemi del Mondo n in A Treasury of 1Vorld Science fJP 349shy350 Edited by Dagobe-rt Hunes London Peter Owen 1962
Glr(l1 o~~tone Cuthbert tTenn-Philippe Ram(~lll His Life and Work London Cassell and Company Ltd 1957
Groves Dictionary of Music and usiciA1s 5th ed S v If Sound by Ll S Lloyd
Harding Hosamond The Oririns of li1usical Time and Expression London Oxford l~iversity Press 1938
Harvard Dictionary of Music 2nd ed Sv AcoustiCS Consonance Dissonance If uGreece U flIntervals iftusic rrheory Savart1f Solmization
Helmholtz Hermann On the Sensations of Tone as a Physiologic amp1 Basis for the Theory of hriusic Translnted by Alexander J Ellis 6th ed New York Patnl ~)nllth 191JB
Henflinrr Konrad Specimen de novo suo systemnte musieo fI
1iseel1anea Rerolinensla 1710 XXVIII
Huygens Christian Oeuvres completes The Ha[ue Nyhoff 1940 Le Cycle lIarmoniauetI Vol 20 pp 141-173
Novus Cyelns Tlarmonicus fI Onera I
varia Leyden 1724 pp 747-754
Jeans Sir tTames Science and Music Cambridge at the University Press 1953
156
L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b
Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945
Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______
1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7
flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696
Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II
Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
Michel LAffilard in 1705 and Louis-Leon Pajot in 1732
neither of whom made chan~es in its mathematical
structu-re
Sauveurs system of 43 rreridians 301 heptamerldians
nno 3010 decllmcridians the equal logarithmic units into
which he divided the octave made possible not only as
close a specification of pitch as could be useful for
acoustical purposes but also provided a satisfactory
approximation to the just scale degrees as well as to
15-comma mean t one t Th e correspondt emperamen ence 0 f
3010 to the loparithm of 2 made possible the calculation
of the number units in an interval by use of logarithmic
tables but Sauveur provided an additional rrethod of
bimodular computation by means of which the use of tables
could be avoided
Sauveur nroposed as am eans of determining the
frequency of vib~ation of a pitch a method employing the
phenomena of beats if two pitches of which the freshy
quencies of vibration are known--2524--beat four times
in a second then the first must make 100 vibrations in
that period while the other makes 96 since a beat occurs
when their pulses coincide Sauveur first gave 100
vibrations in a second as the fixed pitch to which all
others of his system could be referred but later adopted
256 which being a power of 2 permits identification of an
octave by the exuonent of the power of 2 which gives the
flrst pi tch of that octave
vii
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
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Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864
ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963
New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD
Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67
Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195
Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971
-----
157
Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563
Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0
Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451
i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282
bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9
Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498
Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417
Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950
3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI
~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400
1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---
Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966
Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842
AI thouph Sauveur was not the first to ohsArvc tUl t
tones of the harmonic series a~e ei~tte(] when a strinr
vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~
all the values of the harmonics within th~ compass of
five octaves and thus broupht order to earlinr Bcnttered
observations He also noted that a string may vibrate
in several modes at once and aoplied his system a1d his
observations to an explanation of the 1eaninr t0nes of
the morine-trumpet and the huntinv horn His vro~ks n]so
include a system of solmization ~nrl a treatm8nt of vihrntshy
ing strtnTs neither of which lecpived mnch attention
SaUVe1)r was not himself a music theorist a r c1
thus Jean-Philippe Remean CRnnot he snid to have fnlshy
fiJ led Sauveurs intention to found q scIence of fwrvony
Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r
however in a sense father modern aCo11stics and provi r 2
a foundation for the theoretical speculations of otners
viii
bull bull bull
bull bull bull
CONTENTS
INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1
C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25
CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot
CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154
ix
LIST OF ILLUSTKATIONS
1 Division of the Chronometer into thirds of time 37bull
2 Division of the Ch~onometer into thirds of time 38bull
3 Correspondence of the Monnchord and the Pendulum 74
4 CommuniGation of vihrations 98
5 Jodes of the fundamental and the first five harmonics 102
x
LIST OF TABLES
1 Len~ths of strings or of chron0meters (Mersenne) 31
2 Div~nton of the chronomptol 3nto twol ftl of R
n ltcond bull middot middot middot middot bull ~)4
3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00
4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull
6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c
10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO
11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R
12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals
Values from Table 13 in cents bull Sl
torAd notes for each final in 1 a 1) G 1~S
I) JlTrY)nics nne vibratIons p0r Stcopcl JOr
J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv
b
19 Chromatic application of the cycle of 43 bull bull 127
xi
INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS
Joseph Sauveur was born on March 24 1653 at La
F1~che about twenty-five miles southwest of Le Mans His
parents Louis Sauveur an attorney and Renee des Hayes
were according to his biographer Bernard Ie Bovier de
Fontenelle related to the best families of the district rrl
Joseph was through a defect of the organs of the voice 2
absolutely mute until he reached the age of seven and only
slowly after that acquired the use of speech in which he
never did become fluent That he was born deaf as well is
lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite
L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile
i verselle des mu cien s et biblio ra hie el ral e dej
-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems
fron o n ten elle s rr El 0 g e bull If
2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97
1
2
alleged by SCherchen3 although Fontenelle makes only
oblique refepences to Sauveurs inability to hear 4
3Scherchen Nature of Music p 15
4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt
a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull
rSauveur etait sourd avait la voix fausse et netendait ~
rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]
of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make
3
Having displayed an early interest in muchine) unci
physical laws as they are exemplified in siphons water
jets and other related phenomena he was sent to the Jesuit
College at La Fleche5 (which it will be remembered was
attended by both Descartes and Mersenne6 ) His efforts
there were impeded not only by the awkwardness of his voice
but even more by an inability to learn by heart as well
as by his first master who was indifferent to his talent 7
Uninterested in the orations of Cicero and the poetry of
Virgil he nonetheless was fascinated by the arithmetic of
Pelletier of Mans8 which he mastered like other mathematishy
cal works he was to encounter in his youth without a teacher
Aware of the deficiencies in the curriculum at La 1
tleche Sauveur obtained from his uncle canon and grand-
precentor of Tournus an allowance enabling him to pursue
the study of philosophy and theology at Paris During his
study of philosophy he learned in one month and without
master the first six books of Euclid 9 and preferring
mathematics to philosophy and later to t~eology he turned
hls a ttention to the profession of medici ne bull It was in the
course of his studies of anatomy and botany that he attended
5Fontenelle ffEloge p 98
6Scherchen Nature of Music p 25
7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit
9 Ib i d p 99
4
the lectures of RouhaultlO who Fontenelle notes at that
time helped to familiarize people a little with the true
philosophy 11 Houhault s writings in which the new
philosophical spirit c~itical of scholastic principles
is so evident and his rigid methods of research coupled
with his precision in confining himself to a few ill1portnnt
subjects12 made a deep impression on Sauveur in whose
own work so many of the same virtues are apparent
Persuaded by a sage and kindly ecclesiastic that
he should renounce the profession of medicine in Which the
physician uhas almost as often business with the imagination
of his pa tients as with their che ets 13 and the flnancial
support of his uncle having in any case been withdrawn
Sauveur Uturned entirely to the side of mathematics and reshy
solved to teach it14 With the help of several influential
friends he soon achieved a kind of celebrity and being
when he was still only twenty-three years old the geometer
in fashion he attracted Prince Eugene as a student IS
10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)
11 Fontenelle EIage p 99
12Scherchen Nature of Music p 26
13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull
14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner
15F~tis Biographie universelle sv nSauveur
5
An anecdote about the description of Sauveur at
this time in his life related by Fontenelle are parti shy
cularly interesting as they shed indirect Ii Ppt on the
character of his writings
A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it
He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16
If the published papers display a single-mindedness)
a tight organization an absence of the speculative and the
superfluous as well as a paucity of references to other
writers either of antiquity or of the day these qualities
will not seem inconsonant with either the austere simplicity
16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u
II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy
tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit
6
of the Sauveur of this anecdote or the disinclination he
displays here to squander time either on trivial conversashy
tion or even on reading It was indeed his fondness for
pared reasoning and conciseness that had made him seem so
unsuitable a candidate for the profession of medicine--the
bishop ~~d judged
LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l
But traits that might have handicapped a physician freed
the mathematician and geometer for a deeper exploration
of his chosen field
However pure was his interest in mathematics Sauveur
did not disdain to apply his profound intelligence to the
analysis of games of chance18 and expounding before the
king and queen his treatment of the game of basset he was
promptly commissioned to develop similar reductions of
17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl
etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace
lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285
guinguenove hoca and lansguenet all of which he was
successful in converting to algebraic equations19
In 1680 he obtained the title of master of matheshy
matics of the pape boys of the Dauphin20 and in the next
year went to Chantilly to perform experiments on the waters21
It was durinp this same year that Sauveur was first mentioned ~
in the Histoire de lAcademie Royale des Sciences Mr
De La Hire gave the solution of some problems proposed by
Mr Sauveur22 Scherchen notes that this reference shows
him to he already a member of the study circle which had
turned its attention to acoustics although all other
mentions of Sauveur concern mechanical and mathematical
problems bullbullbull until 1700 when the contents listed include
acoustics for the first time as a separate science 1I 23
Fontenelle however ment ions only a consuming int erest
during this period in the theory of fortification which
led him in an attempt to unite theory and practice to
~o to Mons during the siege of that city in 1691 where
flhe took part in the most dangerous operations n24
19Fontenelle Elopetr p 102
20Fetis Biographie universelle sv Sauveur
2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux
22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur
23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696
24Fetis Biographie universelle s v Sauveur1f
8
In 1686 he had obtained a professorship of matheshy
matics at the Royal College where he is reported to have
taught his students with great enthusiasm on several occashy
25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~
of Paris 1hat his attention had by now been turned to
acoustical problems is certain for he remarks in the introshy
ductory paragraphs of his first M~moire (1701) in the
hadT~emoires de l Academie Royale des Sciences that he
attempted to write a Treatise of Speculative Music26
which he presented to the Royal College in 1697 He attribshy
uted his failure to publish this work to the interest of
musicians in only the customary and the immediately useful
to the necessity of establishing a fixed sound a convenient
method for doing vmich he had not yet discovered and to
the new investigations into which he had pursued soveral
phenomena observable in the vibration of strings 27
In 1703 or shortly thereafter Sauveur was appointed
examiner of engineers28 but the papers he published were
devoted with but one exception to acoustical problems
25 Pontenelle Eloge lip 105
26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece
27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below
28Fontenel1e iloge p 106
9
It has been noted that Sauveur was mentioned in
1681 1685 and 1696 in the Histoire de lAcademie 29 In
1700 the year in which Acoustics was first accorded separate
status a full report was given by Fontene1le on the method
SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch
a method wtl1ch he had sought since the abortive aLtempt at
a treatise in 1696 Sauveurs discovery was descrihed by
Scherchen as the first of its kind and for long it was
recognized as the surest method of assessing vibratory
frequenci es 30
In the very next year appeared the first of Sauveurs
published Memoires which purported to be a general system
of intervals and its application to all the systems and
instruments of music31 and in which according to Scherchen
several treatises had to be combined 32 After an introducshy
tion of several paragraphs in which he informs his readers
of the attempts he had previously made in explaining acousshy
tical phenomena and in which he sets forth his belief in
LtlU pOBulblJlt- or a science of sound whl~h he dubbol
29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie
30Scherchen Nature of Music p 29
31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique
32Scherchen Nature of MusiC p 31
10
Acoustics 33 established as firmly and capable of the same
perfection as that of Optics which had recently received
8110h wide recoenition34 he proceeds in the first sectIon
to an examination of the main topic of his paper--the
ratios of sounds (Intervals)
In the course of this examination he makes liboral
use of neologism cOining words where he feels as in 0
virgin forest signposts are necessary Some of these
like the term acoustics itself have been accepted into
regular usage
The fi rRt V[emoire consists of compressed exposi tory
material from which most of the demonstrations belonging
as he notes more properly to a complete treatise of
acoustics have been omitted The result is a paper which
might have been read with equal interest by practical
musicians and theorists the latter supplying by their own
ingenuity those proofs and explanations which the former
would have judged superfluous
33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933
34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326
11
In the first section35 the fundamental terminology
of the science of musical intervals 1s defined wIth great
rigor and thoroughness Much of this terminology correshy
nponds with that then current althol1ph in hln nltnrnpt to
provide his fledgling discipline with an absolutely precise
and logically consistent vocabulary Sauveur introduced a
great number of additional terms which would perhaps have
proved merely an encumbrance in practical use
The second section36 contains an explication of the
37first part of the first table of the general system of
intervals which is included as an appendix to and really
constitutes an epitome of the Memoire Here the reader
is presented with a method for determining the ratio of
an interval and its name according to the system attributed
by Sauveur to Guido dArezzo
The third section38 comprises an intromlction to
the system of 43 meridians and 301 heptameridians into
which the octave is subdivided throughout this Memoire and
its successors a practical procedure by which the number
of heptameridians of an interval may be determined ~rom its
ratio and an introduction to Sauveurs own proposed
35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12
36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below
37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below
38 Sallveur Syst-eme General pp 418-428 see
vol II pp 15-25 below
12
syllables of solmization comprehensive of the most minute
subdivisions of the octave of which his system is capable
In the fourth section39 are propounded the division
and use of the Echometer a rule consisting of several
dl vldod 1 ines which serve as seal es for measuJing the durashy
tion of nOlln(lS and for finding their lntervnls nnd
ratios 40 Included in this Echometer4l are the Chronome lot f
of Loulie divided into 36 equal parts a Chronometer dividBd
into twelfth parts and further into sixtieth parts (thirds)
of a second (of ti me) a monochord on vmich all of the subshy
divisions of the octave possible within the system devised
by Sauveur in the preceding section may be realized a
pendulum which serves to locate the fixed soundn42 and
scales commensurate with the monochord and pendulum and
divided into intervals and ratios as well as a demonstrashy
t10n of the division of Sauveurs chronometer (the only
actual demonstration included in the paper) and directions
for making use of the Echometer
The fifth section43 constitutes a continuation of
the directions for applying Sauveurs General System by
vol 39Sauveur Systeme General pp
II pp 26-33 below 428-436 see
40Sauveur Systeme General II p 428 see vol II p 26 below
41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration
4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below
vol 43Sauveur Syst~me General pp
II pp 33-45 below 436-447 see
13
means of the Echometer in the study of any of the various
established systems of music As an illustration of the
method of application the General System is applied to
the regular diatonic system44 to the system of meun semlshy
tones to the system in which the octave is divided into
55 parta45 and to the systems of the Greeks46 and
ori ontal s 1
In the sixth section48 are explained the applicashy
tions of the General System and Echometer to the keyboards
of both organ and harpsichord and to the chromatic system
of musicians after which are introduced and correlated
with these the new notes and names proposed by Sauveur
49An accompanying chart on which both the familiar and
the new systems are correlated indicates the compasses of
the various voices and instruments
In section seven50 the General System is applied
to Plainchant which is understood by Sauveur to consist
44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below
45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below
I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below
47Sauveur Systeme General pp 444-447 see vol II pp 42-45 below
I
48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below
49 Sauveur Systeme General p 498 see
vol II p 97 below
50 I ISauveur Systeme General n pp 456-463 see
vol II pp 53-60 below
14
of that sort of vo cal music which make s us e only of the
sounds of the diatonic system without modifications in the
notes whether they be longs or breves5l Here the old
names being rejected a case is made for the adoption of
th e new ones which Sauveur argues rna rk in a rondily
cOHlprohonulhle mannor all the properties of the tUlIlpolod
diatonic system n52
53The General System is then in section elght
applied to music which as opposed to plainchant is
defined as the sort of melody that employs the sounds of
the diatonic system with all the possible modifications-shy
with their sharps flats different bars values durations
rests and graces 54 Here again the new system of notes
is favored over the old and in the second division of the
section 55 a new method of representing the values of notes
and rests suitable for use in conjunction with the new notes
and nruooa 1s put forward Similarly the third (U visionbtl
contains a proposed method for signifying the octaves to
5lSauveur Systeme General p 456 see vol II p 53 below
52Sauveur Systeme General p 458 see vol II
p 55 below 53Sauveur Systeme General If pp 463-474 see
vol II pp 60-70 below
54Sauveur Systeme Gen~ral p 463 see vol II p 60 below
55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below
I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below
15
which the notes of a composition belong while the fourth57
sets out a musical example illustrating three alternative
methot1s of notating a melody inoluding directions for the
precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a
summary of Sauveurs discoveries about and obsepvations
concerning harmonies accompanied by a table60 in which the
pitches of the first thirty-two are given in heptameridians
in intervals to the fundamental both reduced to the compass
of one octave and unreduced and in the names of both the
new system and the old Experiments are suggested whereby
the reader can verify the presence of these harmonics in vishy
brating strings and explanations are offered for the obshy
served results of the experiments described Several deducshy
tions are then rrade concerning the positions of nodes and
loops which further oxplain tho obsorvod phonom(nn 11nd
in section ten6l the principles distilled in the previous
section are applied in a very brief treatment of the sounds
produced on the marine trumpet for which Sauvellr insists
no adequate account could hitherto have been given
57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below
58Sauveur Systeme Gen~raln p 498 see vol II p 96 below
59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below
60Sauveur Systeme General p 475 see vol II p 72 below
6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below
16
In the eleventh section62 is presented a means of
detormining whether the sounds of a system relate to any
one of their number taken as fundamental as consonances
or dissonances 63The twelfth section contains two methods of obshy
tain1ng exactly a fixed sound the first one proposed by
Mersenne and merely passed on to the reader by Sauveur
and the second proposed bySauveur as an alternative
method capable of achieving results of greater exactness
In an addition to Section VI appended to tho
M~moire64 Sauveur attempts to bring order into the classishy
fication of vocal compasses and proposes a system of names
by which both the oompass and the oenter of a voice would
be made plain
Sauveurs second Memoire65 was published in the
next year and consists after introductory passages on
lithe construction of the organ the various pipe-materials
the differences of sound due to diameter density of matershy
iul shapo of the pipe and wind-pressure the chElructor1ntlcB
62Sauveur Systeme General pp 484-488 see vol II pp 81-84 below
63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below
64Sauveur Systeme General pp 493-498 see vol II pp 89-94 below
65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below
17
of various stops a rrl dimensions of the longest and shortest
organ pipes66 in an application of both the General System
put forward in the previous Memoire and the theory of harshy
monics also expounded there to the composition of organ
stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro
and of tuning the entire organ with the rule thus obtained
is given in the course of the description of the varlous
types of stops As corroboration of his observations
Sauveur subjoins descriptions of stops composed by Mersenne
and Nivers67 and concludes his paper with an estima te of
the absolute range of sounds 68
69The third Memoire which appeared in 1707 presents
a general method for forming the tempered systems of music
and lays down rules for making a choice among them It
contains four divisions The first of these70 sets out the
familiar disadvantages of the just diatonic system which
result from the differences in size between the various inshy
tervuls due to the divislon of the ditone into two unequal
66scherchen Nature of Music p 39
67 Sauveur II Application p 450 see vol II pp 123-124 below
68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below
69 IJoseph Sauveur Methode generale pour former des
systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707
lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below
70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below
18
rltones and a musical example is nrovided in which if tho
ratios of the just diatonic system are fnithfu]1y nrniorvcd
the final ut will be hipher than the first by two commAS
rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom
havinr been stat ad Sauveur rrooeeds in the second secshy
tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd
sys terns are formed (Phe til ird scctinn73 examines by means
of a table74 constructed for the rnrrnose the systems which
had emerged from the precedin~ analysis as most plausible
those of 31 parts 43 meriltiians and 55 commas as well as
two--the just system and thnt of twelve equal semitones-shy
which are included in the first instance as a basis for
comparison and in the second because of the popula-rity
of equal temperament due accordi ng to Sauve) r to its
simp1ici ty In the fa lJrth section75 several arpurlents are
adriuced for the selection of the system of L1~) merIdians
as ttmiddote mos t perfect and the only one that ShOl11d be reshy
tained to nrofi t from all the advan tages wrdch can be
71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow
72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below
7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~
vol II nne 138-J47 bnlow
4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen
vol II p 15~ below
75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below
19
drawn from the tempored systems in music and even in the
whole of acoustics76
The fourth MemOire published in 1711 is an
answer to a publication by Haefling [siC] a musicologist
from Anspach bull bull bull who proposed a new temperament of 50
8degrees Sauveurs brief treatment consists in a conshy
cise restatement of the method by which Henfling achieved
his 50-fold division his objections to that method and 79
finally a table in which a great many possible systems
are compared and from which as might be expected the
system of 43 meridians is selected--and this time not on~y
for the superiority of the rna thematics which produced it
but also on account of its alleged conformity to the practice
of makers of keyboard instruments
rphe fifth and last Memoire80 on acoustics was pubshy
lished in 171381 without tne benefit of final corrections
76 IISauveur Methode Generale p 281 see vol II
p 150 below
77 tToseph Sauveur Table geneTale des Systemes tem-Ell
per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below
78scherchen Nature of Music pp 43-44
79sauveur Table gen~rale p 416 see vol II p 167 below
130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ
81According to Scherchen it was cOlrL-l~-tgt -1 1shy
c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~
20
It is subdivided into seven sections the first82 of which
sets out several observations on resonant strings--the material
diameter and weight are conside-red in their re1atlonship to
the pitch The second section83 consists of an attempt
to prove that the sounds of the strings of instruments are
1t84in reciprocal proportion to their sags If the preceding
papers--especially the first but the others as well--appeal
simply to the readers general understanning this section
and the one which fol1ows85 demonstrating that simple
pendulums isochronous with the vibrati~ns ~f a resonant
string are of the sag of that stringu86 require a familshy
iarity with mathematical procedures and principles of physics
Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie
82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below
83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below
04 n3auvGur Rapport p 43B sec vol II p 17~)
how
85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below
86Sauveur ftRanport I p 444 see vol II p 178 below
21
while the fourth87 a method for finding the number of
vibrations of a resonant string in a secondn88 might again
be followed by the lay reader The fifth section89 encomshy
passes a number of topics--the determination of fixed sounds
a table of fixed sounds and the construction of an echometer
Sauveur here returns to several of the problems to which he
addressed himself in the M~mo~eof 1701 After proposing
the establishment of 256 vibrations per second as the fixed
pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration
of each pitch in each octave when the fixed sound is taken at
256 vibrations per second The sixth section9l offers
several methods of finding the fixed sounds several more
difficult to construct mechanically than to utilize matheshy
matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section
of the Memoire of 1701 in which several uses were mentioned
for the fixed sound The additional uses consist generally
87Sauveur Rapport pp 448-453 see vol II pp 181-185 below
88Sauveur Rapport p 448 see vol II p 181 below
89sauveur Rapport pp 453-458 see vol II pp 185-190 below
90Sauveur Rapport p 468 see vol II p 203 below
91Sauveur Rapport pp 458-463 see vol II pp 190-195 below
92Sauveur Rapport pp 463-469 see vol II pp 195-201 below
22
in finding the number of vibrations of various vibrating
bodies includ ing bells horns strings and even the
epiglottis
One further paper--devoted to the solution of a
geometrical problem--was published by the Academie but
as it does not directly bear upon acoustical problems it
93hus not boen included here
It can easily be discerned in the course of
t~is brief survey of Sauveurs acoustical papers that
they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and
that they deal only with those aspects of music which are
most general--notational systems systems of intervals
methods for measuring both time and frequencies of vi shy
bration and tne harmonic series--exactly in fact
tla science superior to music u95 (and that not in value
but in logical order) which has as its object sound
in general whereas music has as its object sound
in so fa r as it is agreeable to the hearing u96 There
93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39
94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)
95Sauveur Systeme General II p 403 see vol II p 1 below
96Sauveur Systeme General II p 404 see vol II p 1 below
23
is no attempt anywhere in the corpus to ground a science
of harmony or to provide a basis upon which the merits
of one style or composition might be judged against those
of another style or composition
The close reasoning and tight organization of the
papers become the object of wonderment when it is discovered
that Sauveur did not write out the memoirs he presented to
th(J Irnrlomle they being So well arranged in hill hond Lhlt
Ile had only to let them come out ngrl
Whether or not he was deaf or even hard of hearing
he did rely upon the judgment of a great number of musicians
and makers of musical instruments whose names are scattered
throughout the pages of the texts He also seems to have
enjoyed the friendship of a great many influential men and
women of his time in spite of a rather severe outlook which
manifests itself in two anecdotes related by Fontenelle
Sauveur was so deeply opposed to the frivolous that he reshy
98pented time he had spent constructing magic squares and
so wary of his emotions that he insisted on closjn~ the
mi-tr-riLtge contr-act through a lawyer lest he be carrIed by
his passions into an agreement which might later prove
ur 3Lli table 99
97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n
98 Ibid p 104 Mapic squares areiumbr- --qni 3
_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB
equal Ball History of Mathematics p 118
99Fontenelle Eloge p 104
24
This rather formidable individual nevertheless
fathered two sons by his first wife and a son (who like
his father was mute until the age of seven) and a daughter
by a second lOO
Fontenelle states that although Ur Sauveur had
always enjoyed good health and appeared to be of a robust
Lompor-arncn t ho wai currlod away in two days by u COI1post lon
1I101of the chost he died on July 9 1716 in his 64middotth year
100Ib1d p 107
101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste
11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e
CHAPTER I
THE MEASUREMENT OF TI~I~E
It was necessary in the process of establ j~Jhlng
acoustics as a true science superior to musicu for Sauveur
to devise a system of Bcales to which the multifarious pheshy
nomena which constituted the proper object of his study
might be referred The aggregation of all the instruments
constructed for this purpose was the Echometer which Sauveur
described in the fourth section of the Memoire of 1701 as
U a rule consisting of several divided lines which serve as
scales for measuring the duration of sounds and for finding
their intervals and ratios I The rule is reproduced at
t-e top of the second pInte subioin~d to that Mcmn i re2
and consists of six scales of ~nich the first two--the
Chronometer of Loulie (by universal inches) and the Chronshy
ometer of Sauveur (by twelfth parts of a second and thirds V l
)-shy
are designed for use in the direct measurement of time The
tnird the General Monochord 1s a scale on ihich is
represented length of string which will vibrate at a given
1 l~Sauveur Systeme general II p 428 see vol l
p 26 below
2 ~ ~ Sauveur nSysteme general p 498 see vol I ~
p 96 below for an illustration
3 A third is the sixtieth part of a secon0 as tld
second is the sixtieth part of a minute
25
26
interval from a fundamental divided into 43 meridians
and 301 heptameridians4 corresponding to the same divisions
and subdivisions of the octave lhe fourth is a Pendulum
for the fixed sound and its construction is based upon
tho t of the general Monochord above it The fi ftl scal e
is a ru1e upon which the name of a diatonic interval may
be read from the number of meridians and heptameridians
it contains or the number of meridians and heptflmerldlans
contained can be read from the name of the interval The
sixth scale is divided in such a way that the ratios of
sounds--expressed in intervals or in nurnhers of meridians
or heptameridians from the preceding scale--can be found
Since the third fourth and fifth scales are constructed
primarily for use in the measurement tif intervals they
may be considered more conveniently under that head while
the first and second suitable for such measurements of
time as are usually made in the course of a study of the
durat10ns of individual sounds or of the intervals between
beats in a musical comnosltion are perhaps best
separated from the others for special treatment
The Chronometer of Etienne Loulie was proposed by that
writer in a special section of a general treatise of music
as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in
4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts
27
their absenQe as if they beat the measure of them themselves )
It is described as composed of two parts--a pendulum of
adjustable length and a rule in reference to which the
length of the pendulum can be set
The rule was
bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two
I have made use of the univertal foot because it is known in all sorts of countries
The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King
5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I
ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83
6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze
Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays
Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy
28
It is this scale divided into universal inches
without its pendulum which Sauveur reproduces as the
Chronometer of Loulia he instructs his reader to mark off
AC of 3 feet 8~ lines7 of Paris which will give the length
of a simple pendulum set for seoonds
It will be noted first that the foot of Paris
referred to by Sauveur is identical to the foot of the King
rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy
5Loulie to 12 inches 26 lines which gi ves three universal
feet of 36 inches 8~ lines preoisely the number of inches
and lines of the foot of Paris equated by Sauveur to the
36 inches of the universal foot into which he directs that
the Chronometer of Loulie in his own Echometer be divided
In addition the astronomical inches referred to by Sauveur
in the Memoire of 1713 must be identical to the universal
inches in the Memoire of 1701 for the 36 astronomical inches
are equated to 36 inches 8~ lines of the foot of Paris 8
As the foot of the King measures 325 mm9 the universal
foot re1orred to must equal 3313 mm which is substantially
larger than the 3048 mm foot of the system currently in
use Second the simple pendulum of which Sauveur speaks
is one which executes since the mass of the oscillating
body is small and compact harmonic motion defined by
7A line is the twelfth part of an inch
8Sauveur Rapport n p 434 see vol II p 169 below
9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8
29
Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period
of the simple pendulum described by Sauveur will be two
seconds since the period of a pendulum is the time required 11
for a complete cycle and the complete cycle of Sauveurs
pendulum requires two seconds
Sauveur supplies the lack of a pendulum in his
version of Loulies Chronometer with a set of instructions
on tho correct use of the scale he directs tho ronclol to
lengthen or shorten a simple pendulum until each vibration
is isochronous with or equal to the movement of the hand
then to measure the length of this pendulum from the point
of suspension to the center of the ball u12 Referring this
leneth to the first scale of the Echometer--the Chronometer
of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the
most celebrated attempt to make a machine for counting
musical ti me before that of Malzel and was Ufrequently
referred to in musical books of the eighte3nth centuryu14
Sir John Hawkins and Alexander Malcolm nbo~h thought it
10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum
llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25
12Sauveur trSyst~me General p 432 see vol ~ p 30 below
13Ibid bull
14Hardlng 0 r i g1nsmiddot p 9 bull
30
~ 5 sufficiently interesting to give a careful description Ill
Nonetheless Sauveur dissatisfied with it because the
durations of notes were not marked in any known relation
to the duration of a second the periods of vibration of
its pendulum being flro r the most part incommensurable with
a secondu16 proceeded to construct his own chronometer on
the basis of a law stated by Galileo Galilei in the
Dialogo sopra i due Massimi Slstemi del rTondo of 1632
As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17
Mersenne bad on the basis of th is law construc ted
a table which correlated the lengths of a gtendulum and half
its period (Table 1) so that in the fi rst olumn are found
the times of the half-periods in seconds~n the second
tt~e square of the corresponding number fron the first
column to whic h the lengths are by Galileo t slaw
151bid bull
16 I ISauveur Systeme General pp 435-436 seD vol
r J J 33 bel OVI bull
17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in
fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350
31
TABLE 1
TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS
[FROM MERSENNE HARMONIE UNIVEHSELLE]
I II III
feet
1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J
2
8 64 224 9 81 283~
10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016
f)1B71middot25 625 tJ ~ shy ~~
26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865
proportional and in the third the lengths of a pendulum
with the half-periods indicated in the first column
For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC
because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last
32
2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18
But Sauveur required an exnmplo smallor still for
the Chronometer he envisioned was to be capable of measurshy
ing durations smaller than one second and of measuring
more closely than to the nearest second
It is thus that the chronometer nroposed by Sauveur
was divided proportionally so that it could be read in
twelfths of a second and even thirds 19 The numbers of
the points of division at which it was necessary for
Sauveur to arrive in the chronometer ruled in twelfth parts
of a second and thirds may be determined by calculation
of an extension of the table of Mersenne with appropriate
adjustments
If the formula T bull 2~ is applied to the determinashy
tion of these point s of di vision the constan ts 2 1 and r-
G may be represented by K giving T bull K~L But since the
18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2
19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second
33
length of the pendulum set for seconds is given as 36
inches20 then 1 = 6K or K = ~ With the formula thus
obtained--T = ~ or 6T =L or L = 36T2_-it is possible
to determine the length of the pendulum in inches for
each of the twelve twelfths of a second (T) demanded by
the construction (Table 2)
All of the lengths of column L are squares In
the fourth column L2 the improper fractions have been reshy
duced to integers where it was possible to do so The
values of L2 for T of 2 4 6 8 10 and 12 twelfths of
a second are the squares 1 4 9 16 25 and 36 while
the values of L2 for T of 1 3 5 7 9 and 11 twelfths
of a second are 1 4 9 16 25 and 36 with the increments
respectively
Sauveurs procedure is thus clear He directs that
the reader to take Hon the first scale AB 1 4 9 16
25 36 49 64 and so forth inches and carry these
intervals from the end of the rule D to E and rrmark
on these divisions the even numbers 0 2 4 6 8 10
12 14 16 and so forth n2l These values correspond
to the even numbered twelfths of a second in Table 2
He further directs that the first inch (any univeYsal
inch would do) of AB be divided into quarters and
that the reader carry the intervals - It 2~ 3~ 4i 5-4-
20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below
21 Ibid bull
34
TABLE 2
T L L2
(in integers + inc rome nt3 )
12 144~1~)2 3612 ~
11 121(1~)2 25 t 5i12 ~
10 100 12
(1~)2 ~
25
9 81(~) 2 16 + 412 4
8 64(~) 2 1612 4
7 (7)2 49 9 + 3t12 2 4
6 (~)2 36 912 4
5 (5)2 25 4 + 2-t12 2 4
4 16(~) 2 412 4
3 9(~) 2 1 Ii12 4 2 (~)2 4 I
12 4
1 1 + l(~) 2 0 412 4
6t 7t and so forth over after the divisions of the
even numbers beginning at the end D and that he mark
on these new divisions the odd numbers 1 3 5 7 9 11 13
15 and so forthrr22 which values correspond to those
22Sauveur rtSysteme General p 420 see vol II pp 26-27 below
35
of Table 2 for the odd-numbered twelfths of u second
Thus is obtained Sauveurs fi rst CIlronome ter div ided into
twelfth parts of a second (of time) n23
The demonstration of the manner of dividing the
chronometer24 is the only proof given in the M~moire of 1701
Sauveur first recapitulates the conditions which he stated
in his description of the division itself DF of 3 feet 8
lines (of Paris) is to be taken and this represents the
length of a pendulum set for seconds After stating the law
by which the period and length of a pendulum are related he
observes that since a pendulum set for 1 6
second must thus be
13b of AC (or DF)--an inch--then 0 1 4 9 and so forth
inches will gi ve the lengths of 0 1 2 3 and so forth
sixths of a second or 0 2 4 6 and so forth twelfths
Adding to these numbers i 1-14 2t 3i and- so forth the
sums will be squares (as can be seen in Table 2) of
which the square root will give the number of sixths in
(or half the number of twelfths) of a second 25 All this
is clear also from Table 2
The numbers of the point s of eli vis ion at which it
WIlS necessary for Sauveur to arrive in his dlvis10n of the
chronometer into thirds may be determined in a way analogotls
to the way in which the numbe])s of the pOints of division
of the chronometer into twe1fths of a second were determined
23Sauveur Systeme General p 420 see vol II pp 26-27 below
24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below
25Ibid bull
36
Since the construction is described 1n ~eneral ternls but
11111strnted between the numbers 14 and 15 the tahle
below will determine the numbers for the points of
division only between 14 and 15 (Table 3)
The formula L = 36T2 is still applicable The
values sought are those for the sixtieths of a second between
the 14th and 15th twelfths of a second or the 70th 7lst
72nd 73rd 74th and 75th sixtieths of a second
TABLE 3
T L Ll
70 4900(ig)260 155
71 5041(i~260 100
72 5184G)260 155
73 5329(ig)260 100
74 5476(ia)260 155
75 G~)2 5625 60 100
These values of L1 as may be seen from their
equivalents in Column L are squares
Sauveur directs the reader to take at the rot ght
of one division by twelfths Ey of i of an inch and
divide the remainder JE into 5 equal parts u26
( ~ig1Jr e 1)
26 Sauveur Systeme General p 420 see vol II p 27 below
37
P P1 4l 3
I I- ~ 1
I I I
d K A M E rr
Fig 1
In the figure P and PI represent two consecutive points
of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the
points of di vision of crE into 5 equal parts One-fourth
inch having been divided into 25 small equal parts
Sauveur instructs the reader to
take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27
This procedure has been approximated in Fig 1 The four
points K A fA and y will according to SauvenT divide
[y into 5 parts from which we will obtain the divisions
of our chronometer in thirds28
Taking P of 14 (or ~g of a second) PI will equal
15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative
positions of the diVisions between 14 and 15
The quarter inch at the right having been subshy
700tracted the remainder 100 is divided into five equal
parts of i6g each To these five parts are added the small
- -
38
0 )
T-1--W I
cleT2
T deg1 0
00 rt-degIQ
shy
deg1degpound
CIOr0
01deg~
I J 1 CL l~
39
parts obtained by dividing a quarter inch into 25 equal
parts in the quantities 149 and 16 respectively This
addition gives results enumerated in Table 4
TABLE 4
IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH
tEW UmGTH)4~)OO
-f -100
P to 1 140 1 141 P to Y 5041 100 roo 100 100
P to 2 280 4 284 5184P to 100 100 100 100
P to 3 420 9 429 P to fA 5329 100 100 100 100
p to 4 560 16 576 p to y- 5476 100 100 roo 100
The four lengths thus constructed correspond preshy
cisely to the four found previously by us e of the formula
and set out in Table 3
It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the
difference between the square numbers representing the
lengths is consistently i (a~ can be seen clearly in
rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)
represerting the quarter inch taken at the right in
Ftp 2 A was then di vided into f 1 ve parts each of
which equa Is g To n of these 4 parts were added in
40
2 nturn 100 small parts so that the trinomial expressing 22 An n
the length of the pendulum ruled in thirds is A 5 100
The demonstration of the construction to which
Sauveur refers the reader29 differs from this one in that
Sauveur states that the difference 6[ is 2A + 1 which would
be true only if the difference between themiddot successive
numbers squared in L of Table 2 were 1 instead of~ But
Sauveurs expression A2+ 2~n t- ~~ is equivalent to the
one given above (A2+ AS +l~~) if as he states tho 1 of
(2A 1) is taken to be inch and with this stipulation
his somewhat roundabout proof becomes wholly intelligible
The chronometer thus correctly divided into twelfth
parts of a second and thirds is not subject to the criticism
which Sauveur levelled against the chronometer of Loulie-shy
that it did not umark the duration of notes in any known
relation to the duration of a second because the periods
of vibration of its pendulum are for the most part incomshy
mensurable with a second30 FonteneJles report on
Sauveurs work of 1701 in the Histoire de lAcademie31
comprehends only the system of 43 meridians and 301
heptamerldians and the theory of harmonics making no
29Sauveur Systeme General pp432-433 see vol II pp 39-31 below
30 Sauveur uSysteme General pp 435-436 see vol II p 33 below
31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180
41
mention of the Echometer or any of its scales nevertheless
it was the first practical instrument--the string lengths
required by Mersennes calculations made the use of
pendulums adiusted to them awkward--which took account of
the proportional laws of length and time Within the next
few decades a number of theorists based thei r wri tings
on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol
LAffilard and Louis-Leon Pajot Cheva1ier32 but they
will perhaps best be considered in connection with
others who coming after Sauveur drew upon his acoustical
discoveries in the course of elaborating theories of
music both practical and speculative
32Harding Origins pp 11-12
CHAPTER II
THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS
Sauveurs Memoire of 17011 is concerned as its
title implies principally with the elaboration of a system
of measurement classification nomenclature and notation
of intervals and sounds and with examples of the supershy
imposition of this system on existing systems as well as
its application to all the instruments of music This
program is carried over into the subsequent papers which
are devoted in large part to expansion and clarification
of the first
The consideration of intervals begins with the most
fundamental observation about sonorous bodies that if
two of these
make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2
This prinCiple discovered only about seventy years
lSauveur Systeme General
2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below
42
43
earlier by both Mersenne and Galileo3 is one of the
foundation stones upon which Sauveurs system is built
The intervals are there assigned names according to the
relative numbers of vibrations of the sounds of which they
are composed and these names partly conform to usage and
partly do not the intervals which fall within the compass
of one octave are called by their usual names but the
vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono
more than an oc tave above a fundamental are designs ted as
belonging to the acute octaves and those falling below are
said to belong to the grave octaves 4 The intervals
reaching into these acute and grave octaves are called
replicas triplicas and so forth or sub-replicas
sub-triplicas and so forth
This system however does not completely satisfy
Sauveur the interval names are ambiguous (there are for
example many sizes of thirds) the intervals are not
dOllhled when their names are dOllbled--n slxth for oxnmplo
is not two thirds multiplying an element does not yield
an acceptable interval and the comma 1s not an aliquot
part of any interval Sauveur illustrates the third of
these difficulties by pointing out the unacceptability of
intervals constituted by multiplication of the major tone
3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1
1877 (reprint ed New York Dover Publications 1945)
4Sauveur Systeme General It p 409 see vol IIJ p 6 below
44
But the Pythagorean third is such an interval composed
of two major tones and so it is clear here as elsewhere
too t the eli atonic system to which Sauveur refers is that
of jus t intona tion
rrhe Just intervuls 1n fact are omployod by
Sauveur as a standard in comparing the various temperaments
he considers throughout his work and in the Memoire of
1707 he defines the di atonic system as the one which we
follow in Europe and which we consider most natural bullbullbull
which divides the octave by the major semi tone and by the
major and minor tone s 5 so that it is clear that the
diatonic system and the just diatonic system to which
Sauveur frequently refers are one and the same
Nevertheless the system of just intonation like
that of the traditional names of the intervals was found
inadequate by Sauveur for reasons which he enumerated in
the Memo ire of 1707 His first table of tha t paper
reproduced below sets out the names of the sounds of two
adjacent octaves with numbers ratios of which represhy
sent the intervals between the various pairs o~ sounds
24 27 30 32 36 40 45 48 54 60 64 72 80 90 98
UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut
T t S T t T S T t S T t T S
lie supposes th1s table to represent the just diatonic
system in which he notes several serious defects
I 5sauveur UMethode Generale p 259 see vol II p 128 below
7
45
The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6
The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS
A melody composed in this system could not he aTpoundTues be
performed on an organ or harpsichord and devices the sounns
of which depend solely on the keys of a keyboa~d without
the players being able to correct them8 for if after
a sound you are to make an interval which is altered by
a commu--for example if after LA you aroto rise by a
fourth to re you cannot do so for the fourth LAre is
too large by a comma 9 rrhe same difficulties would beset
performers on trumpets flut es oboes bass viols theorbos
and gui tars the sound of which 1s ruled by projections
holes or keys 1110 or singers and Violinists who could
6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below
7 Ibid bull
n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483
I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below
I IlOSauveur Methode Generale p 262 see vol II p 132 below
46
not for lack perhaps of a fine ear make the necessary
corrections But even the most skilled amont the pershy
formers on wind and stringed instruments and the best
11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c
system because of the discrepancies in interval s1za and
he subjoins an example of plainchant in which if the
intervals are sung just the last ut will be higher than
the first by 2 commasll so that if the litany is sung
55 times the final ut of the 55th repetition will be
higher than the fi rst ut by 110 commas or by two octaves 12
To preserve the identity of the final throughout
the composition Sauveur argues the intervals must be
changed imperceptibly and it is this necessity which leads
13to the introduc tion of t he various tempered ays ternf
After introducing to the reader the tables of the
general system in the first Memoire of 1701 Sauveur proshy
ceeds in the third section14 to set out ~is division of
the octave into 43 equal intervals which he calls
llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20
12Ibid bull
I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below
14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below
47
meridians and the division of each meridian into seven
equal intervals which he calls Ifheptameridians
The number of meridians in each just interval appears
in the center column of Sauveurs first table15 and the
number of heptameridians which in some instances approaches
more nearly the ratio of the just interval is indicated
in parentheses on th e corresponding line of Sauveur t s
second table
Even the use of heptameridians however is not
sufficient to indicate the intervals exactly and although
Sauveur is of the opinion that the discrepancies are too
small to be perceptible in practice16 he suggests a
further subdivision--of the heptameridian into 10 equal
decameridians The octave then consists of 43
meridians or 301 heptameridja ns or 3010 decal11eridians
rihis number of small parts is ospecially well
chosen if for no more than purely mathematical reasons
Since the ratio of vibrations of the octave is 2 to 1 in
order to divide the octave into 43 equal p~rts it is
necessary to find 42 mean proportionals between 1 and 217
15Sauveur Systeme General p 498 see vol II p 95 below
16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y
G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian
17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below
48
The task of finding a large number of mean proportionals
lIunknown to the majority of those who are fond of music
am uvery laborious to others u18 was greatly facilitated
by the invention of logarithms--which having been developed
at the end of the sixteenth century by John Napier (1550shy
1617)19 made possible the construction of a grent number
01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront
practical difficulties In the problem of constructing
43 proportionals however the values are patticularly
easy to determine because as 43 is a prime factor of 301
and as the first seven digits of the common logarithm of
2 are 3010300 by diminishing the mantissa of the logarithm
by 300 3010000 remains which is divisible by 43 Each
of the 43 steps of Sauveur may thus be subdivided into 7-shy
which small parts he called heptameridians--and further
Sllbdlvlded into 10 after which the number of decnmoridlans
or heptameridians of an interval the ratio of which
reduced to the compass of an octave 1s known can convenshy
iently be found in a table of mantissas while the number
of meridians will be obtained by dividing vhe appropriate
mantissa by seven
l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below
19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617
49
The cycle of 301 takes its place in a series of
cycles which are sometime s extremely useful fo r the purshy
20poses of calculation lt the cycle of 30103 jots attribshy
uted to de Morgan the cycle of 3010 degrees--which Is
in fact that of Sauveurs decameridians--and Sauveurs
cycl0 01 001 heptamerldians all based on the mllnLlsln of
the logarithm of 2 21 The system of decameridlans is of
course a more accurate one for the measurement of musical
intervals than cents if not so convenient as cents in
certain other ways
The simplici ty of the system of 301 heptameridians
1s purchased of course at the cost of accuracy and
Sauveur was aware that the logarithms he used were not
absolutely exact ubecause they are almost all incommensurshy
ablo but tho grnntor the nurnbor of flputon tho
smaller the error which does not amount to half of the
unity of the last figure because if the figures stricken
off are smaller than half of this unity you di sregard
them and if they are greater you increase the last
fif~ure by 1 1122 The error in employing seven figures of
1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600
octave or finally than one vibration out of 86n5800
~OHelmhol tz) Sensatlons of Tone p 457
21 Ibid bull
22Sauveur Methode Generale p 275 see vol II p 143 below
50
n23which is of absolutely no consequence The error in
striking off 3 fir-ures as was done in forming decameridians
rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of
868524 and the error in striking off the last four
figures as was done in forming the heptameridians will
be at the greatest only ~ heptamerldian or Ii of a
1 25 eomma or 602 of an octave or lout of 870 vlbration
rhls last error--l out of 870 vibrations--Sauveur had
found tolerable in his M~moire of 1701 26
Despite the alluring ease with which the values
of the points of division may be calculated Sauveur 1nshy
sists that he had a different process in mind in making
it Observing that there are 3T2t and 2s27 in the
octave of the diatonic system he finds that in order to
temper the system a mean tone must be found five of which
with two semitones will equal the octave The ratio of
trIO tones semltones and octaves will be found by dlvldlnp
the octave into equal parts the tones containing a cershy
tain number of them and the semi tones ano ther n28
23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below
24Sauveur Methode GenEsectrale p 275 see vol II p 144 below
25 Ibid bull
26 Sauveur Systeme General p 422 see vol II p 19 below
2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone
28Sauveur MEthode Generale p 265 see vol II p 135 below
51
If T - S is s (the minor semitone) and S - s is taken as
the comma c then T is equal to 28 t 0 and the octave
of 5T (here mean tones) and 2S will be expressed by
128t 7c and the formula is thus derived by which he conshy
structs the temperaments presented here and in the Memoire
of 1711
Sau veul proceeds by determining the ratios of c
to s by obtaining two values each (in heptameridians) for
s and c the tone 28 + 0 has two values 511525 and
457575 and thus when the major semitone s + 0--280287-shy
is subtracted from it s the remainder will assume two
values 231238 and 177288 Subtracting each value of
s from s + 0 0 will also assume two values 102999 and
49049 To obtain the limits of the ratio of s to c the
largest s is divided by the smallest 0 and the smallest s
by the largest c yielding two limiting ratlos 29
31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range
between l~ and 4~ and the octave 12s+70 will 11e30 between
2774 and 6374 bull For simplicity he settles on the approximate
2 2limits of 1 to between 13 and 43 for c and s so that if
o 1s set equal to 1 s will range between 2 and 4 and the
29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284
30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2
35 35
52
octave will be 31 43 and 55 With a c of 2 s will fall
between 4 and 9 and the octave will be 62748698110
31 or 122 and so forth
From among these possible systems Sauveur selects
three for serious consideration
lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32
Barbour has written of Sauveur and his method that
to him
the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33
31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament
2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below
33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3
53
The formula implied in Barbours calculations is
5 (S +s) +28 which is equlvalent to Sauveur t s formula
12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since
73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70
The superparticular ratios 32 43 54 and so forth
represont ratios of S to s when c is equal to 1 and so
n +1the sugrested - series is an instance of the more genshyn
eral serie s s + c when C is equal to one As n increases s
the fraction 7n+4 representative of the fifthl2n+7
approaches 127 as its limit or the fifth of equal temperashy11 ~S4
mont from below when n =1 the fraction equals 19
which corresponds to 69473 or 695 cents while the 11mitshy
7lng value 12 corresponds to 700 cents Similarly
4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this
study has shown however Sauveur had no intention of
allowing n to increase beyond 4 although the reason he
gave in restricting its range was not that the thirds
would otherwise become intolerably sharp but rather that
the system would become unwieldy with the progressive
mUltiplication of its parts Neverthelesf with the
34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6
5
35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128
54
limitation Sauveur set on the range of s his system seems
immune to the criticism levelled at it by Barbour
It is perhaps appropriate to note here that for
any values of sand c in which s is greater than c the
7s + 4cfrac tion representing the fifth l2s + 7c will be smaller
than l~ Thus a1l of Suuveurs systems will be nngative-shy
the fifths of all will be flatter than the just flfth 36
Of the three systems which Sauveur singled out for
special consideration in the Memoire of 1707 the cycles
of 31 43 and 55 parts (he also discusses the cycle of
12 parts because being very simple it has had its
partisans u37 )--he attributed the first to both Mersenne
and Salinas and fi nally to Huygens who found tile
intervals of the system exactly38 the second to his own
invention and the third to the use of ordinary musicians 39
A choice among them Sauveur observed should be made
36Ib i d p xi
37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below
38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does
not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118
39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone
6
55
partly on the basis of the relative correspondence of each
to the diatonic system and for this purpose he appended
to the Memoire of 1707 a rable for comparing the tempered
systems with the just diatonic system40 in Which the
differences of the logarithms of the various degrees of
the systems of 12 31 43 and 55 to those of the same
degrees in just intonation are set out
Since cents are in common use the tables below
contain the same differences expressed in that measure
Table 5 is that of just intonation and contains in its
first column the interval name assigned to it by Sauveur41
in the second the ratio in the third the logarithm of
the ratio given by Sauveur42 in the fourth the number
of cents computed from the logarithm by application of
the formula Cents = 3986 log I where I represents the
ratio of the interval in question43 and in the fifth
the cents rounded to the nearest unit (Table 5)
temperament favored by Silberman Barbour Tuning and Temperament p 126
40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below
41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths
42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column
43John Backus Acoustical Foundations p 292
56
TABLE 5
JUST INTONATION
INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)
VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB
1 169 2498775 99601 996
VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610
IV 4532 1480625 59018 590 4 43 1249387 49800 498
III 54 0969100 38628 386 3 65middot 0791812 31561 316
112 98 0511525 20389 204
III 109 0457575 18239 182
2 1615 0280287 11172 112
The first column of Table 6 gives the name of the
interval the second the number of parts of the system
of 12 which are given by Sauveur44 as constituting the
corresponding interval in the third the size of the
number of parts given in the second column in cents in
trIo fourth column tbo difference between the size of the
just interval in cents (taken from Table 5)45 and the
size of the parts given in the third column and in the
fifth Sauveurs difference calculated in cents by
44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below
45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s
57
application of the formula cents = 3986 log I but
rounded to the nearest cent
rABLE 6
SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE
VII 11 1100 +12 +12 72 71
10 1000 -IS + 4
-18 + 4
VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5
JV 6 600 -10 +10
-10 flO
4 5 500 + 2 + 2 III 4 400 +14 +14
3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS
2 1 100 -12 -12
It will be noted that tithe interval and it s comshy
plement have the same difference except that in one it
is positlve and in the other it is negative tl46 The sum
of differences of the tempered second to the two of just
intonation is as would be expected a comma (about
22 cents)
The same type of table may be constructed for the
systems of 3143 and 55
For the system of 31 the values are given in
Table 7
46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below
58
TABLE 7
THE SYSTEM OF 31
SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE
VII 28 1084 - 4 - 4 72 71 26 1006
-12 +10
-11 +10
VI 6
23 21
890 813
--
6 1
- 6 - 1
V 18 697 - 5 - 5 5 16 619 + 9 10
IV 15 581 - 9 -10 4 13 503 + 5 + 5
III 10 387 + 1 + 1 3 8 310 - 6 - 6
112 III
5 194 -10 +12
-10 11
2 3 116 4 + 4
The small discrepancies of one cent between
Sauveurs calculation and those in the fourth column result
from the rounding to cents in the calculations performed
in the computation of the values of the third and fourth
columns
For the system of 43 the value s are given in
Table 8 (Table 8)
lhe several discrepancies appearlnt~ in thln tnblu
are explained by the fact that in the tables for the
systems of 12 31 43 and 55 the logarithms representing
the parts were used by Sauveur in calculating his differshy
encss while in his table for the system of 43 he employed
heptameridians instead which are rounded logarithms rEha
values of 6 V and IV are obviously incorrectly given by
59
Sauveur as can be noted in his table 47 The corrections
are noted in brackets
TABLE 8
THE SYSTEM OF 43
SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE
VII 39 1088 0 0 -13 -1372 36 1005
71 + 9 + 8
VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4
IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4
III 14 391 5 + 4 3 11 307 9 - 9-
112 - 9 -117 195 III +13 +13
2 4 112 0 0
For the system of 55 the values are given in
Table 9 (Table 9)
The values of the various differences are
collected in Table 10 of which the first column contains
the name of the interval the second third fourth and
fifth the differences from the fourth columns of
(ables 6 7 8 and 9 respectively The differences of
~)auveur where they vary from those of the third columns
are given in brackets In the column for the system of
43 the corrected values of Sauveur are given where they
[~re appropriate in brackets
47 IISauveur Methode Generale p 276 see vol I~ p 145 below
60
TABLE 9
ThE SYSTEM OF 55
SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE
VII 50 1091 3 -+ 3 72
71 46 1004
-14 + 8
-14
+ 8
VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5
32 28
698 611
- 4 + 1
- 4 +shy 1
IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4
III 18 393 + 7 + 6 3 14 305 -11 -10
112 III
9 196 - 8 +14
- 8 +14
2 5 109 - 3 - 3
TABLE 10
DIFFEHENCES
SYSTEMS
INTERVAL 12 31 43 55
VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14
71 + 4 +10 9 ~8] 8
VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5
IV 4
III
- 2 -10 +10 + 2 +14
- 5 + 9 [+101 - 9 F-10] 1shy 5 1
- 4 + 4 - 4+ 4 _ + 5 L+41
4 1 - 1 + 4 7 8shy 6]
3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14
2 -12 4 0 - 3
61
Sauveur notes that the differences for each intershy
val are largest in the extreme systems of the three 31
43 55 and that the smallest differences occur in the
fourths and fifths in the system of 55 J at the thirds
and sixths in the system of 31 and at the minor second
and major seventh in the system of 4348
After layin~ out these differences he f1nally
proceeds to the selection of a system The principles
have in part been stated previously those systems are
rejected in which the ratio of c to s falls outside the
limits of 1 to l and 4~ Thus the system of 12 in which
c = s falls the more so as the differences of the
thirds and sixths are about ~ of a comma 1t49
This last observation will perhaps seem arbitrary
Binee the very system he rejects is often used fiS a
standard by which others are judged inferior But Sauveur
was endeavoring to achieve a tempered system which would
preserve within the conditions he set down the pure
diatonic system of just intonation
The second requirement--that the system be simple-shy
had led him previously to limit his attention to systems
in which c = 1
His third principle
that the tempered or equally altered consonances do not offend the ear so much as consonances more altered
48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below
49Sauveur Methode Generale n p 278 see vol II p 147 below
62
mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50
is one of the very few arbitrary aesthetic judgments which
Sauveur allows to influence his decisions The prinCiple
of course favors the adoption of the system of 43 which
it will be remembered had generally smaller differences
to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy
the differences of the columns for the systems of 31 43
and 55 in Table 10 add respectively to 94 80 and 90
A second perhaps somewhat arbitrary aesthetic
judgment that he aJlows to influence his reasoning is that
a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01
The popularity of the mean-tone temperaments however
with their attempt to achieve p1re thirds at the expense
of the fifths WJuld seem to belie this pronouncement 52
The choice of the system of 43 having been made
as Sauveur insists on the basis of the preceding princishy
pIes J it is confirmed by the facility gained by the corshy
~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s
the ~antissa of the logarithm of 2 and even more from
the fa ct t1at
)oSal1veur M~thode Generale p 278 see vol II p 148 below
51Sauvenr UMethocle Generale n p 279 see vol II p 148 below
52Barbour Tuning and Temperament p 11 and passim
63
the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03
In 1711 Sauveur p11blished a Memolre)4 in rep] y
to Konrad Benfling Nho in 1708 constructed a system of
50 equal parts a description of which Was pubJisheci in
17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht
of as an octave comnosed of ditonic commas since
122 ~ 24 = 5056 That system was constructed according
to Sauveur by reciprocal additions and subtractions of
the octave fifth and major third and 18 bused upon
the principle that a legitimate system of music ought to
have its intervals tempered between the just interval and
n57that which he has found different by a comma
Sauveur objects that a system would be very imperfect if
one of its te~pered intervals deviated from the ~ust ones
53Sauveur Methode Gene~ale p 273 see vol II p 141 below
54SnuvelJr Tahle Gen~rn1e II
55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII
56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)
57Sauveur Table Gen6rale1I p 407 see vol II p 155 below
64
even by a half-comma 58 and further that although
Ilenflinr wnnts the tempered one [interval] to ho betwoen
the just an d exceeding one s 1 t could just as reasonabJ y
be below 59
In support of claims and to save himself the trolJhle
of respondi ng in detail to all those who might wi sh to proshy
pose new systems Sauveur prepared a table which includes
nIl the tempered systems of music60 a claim which seems
a bit exaggerated 1n view of the fact that
all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt
l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61
The positive systems forming their thirds by 8 fifths r
dowl for their fifths being larger than E T LEqual
TemperamentJ fifths depress the pitch bel~w E T when
tuned downwardsrt so that the third of A should he nb
58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72
L J
rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )
59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below
60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11
67 below
61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246
65
which is inconsistent wi~h musical usage require a
62 separate notation Sauveur was according to Barbour
uflahlc to npprecinto the splondid vn]uo of tho third)
of the latter [the system of 53J since accordinp to his
theory its thirds would have to be as large as Pythagorean
thi rds 63 arei a glance at the table provided wi th
f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT
considered the third of the system of 53 to be thnt of 18
steps or 408 cents which is precisely the size of the
Pythagorean third or in Sauveurs table 55 decameridians
(about 21 cents) sharp rather than the nearly perfect
third of 17 steps or 385 cents formed by 8 descending fifths
The rest of the 25 systems included by Sauveur in
his table are rejected by him either because they consist
of too many parts or because the differences of their
intervals to those of just intonation are too Rro~t bull
flhemiddot reasoning which was adumbrat ed in the flemoire
of 1701 and presented more fully in those of 1707 and
1711 led Sauveur to adopt the system of 43 meridians
301 heptameridians and 3010 decameridians
This system of 43 is put forward confident1y by
Sauveur as a counterpart of the 360 degrees into which the
circle ls djvlded and the 10000000 parts into which the
62RHlIT Bosanquet Temperament or the di vision
of the Octave Musical Association Proceedings 1874shy75 p 13
63Barbour Tuning and Temperament p 125
64Sauveur Table Gen6rale p 416 see vol II p 167 below
66
whole sine is divided--as that is a uniform language
which is absolutely necessary for the advancement of that
science bull 65
A feature of the system which Sauveur describes
but does not explain is the ease with which the rntios of
intervals may be converted to it The process is describod
661n tilO Memolre of 1701 in the course of a sories of
directions perhaps directed to practical musicians rathor
than to mathematicians in order to find the number of
heptameridians of an interval the ratio of which is known
it is necessary only to add the numbers of the ratio
(a T b for example of the ratio ~ which here shall
represent an improper fraction) subtract them (a - b)
multiply their difference by 875 divide the product
875(a of- b) by the sum and 875(a - b) having thus been(a + b)
obtained is the number of heptameridians sought 67
Since the number of heptamerldians is sin1ply the
first three places of the logarithm of the ratio Sauveurs
II
65Sauveur Table Generale n p 406 see vol II p 154 below
66~3auveur
I Systeme Generale pp 421-422 see vol pp 18-20 below
67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since
v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave
67
process amounts to nothing less than a means of finding
the logarithm of the ratio of a musical interval In
fact Alexander Ellis who later developed the bimodular
calculation of logarithms notes in the supplementary
material appended to his translation of Helmholtzs
Sensations of Tone that Sauveur was the first to his
knowledge to employ the bimodular method of finding
68logarithms The success of the process depends upon
the fact that the bimodulus which is a constant
Uexactly double of the modulus of any system of logashy
rithms is so rela ted to the antilogari thms of the
system that when the difference of two numbers is small
the difference of their logarithms 1s nearly equal to the
bimodulus multiplied by the difference and di vided by the
sum of the numbers themselves69 The bimodulus chosen
by Sauveur--875--has been augmented by 6 (from 869) since
with the use of the bimodulus 869 without its increment
constant additive corrections would have been necessary70
The heptameridians converted to c)nt s obtained
by use of Sau veur I s method are shown in Tub1e 11
68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447
69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other
70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447
68
TABLE 11
INT~RVAL RATIO SIZE (BYBIMODULAR
JUST RATIO IN CENTS
DIFFERENCE
COMPUTATION)
IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o
III 54 387 386 t 1 3 65 317 316 + 1
112 98 205 204 + 1
III 109 184 182 t 2 2 1615 113 112 + 1
In this table the size of the interval calculated by
means of the bimodu1ar method recommended by Sauveur is
seen to be very close to that found by other means and
the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s
method devised later was correct for the Major Third
The system of 43 meridians wi th it s variolls
processes--the further di vision into 301 heptame ridlans
and 3010 decameridians as well as the bimodular method of
comput ing the number of heptameridians di rt9ctly from the
ratio of the proposed interva1--had as a nncessary adshy
iunct in the wri tings of Sauveur the estSblishment of
a fixed pitch by the employment of which together with
71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho
I~ )rutlo of the tritone is given by Sauveur as 32) The
tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp
69
the system of 43 the name of any pitch could be determined
to within the range of a half-decameridian or about 02
of a cent 72 It had been partly for Jack of such n flxod
tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve
Munic of 1697 so deficient that he could not in conscience
publish it73 Having addressed that problem he came forth
in 1700 with a means of finding the fixed sound a
description of which is given in the Histoire de lAcademie
of the year 1700 Together with the system of decameridshy
ians the fixed sound placed at Sauveurs disposal a menns
for moasuring pitch with scientific accuracy complementary I
to the system he put forward for the meaSurement of time
in his Chronometer
Fontenelles report of Sauveurs method of detershy
mining the fixed sound begins with the assertion that
vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74
72A decameridian equals about 039 cents and half a decameridi an about 019 cents
73Sauveur trSyst~me Generale p 405 see vol II p 3 below
74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales
lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux
~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les
I nombres qui expriment 11inegal1te des cordes sont plus grands II
70
For example if the lengths are 2 and I the shorter string
makes 2 vibrations while the longer makes 1 If the lengths
are 25 and 24 the longer will make 24 vibrations while
the shorte~ makes 25
Sauveur had noticed that when you hear Organs tuned
am when two pipes which are nearly in unison are plnyan
to[~cthor tnere are certain instants when the common sOllnd
thoy rendor is stronrer and these instances scem to locUr
75at equal intervals and gave as an explanation of this
phenomenon the theory that the sound of the two pipes
together must have greater force when their vibrations
after having been separated for some time come to reunite
and harmonize in striking the ear at the same moment 76
As the pipes come closer to unison the numberS expressin~
their ratio become larger and the beats which are rarer
are more easily distinguished by the ear
In the next paragraph Fontenelle sets out the deshy
duction made by Sauveur from these observations which
made possible the establishment of the fixed sound
If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it
75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux
76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup
71
made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In
r7 middotthl fl gl ven t 1me bull
Having found the means of establishing the number
of vibrations of a sound Sauveur settled upon 100 as the
number of vibrations which the fixed sound to which all
others could be referred in comparison makes In one
second
Sauveur also estimated the number of beats perceivshy
able in a second about six in a second can be distinguished
01[11] y onollph 78 A grenter numbor would not bo dlnshy
tinguishable in one second but smaller numbers of beats
77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u
78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite
72
in a second Vlould be distinguished with greater and rreater
ease This finding makes it necessary to lower by octaves
the pipes employed in finding the number of vibrations in
a second of a given pitch in reference to the fixed tone
in order to reduce the number of beats in a second to a
countable number
In the Memoire of 1701 Sauvellr returned to the
problem of establishing the fixed sound and gave a very
careful ctescription of the method by which it could be
obtained 79 He first paid tribute to Mersenne who in
Harmonie universelle had attempted to demonstrate that
a string seventeen feet long and held by a weight eight
pounds would make 8 vibrations in a second80--from which
could be deduced the length of string necessary to make
100 vibrations per second But the method which Sauveur
took as trle truer and more reliable was a refinement of
the one that he had presented through Fontenelle in 1700
Three organ pipes must be tuned to PA and pa (UT
and ut) and BOr or BOra (SOL)81 Then the major thlrd PA
GAna (UTMI) the minor third PA go e (UTMlb) and
fin2l1y the minor senitone go~ GAna (MlbMI) which
79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below
80IJIersenne Harmonie univergtsel1e 11117 pp 140-146
81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below
73
has a ratio of 24 to 25 A beating will occur at each
25th vibra tion of the sha rper one GAna (MI) 82
To obtain beats at each 50th vibration of the highshy
est Uemploy a mean g~ca between these two pipes po~ and
GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in
the same time 48 59 and 50 vibrationSj83 and to obtain
beats at each lOath vibration of the highest the mean ga~
should be placed between the pipes g~ca and GAna and the v
mean gu between go~ and g~ca These five pipes gose
v Jgu g~~ ga~ and GA~ will make their beats at 96 97
middot 98 99 and 100 vibrations84 The duration of the beats
is me asured by use of a pendulum and a scale especially
rra rke d in me ridia ns and heptameridians so tha t from it can
be determined the distance from GAna to the fixed sound
in those units
The construction of this scale is considered along
with the construction of the third fourth fifth and
~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l
bo remembered were devised for the measurement of temporal
du rations to the nearest third The third scale is the
General Monochord It is divided into meridians and heptashy
meridians by carrying the decimal ratios of the intervals
in meridians to an octave (divided into 1000 pa~ts) of the
monochord The process is repeated with all distances
82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow
83Ibid bull The mean required is the geometric mean
84Ibid bull
v
74
halved for the higher octaves and doubled for the lower
85octaves The third scale or the pendulum for the fixed
sound employed above to determine the distance of GAna
from the fixed sound was constructed by bringing down
from the Monochord every other merldian and numbering
to both the left and right from a point 0 at R which marks
off 36 unlvornul inches from P
rphe reason for thi s division into unit s one of
which is equal to two on the Monochord may easily be inshy
ferred from Fig 3 below
D B
(86) (43) (0 )
Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~
(43) (215)
Fig 3
C bisects AB an d 01 besects AIBI likewi se D hi sects AC
und Dl bisects AlGI- If AB is a monochord there will
be one octave or 43 meridians between B and C one octave
85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian
75
or 43 more between C and D and so forth toward A If
AB and AIBI are 36 universal inches each then the period
of vibration of AIBl as a pendulum will be 2 seconds
and the half period with which Sauveur measured~ will
be 1 second Sauveur wishes his reader to use this
pendulum to measure the time in which 100 vibrations are
mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4
vibratlons of pipes in the ratio 2524) If the pendulum
is AIBI in length there will be 100 vihrations in 1
second If the pendulu111 is AlGI in length or tAIBI
1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee
the period of a pendulum is proportional to the square root
of its length There will then be 100-12 vibrations in one 100
second (since 2 =~ where x represents the number of
2
vibrations in one second) or 14142135 vibrations in one
second The ratio of e vibrations will then be 14142135
to 100 or 14142135 to 1 which is the ratio of the tritone
or ahout 21i meridians Dl is found by the same process to
mark 43 meridians and from this it can be seen that the
numhers on scale AIBI will be half of those on AB which
is the proportion specified by Sauveur
rrne fifth scale indicates the intervals in meridshy
lans and heptameridJans as well as in intervals of the
diatonic system 1I86 It is divided independently of the
f ~3t fonr and consists of equal divisionsJ each
86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below
76
representing a meridian and each further divisible into
7 heptameridians or 70 decameridians On these divisions
are marked on one side of the scale the numbers of
meridians and on the other the diatonic intervals the
numbers of meridians and heptameridians of which can be I I
found in Sauveurs Table I of the Systeme General
rrhe sixth scale is a sCale of ra tios of sounds
nncl is to be divided for use with the fifth scale First
100 meridians are carried down from the fifth scale then
these pl rts having been subdivided into 10 and finally
100 each the logarithms between 100 and 500 are marked
off consecutively on the scale and the small resulting
parts are numbered from 1 to 5000
These last two scales may be used Uto compare the
ra tios of sounds wi th their 1nt ervals 87 Sauveur directs
the reader to take the distance representinp the ratIo
from tbe sixth scale with compasses and to transfer it to
the fifth scale Ratios will thus be converted to meridians
and heptameridia ns Sauveur adds tha t if the numberS markshy
ing the ratios of these sounds falling between 50 and 100
are not in the sixth scale take half of them or double
themn88 after which it will be possible to find them on
the scale
Ihe process by which the ratio can be determined
from the number of meridians or heptameridians or from
87Sauveur USysteme General fI p 434 see vol II p 32 below
I I88Sauveur nSyst~me General p 435 seo vol II p 02 below
77
an interval of the diatonic system is the reverse of the
process for determining the number of meridians from the
ratio The interval is taken with compasses on the fifth
scale and the length is transferred to the sixth scale
where placing one point on any number you please the
other will give the second number of the ratio The
process Can be modified so that the ratio will be obtainoo
in tho smallest whole numbers
bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89
Suuveur reports at the end of the fourth section shy
of the Memoire of 1701 tha t Chapotot one of the most
skilled engineers of mathematical instruments in Paris
has constructed Echometers and that he has made one of
them from copper for His Royal Highness th3 Duke of
Orleans 90 Since the fifth and sixth scale s could be
used as slide rules as well as with compas5es as the
scale of the sixth line is logarithmic and as Sauveurs
above romarl indicates that he hud had Echometer rulos
prepared from copper it is possible that the slide rule
89Sauveur Systeme General I p 435 see vol II
p 33 below
90 ISauveur Systeme General pp 435-436 see vol II p 33 below
78
which Cajori in his Historz of the Logarithmic Slide Rule91
reports Sauveur to have commissioned from the artisans Gevin
am Le Bas having slides like thos e of Seth Partridge u92
may have been musical slide rules or scales of the Echo-
meter This conclusion seems particularly apt since Sauveur
hnd tornod his attontion to Acoustlcnl problems ovnn boforo
hIs admission to the Acad~mie93 and perhaps helps to
oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in
his Dictionnaire universel de mathematigue at de physique
that before 1753 R P Pezenas was the only author to
discuss these kinds of scales [slide rules] 94 thus overshy
looking Sauveur as well as several others but Sauveurs
rule may have been a musical one divided although
logarithmically into intervals and ratios rather than
into antilogaritr~s
In the Memoire of 171395 Sauveur returned to the
subject of the fixed pitch noting at the very outset of
his remarks on the subject that in 1701 being occupied
wi th his general system of intervals he tcok the number
91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)
92Ib1 d p 43 bull
93Scherchen Nature of Music p 26
94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es
95Sauveur J Rapport It
79
100 vibrations in a seoond only provisionally and having
determined independently that the C-SOL-UT in practice
makes about 243~ vibrations per second and constructing
Table 12 below he chose 256 as the fundamental or
fixed sound
TABLE 12
1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1
2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214
32768 65536
215 216
With this fixed sound the octaves can be convenshy
iently numbered by taking the power of 2 which represents
the number of vibrations of the fundamental of each octave
as the nmnber of that octave
The intervals of the fundamentals of the octaves
can be found by multiplying 3010300 by the exponents of
the double progression or by the number of the octave
which will be equal to the exponent of the expression reshy
presenting the number of vibrations of the various func1ashy
mentals By striking off the 3 or 4 last figures of this
intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians
can be added to obtain all the meridians and intervals
of each octave
96 Ibid p 454 see vol II p 186 below
80
To render all of this more comprehensible Sauveur
offers a General table of fixed sounds97 which gives
in 13 columns the numbers of vibrations per second from
8 to 65536 or from the third octave to the sixteenth
meridian by meridian 98
Sauveur discovered in the course of his experiments
with vibra ting strings that the same sound males twice
as many vibrations with strings as with pipes and con-
eluded that
in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99
It will be remembered that even in the discllssion of
pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy
tion taken as a vibration half of a period lOO
rlho th cory of fixed tone thon and thB te-rrnlnolopy
of vibrations were elaborated and refined respectively
in the M~moire of 1713
97 Sauveur Rapport lip 468 see vol II p 203 below
98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly
99sauveur uRapport pp 450-451 see vol II p 183 below
lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below
81
The applications which Sauveur made of his system
of measurement comprising the echometer and the cycle
of 43 meridians and its subdivisions were illustrated ~
first in the fifth and sixth sections of the Memoire of
1701
In the fifth section Sauveur shows how all of the
varIous systems of music whether their sounas aro oxprossoc1
by lithe ratios of their vibrations or by the different
lengths of the strings of a monochord which renders the
proposed system--or finally by the ratios of the intervals
01 one sound to the others 101 can be converted to corshy
responding systems in meridians or their subdivisions
expressed in the special syllables of solmization for the
general system
The first example he gives is that of the regular
diatonic system or the system of just intonation of which
the ratios are known
24 27 30 32 36 40 ) 484
I II III IV v VI VII VIII
He directs that four zeros be added to each of these
numhors and that they all be divided by tho ~Jmulle3t
240000 The quotient can be found as ratios in the tables
he provides and the corresponding number of meridians
a~d heptameridians will be found in the corresponding
lOlSauveur Systeme General p 436 see vol II pp 33-34 below
82
locations of the tables of names meridians and heptashy
meridians
The Echometer can also be applied to the diatonic
system The reader is instructed to take
the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02
If one point is placed on 0 the other will give the
intervals in meridians and heptameridians bull bull bull as well
as the interval bullbullbull of the diatonic system 103
He next considers a system in which lengths of a
monochord are given rather than ratios Again rntios
are found by division of all the string lengths by the
shortest but since string length is inversely proportional
to the number of vibrations a string makes in a second
and hence to the pitch of the string the numbers of
heptameridians obtained from the ratios of the lengths
of the monochord must all be subtracted from 301 to obtain
tne inverses OT octave complements which Iru1y represent
trIO intervals in meridians and heptamerldlnns which corshy
respond to the given lengths of the strings
A third example is the system of 55 commas Sauveur
directs the reader to find the number of elements which
each interval comprises and to divide 301 into 55 equal
102 ISauveur Systeme General pp 438-439 see vol II p 37 below
l03Sauveur Systeme General p 439 see vol II p 37 below
83
26parts The quotient will give 555 as the value of one
of these parts 104 which value multiplied by the numher
of parts of each interval previously determined yields
the number of meridians or heptameridians of each interval
Demonstrating the universality of application of
hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct
two systems foreign to the usage of his time one ancient
and one orlental The ancient system if that of the
Greeks reported by Mersenne in which of three genres
the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105
Sauveurs reconstruction of Mersennes Greek system gives
tl1C diatonic system with steps at 0 28 78 and 125 heptashy
meridians the chromatic system with steps at 0 28 46
and 125 heptameridians and the enharmonic system with
steps at 0 14 28 and 125 heptameridians In the
chromatic system the two semi tones 0-28 and 28-46 differ
widely in size the first being about 112 cents and the
other only about 72 cents although perhaps not much can
be made of this difference since Sauveur warns thnt
104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below
105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure
84
each of these [the genres] has been d1 vided differently
by different authors nlD6
The system of the orientalsl07 appears under
scrutiny to have been composed of two elements--the
baqya of abou t 23 heptamerldl ans or about 92 cen ts and
lOSthe comma of about 5 heptamerldlans or 20 cents
SnUV0Ul adds that
having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer
In the sixth section applications are made of the
system and the Echometer to the voice and the instruments
of music With C-SOL-UT as the fundamental sound Sauveur
presents in the third plate appended to tpe Memoire a
diagram on which are represented the keys of a keyboard
of organ or harpsichord the clef and traditional names
of the notes played on them as well as the syllables of
solmization when C is UT and when C is SOL After preshy
senting his own system of solmization and notes he preshy
sents a tab~e of ranges of the various voices in general
and of some of the well-known singers of his day followed
106Sauveur II Systeme General p 444 see vol II p 42 below
107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians
lOSSauveur Systeme General p 445 see vol II p 43 below
I IlO9Sauveur Systeme General p 447 see vol II p 45 below
85
by similar tables for both wind and stringed instruments
including the guitar of 10 frets
In an addition to the sixth section appended to
110the Memoire Sauveur sets forth his own system of
classification of the ranges of voices The compass of
a voice being defined as the series of sounds of the
diatonic system which it can traverse in sinping II
marked by the diatonic intervals III he proposes that the
compass be designated by two times the half of this
interval112 which can be found by adding 1 and dividing
by 2 and prefixing half to the number obtained The
first procedure is illustrated by V which is 5 ~ 1 or
two thirds the second by VI which is half 6 2 or a
half-fourth or a fourth above and third below
To this numerical designation are added syllables
of solmization which indicate the center of the range
of the voice
Sauveur deduces from this that there can be ttas
many parts among the voices as notes of the diatonic system
which can be the middles of all possible volces113
If the range of voices be assumed to rise to bis-PA (UT)
which 1s c and to descend to subbis-PA which is C-shy
110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below
lllSauveur Systeme General p 493 see vol II p 89 below
l12Ibid bull
II p
113Sauveur
90 below
ISysteme General p 494 see vol
86
four octaves in all--PA or a SOL UT or a will be the
middle of all possible voices and Sauveur contends that
as the compass of the voice nis supposed in the staves
of plainchant to be of a IXth or of two Vths and in the
staves of music to be an Xlth or two Vlthsnl14 and as
the ordinary compass of a voice 1s an Xlth or two Vlths
then by subtracting a sixth from bis-PA and adrllnp a
sixth to subbis-PA the range of the centers and hence
their number will be found to be subbis-LO(A) to Sem-GA
(e) a compass ofaXIXth or two Xths or finally
19 notes tll15 These 19 notes are the centers of the 19
possible voices which constitute Sauveurs systeml16 of
classification
1 sem-GA( MI)
2 bull sem-RA(RE) very high treble
3 sem-PA(octave of C SOL UT) high treble or first treble
4 DO( S1)
5 LO(LA) low treble or second treble
6 BO(G RE SOL)
7 SO(octave of F FA TIT)
8 G(MI) very high counter-tenor
9 RA(RE) counter-tenor
10 PA(C SOL UT) very high tenor
114Ibid 115Sauveur Systeme General p 495 see vol
II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol
II pp 91-92 below
87
11 sub-DO(SI) high tenor
12 sub-LO(LA) tenor
13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth
14 sub-SOC F JA UT) low tenor
15 sub-FA( NIl)
16 sub-HAC HE) lower tenor
17 sub-PA(sub-octave of C SOL TIT)
18 subbis-DO(SI) bass
19 subbis-LO(LA)
The M~moire of 1713 contains several suggestions
which supplement the tables of the ranges of voices and
instruments and the system of classification which appear
in the fifth and sixth chapters of the M6moire of 1701
By use of the fixed tone of which the number of vlbrashy
tions in a second is known the reader can determine
from the table of fixed sounds the number of vibrations
of a resonant body so that it will be possible to discover
how many vibrations the lowest tone of a bass voice and
the hif~hest tone of a treble voice make s 117 as well as
the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118
Sauveur gives in the notes of his system the tones of
various church bells which he had drawn from a Ivl0rno 1 re
u117Sauveur Rapnort p 464 see vol III
p 196 below
l18Sauveur Rapport1f p 464 see vol II pp 196-197 below
88
on the tones of bells given him by an Honorary Canon of
Paris Chastelain and he appends a system for determinshy
ing from the tones of the bells their weights 119
Sauveur had enumerated the possibility of notating
pitches exactly and learning the precise number of vibrashy
tions of a resonant body in his Memoire of 1701 in which
he gave as uses for the fixed sound the ascertainment of
the name and number of vibrations 1n a second of the sounds
of resonant bodies the determination from changes in
the sound of such a body of the changes which could have
taken place in its substance and the discovery of the
limits of hearing--the highest and the lowest sounds
which may yet be perceived by the ear 120
In the eleventh section of the Memoire of 1701
Sauveur suggested a procedure by which taking a particshy
ular sound of a system or instrument as fundamental the
consonance or dissonance of the other intervals to that
fundamental could be easily discerned by which the sound
offering the greatest number of consonances when selected
as fundamental could be determined and by which the
sounds which by adjustment could be rendered just might
be identified 121 This procedure requires the use of reshy
ciprocal (or mutual) intervals which Sauveur defines as
119Sauveur Rapport rr p 466 see vol II p 199 below
120Sauveur Systeme General p 492 see vol II p 88 below
121Sauveur Systeme General p 488 see vol II p 84 below
89
the interval of each sound of a system or instrument to
each of those which follow it with the compass of an
octave 122
Sauveur directs the ~eader to obtain the reciproshy
cal intervals by first marking one af~er another the
numbers of meridians and heptameridians of a system in
two octaves and the numbers of those of an instrument
throughout its whole compass rr123 These numbers marked
the reciprocal intervals are the remainders when the numshy
ber of meridians and heptameridians of each sound is subshy
tracted from that of every other sound
As an example Sauveur obtains the reciprocal
intervals of the sounds of the diatonic system of just
intonation imagining them to represent sounds available
on the keyboard of an ordinary harpsiohord
From the intervals of the sounds of the keyboard
expressed in meridians
I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39
VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82
he constructs a table124 (Table 13) in which when the
l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below
123Sauveur Systeme GeniJral p 485 see vol II p 81 below
I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below
90
sounds in the left-hand column are taken as fundamental
the sounds which bear to it the relationship marked by the
intervals I 2 II 3 and so forth may be read in the
line extending to the right of the name
TABLE 13
RECIPHOCAL INT~RVALS
Diatonic intervals
I 2 II 3 III 4 IV (5)
V 6 VI 7 VIr VIrI
Old names UT d RE b MI FA d SOL d U b 51 VT
New names PA pi RA go GA SO sa BO ba LO de DO FA
UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113
cJ)
r-i ro gtH OJ
+gt c middotrl
r-i co u 0 ~-I 0
-1 u (I)
H
Q)
J+l
d pi
HE RA
b go
MI GA
FA SO
d sa
0 4
0 4
0 (3)
a 4
0 (3)
0 4
(8) 11
7 11
7 (10)
7 11
7 (10)
7 11
(15)
14
14
14
14
( 15)
18
18
(17)
18
18
18
(22)
21
21
(22)
21
(22)
25
25
25
25
25
25
29
29
(28)
29
(28)
29
(33)
32
32
32
32
(33)
36
36
(35)
36
36
36
(40)
39
39
(40)
3()
(10 )
43
43
43
43
Il]
43
4-lt1 0
SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43
cJ) -t ro +gt C (1)
E~ ro T~ c J
u
d sa
LA LO
b de
5I DO
0 4
a 4
a (3)
0 4
(8) 11
7 11
7 (10)
7 11
(15)
14
14
(15)
18
18
18
18
(22)
(22)
21
(22)
(26)
25
25
25
29
29
(28)
29
(33)
32
32
32
36
36
(35)
36
(40)
3lt)
39
(40)
43
43
43
43
It will be seen that the original octave presented
b ~ bis that of C C D E F F G G A B B and C
since 3 meridians represent the chromatic semitone and 4
91
the diatonic one whichas Barbour notes was considered
by Sauveur to be the larger of the two 125 Table 14 gives
the values in cents of both the just intervals from
Sauveurs table (Table 13) and the altered intervals which
are included there between brackets as well as wherever
possible the names of the notes in the diatonic system
TABLE 14
VALUES FROM TABLE 13 IN CENTS
INTERVAL MERIDIANS CENTS NAME
(2) (3) 84 (C )
2 4 112 Db II 7 195 D
(II) (8 ) 223 (Ebb) (3 ) 3
(10) 11
279 3Q7
(DII) Eb
III 14 391 E (III)
(4 ) (15) (17 )
419 474
Fb (w)
4 18 502 F IV 21 586 FlI
(IV) V
(22) 25
614 698
(Gb) G
(V) (26) 725 (Abb) (6) (28) 781 (G)
6 29 809 Ab VI 32 893 A
(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb
VII 39 1088 B (VII) (40) 1116 (Cb )
The names were assigned in Table 14 on the assumpshy
tion that 3 meridians represent the chromatic semitone
125Barbour Tuning and Temperament p 128
92
and 4 the diatonic semi tone and with the rreatest simshy
plicity possible--8 meridians was thus taken as 3 meridians
or a chromatic semitone--lower than 11 meridians or Eb
With Table 14 Sauveurs remarks on the selection may be
scrutinized
If RA or LO is taken for the final--D or A--all
the tempered diatonic intervals are exact tr 126_-and will
be D Eb E F F G G A Bb B e ell and D for the
~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c
and A for the final on A Nhen another tone is taken as
the final however there are fewer exact diatonic notes
Bbbwith Ab for example the notes of the scale are Ab
cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with
values of 0 112 223 304 419 502 614 725 809 921
1004 1116 and 1200 in cents The fifth of 725 cents and
the major third of 419 howl like wolves
The number of altered notes for each final are given
in Table 15
TABLE 15
ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13
C v rtil D Eb E F Fil G Gtt A Bb B
2 5 0 5 2 3 4 1 6 1 4 3
An arrangement can be made to show the pattern of
finals which offer relatively pure series
126SauveurI Systeme General II p 488 see vol
II p 84 below
1
93
c GD A E B F C G
1 2 3 4 3 25middot 6
The number of altered notes is thus seen to increase as
the finals ascend by fifths and having reached a
maximum of six begins to decrease after G as the flats
which are substituted for sharps decrease in number the
finals meanwhile continuing their ascent by fifths
The method of reciplocal intervals would enable
a performer to select the most serviceable keys on an inshy
strument or in a system of tuning or temperament to alter
those notes of an instrument to make variolJs keys playable
and to make the necessary adjustments when two instruments
of different tunings are to be played simultaneously
The system of 43 the echometer the fixed sound
and the method of reciprocal intervals together with the
system of classification of vocal parts constitute a
comprehensive system for the measurement of musical tones
and their intervals
CHAPTER III
THE OVERTONE SERIES
In tho ninth section of the M6moire of 17011
Sauveur published discoveries he had made concerning
and terminology he had developed for use in discussing
what is now known as the overtone series and in the
tenth section of the same Mernoire2 he made an application
of the discoveries set forth in the preceding chapter
while in 1702 he published his second Memoire3 which was
devoted almost wholly to the application of the discovershy
ies of the previous year to the construction of organ
stops
The ninth section of the first M~moire entitled
The Harmonics begins with a definition of the term-shy
Ira hatmonic of the fundamental [is that which makes sevshy
eral vibrations while the fundamental makes only one rr4 -shy
which thus has the same extension as the ~erm overtone
strictly defined but unlike the term harmonic as it
lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below
3 Sauveur Application II
4Sauveur Systeme General9 p 474 see vol II p 70 below
94
95
is used today does not include the fundamental itself5
nor does the definition of the term provide for the disshy
tinction which is drawn today between harmonics and parshy
tials of which the second term has Ifin scientific studies
a wider significance since it also includes nonharmonic
overtones like those that occur in bells and in the comshy
plex sounds called noises6 In this latter distinction
the term harmonic is employed in the strict mathematical
sense in which it is also used to denote a progression in
which the denominators are in arithmetical progression
as f ~ ~ ~ and so forth
Having given a definition of the term Ifharmonic n
Sauveur provides a table in which are given all of the
harmonics included within five octaves of a fundamental
8UT or C and these are given in ratios to the vibrations
of the fundamental in intervals of octaves meridians
and heptameridians in di~tonic intervals from the first
sound of each octave in diatonic intervals to the fundashy
mental sOlJno in the new names of his proposed system of
solmization as well as in the old Guidonian names
5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on
6Ibid bull
7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If
8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below
96
The harmonics as they appear from the defn--~ tior
and in the table are no more than proportions ~n~ it is
Juuveurs program in the remainder of the ninth sect ton
to make them sensible to the hearing and even to the
slvht and to indicate their properties 9 Por tlLl El purshy
pose Sauveur directs the reader to divide the string of
(l lillHloctlord into equal pnrts into b for intlLnnco find
pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under
a sound that I call the fundamental of that strinplO
flhen a thin obstacle is placed on one of the points of
division of the string into equal parts the disturbshy
ance bull bull bull of the string is communicated to both sides of
the obstaclell and the string will render the 5th harshy
monic or if the fundamental is C E Sauveur explains
tnis effect as a result of the communication of the v1brashy
tions of the part which is of the length of the string
to the neighboring parts into which the remainder of the
ntring will (11 vi de i taelf each of which is elt11101 to tllO
r~rst he concludes from this that the string vibrating
in 5 parts produces the 5th ha~nonic and he calls
these partial and separate vibrations undulations tneir
immObile points Nodes and the midpoints of each vibrashy
tion where consequently the motion is greatest the
9 bull ISauveur Systeme General p 476 see vol II
p 73 below
I IlOSauveur Systeme General If pp 476-477 S6B
vol II p 73 below
11Sauveur nSysteme General n p 477 see vol p 73 below
97
bulges12 terms which Fontenelle suggests were drawn
from Astronomy and principally from the movement of the
moon 1113
Sauveur proceeds to show that if the thin obstacle
is placed at the second instead of the first rlivlsion
hy fifths the string will produce the fifth harmonic
for tho string will be divided into two unequal pn rts
AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy
municate its vibrations to CG leaving GB which vibrashy
ting twice as fast as either AC or CG will communicate
its vibrations from FG to FE through DA (Fig 4)
The undulations are audible and visible as well
Sauveur suggests that small black and white paper riders
be attached to the nodes and bulges respectively in orcler
tnat the movements of the various parts of the string mirht
be observed by the eye This experiment as Sauveur notes
nad been performed as early as 1673 by John iJallls who
later published the results in the first paper on muslshy
cal acoustics to appear in the transactions of the society
( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy
bJing of Consonant Strings a New Musical Discovery 14
- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r
-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de
c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation
-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II
Ii Groves Dictionary of Music and Mus c1 rtn3
ej s v S)und by LI S Lloyd
98
B
n
E
A c B
lig 4 Communication of vibrations
Wallis httd tuned two strings an octave apart and bowing
ttJe hipher found that the same note was sounderl hy the
oLhor strinr which was found to be vihratyening in two
Lalves for a paper rider at its mid-point was motionless16
lie then tuned the higher string to the twefth of the lower
and lIagain found the other one sounding thjs hi~her note
but now vibrating in thirds of its whole lemiddot1gth wi th Cwo
places at which a paper rider was motionless l6 Accordng
to iontenelle Sauveur made a report to t
the existence of harmonics produced in a string vibrating
in small parts and
15Ibid bull
16Ibid
99
someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l
Sauveur drew from his experiments a series of conshy
clusions a summary of which constitutes the second half
of the ninth section of his first M6mnire He proposed
first that a harmonic formed by the placement of a thin
obstacle on a potential nodal point will continue to
sound when the thin obstacle is re-r1oved Second he noted
that if a string is already vibratin~ in five parts and
a thin obstacle on the bulge of an undulation dividing
it for instance into 3 it will itself form a 3rd harshy
monic of the first harmonic --the 15th harmon5_c of the
fundamental nIB This conclusion seems natnral in view
of the discovery of the communication of vibrations from
one small aliquot part of the string to others His
third observation--that a hlrmonic can he indllced in a
string either by setting another string nearby at the
unison of one of its harmonics19 or he conjectured by
setting the nearby string for such a sound that they can
lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public
p
18 Sauveur 77 below
ItS ysteme G Ifeneral p 480 see vol II
19Ibid bull
100
divide by their undulations into harmonics Wilich will be
the greatest common measure of the fundamentals of the
two strings 20__was in part anticipated by tTohn Vallis
Wallis describing several experiments in which harmonics
were oxcttod to sympathetIc vibration noted that ~tt hnd
lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2
Sauveur assumed fourth that the harmonics of a
string three feet long could be heard only to the fifth
octave (which was also the limit of the harmonics he preshy
sented in the table of harmonics) a1 though it seems that
he made this assumption only to make cleare~ his ensuing
discussion of the positions of the nodal points along the
string since he suggests tha t harmonic s beyond ti1e 128th
are audible
rrhe presence of harmonics up to the ~S2nd or the
fIfth octavo having been assumed Sauveur proceeds to
his fifth conclusion which like the sixth and seventh
is the result of geometrical analysis rather than of
observation that
every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from
90 f-J Ibid As when one is at the fourth of the other
and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union
2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839
101
the nearest node of other ha2~onics by at least a 32nd part of its undulation
This is easiJy understood since the successive
thirty-seconds of the string as well as the successive
thirds of the string may be expressed as fractions with
96 as the denominator Sauveur concludes from thIs that
the lower numbered harmonics will have considerah1e lenrth
11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or
memhe~s will have little--a conclusion which seems reasonshy
able in view of the fourth deduction that the node of a
harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so
t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)
and so forth give the minimum lengths by which a neighborshy
ing node must be removed from the nodes of the fundamental
and consecutive harmonics The conclusion that the nodes
of harmonics bearing higher numbers are packed more
tightly may be illustrated by the division of the string
1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s
lying helow the points of division represent sixtieths of
the length of the string and the numbers below them their
differences (in sixtieths) while the fractions lying
above the line represent the lengths of string to those
( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below
23Sauveur Systeme General p 482 see vol II p 78 below
T24Sauveur Systeme General p 481 see vol LJ
pp 77-78 below
102
points of division It will be seen that the greatest
differences appear adjacent to fractions expressing
divisions of the diagrammatic string into the greatest
number of parts
3o
3110 l~ IS 30 10
10
Fig 5 Nodes of the fundamental and the first five harmonics
11rom this ~eometrical analysis Sauvcllr con JeeturO1
that if the node of a small harmonic is a neighbor of two
nodes of greater sounds the smaller one wi]l be effaced
25by them by which he perhaps hoped to explain weakness
of the hipher harmonics in comparison with lower ones
The conclusions however which were to be of
inunediate practical application were those which concerned
the existence and nature of the harmonics ~roduced by
musical instruments Sauveur observes tha if you slip
the thin bar all along [a plucked] string you will hear
a chirping of harmonics of which the order will appear
confused but can nevertheless be determined by the princishy
ples we have established26 and makes application of
25 IISauveur Systeme General p 482 see vol II p 79 below
26Ibid bull
10
103
the established principles illustrated to the explanation
of the tones of the marine trurnpet and of instruments
the sounds of which las for example the hunting horn
and the large wind instruments] go by leaps n27 His obshy
servation that earlier explanations of the leaping tones
of these instruments had been very imperfect because the
principle of harmonics had been previously unknown appears
to 1)6 somewhat m1sleading in the light of the discoverlos
published by Francis Roberts in 1692 28
Roberts had found the first sixteen notes of the
trumpet to be C c g c e g bb (over which he
d ilmarked an f to show that it needed sharpening c e
f (over which he marked I to show that the corresponding
b l note needed flattening) gtl a (with an f) b (with an
f) and c H and from a subse()uent examination of the notes
of the marine trumpet he found that the lengths necessary
to produce the notes of the trumpet--even the 7th 11th
III13th and 14th which were out of tune were 2 3 4 and
so forth of the entire string He continued explaining
the 1 eaps
it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a
27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below
28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~
104
corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29
In this explanation he seems to have anticipated
hlUVOll r wno wrot e thu t
the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D
In 1702 Sauveur turned his attention to the apshy
plication of harmonics to the constMlction of organ stops
as the result of a conversatlon with Deslandes which made
him notice that harmonics serve as the basis for the comshy
position of organ stops and for the mixtures that organshy
ists make with these stops which will be explained in a I
few words u3l Of the Memoire of 1702 in which these
findings are reported the first part is devoted to a
description of the organ--its keyboards pipes mechanisms
and the characteristics of its various stops To this
is appended a table of organ stops32 in which are
arrayed the octaves thirds and fifths of each of five
octaves together with the harmoniC which the first pipe
of the stop renders and the last as well as the names
29 Ibid bull
30Sauveur Systeme General p 483 see vol II p 79 below
31 Sauveur uApplicationn p 425 see vol II p 98 below
32Sauveur Application p 450 see vol II p 126 below
105
of the various stops A second table33 includes the
harmonics of all the keys of the organ for all the simple
and compound stops1I34
rrhe first four columns of this second table five
the diatonic intervals of each stop to the fundamental
or the sound of the pipe of 32 feet the same intervaJs
by octaves the corresponding lengths of open pipes and
the number of the harmonic uroduced In the remnincier
of the table the lines represent the sounds of the keys
of the stop Sauveur asks the reader to note that
the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic
Sauveur notes as well til at the sounds of all the
octaves in the lines are harmonic--or in double proportion
rrhe first observation can ea 1y he verified by
selecting a column and dividing the lar~er numbers by
the smallest The results for the column of sub-RE or
d are given in Table 16 (Table 16)
For a column like that of PI(C) in whiCh such
division produces fractions the first note must be conshy
sidered as itself a harmonic and the fundamental found
the series will appear to be harmonic 36
33Sauveur Application p 450 see vol II p 127 below
34Sauveur Anplication If p 434 see vol II p 107 below
35Sauveur IIApplication p 436 see vol II p 109 below
36The method by which the fundamental is found in
106
TABLE 16
SOUNDS OR HARMONICSsom~DS 9
9 1 18 2 27 3 36 4 45 5 54 6 72 n
] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96
Principally from these observotions he d~aws the
conclusion that the compo tion of organ stops is harronic
tha t the mixture of organ stops shollld be harmonic and
tflat if deviations are made flit is a spec1es of ctlssonance
this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave
107
in the harmonics which has some relation with the disshy
sonances employed in music u37
Sauveur noted that the organ in its mixture of
stops only imitated
the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38
At the end of the Memoire of 1702 Sauveur attempted
to establish the limits of all sounds as well as of those
which are clearly perceptible observing that the compass
of the notes available on the organ from that of a pipe
of 32 feet to that of a nipe of 4t lines is 10 octaves
estimated that to that compass about two more octaves could
be added increasing the absolute range of sounds to
twelve octaves Of these he remarks that organ builders
distinguish most easily those from the 8th harmonic to the
l28th Sauveurs Table of Fixed Sounds subioined to his
M~moire of 171339 made it clear that the twelve octaves
to which he had referred eleven years earlier wore those
from 8 vibrations in a second to 32768 vibrations in a
second
Whether or not Sauveur discovered independently
all of the various phenomena which his theory comprehends
37Sauveur Application p 450 see vol II p 124 below
38sauveur Application pp 450-451 see vol II p 124 below
39Sauveur Rapnort p 468 see vol II p 203 below
108
he seems to have made an important contribution to the
development of the theory of overtones of which he is
usually named as the originator 40
Descartes notes in the Comeendiurn Musicae that we
never hear a sound without hearing also its octave4l and
Sauveur made a similar observation at the beginning of
his M~moire of 1701
While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42
It is true as well that Wallis and Roberts had antici shy
pated the discovery of Sauveur that strings will vibrate
in aliquot parts as has been seen But Sauveur brought
all these scattered observations together in a coherent
theory in which it was proposed that the harmonlc s are
sounded by strings the numbers of vibrations of which
in a given time are integral multiples of the numbers of
vibrations of the fundamental in that same time Sauveur
having devised a means of determining absolutely rather
40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii
4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20
42Sauveur Systeme General p 405 see vol II p 3 below
109
than relati vely the number of vibra tions eXfcuted by a
string in a second this definition of harmonics with
reference to numbers of vibrations could be applied
directly to the explanation of the phenomena ohserved in
the vibration of strings His table of harmonics in
which he set Ollt all the harmonics within the ranpe of
fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB
brought system to the diversity of phenomena previolls1y
recognized and his work unlike that of Wallis and
Roberts in which it was merely observed that a string
the vibrations of which were divided into equal parts proshy
ducod the same sounds as shorter strIngs vlbrutlnr~ us
wholes suggested that a string was capable not only of
produc ing the harmonics of a fundamental indi vidlJally but
that it could produce these vibrations simultaneously as
well Sauveur thus claims the distinction of having
noted the important fact that a vibrating string could
produce the sounds corresponding to several of its harshy
monics at the same time43
Besides the discoveries observations and the
order which he brought to them Sauveur also made appli shy
ca tions of his theories in the explanation of the lnrmonic
structure of the notes rendered by the marine trumpet
various wind instruments and the organ--explanations
which were the richer for the improvements Sauveur made
through the formulation of his theory with reference to
43Lindsay Introduction to Rayleigh rpheory of Sound p xv
110
numbers of vibrations rather than to lengths of strings
and proportions
Sauveur aJso contributed a number of terms to the
s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s
one which was first used by Sauveur to describe phenomena
observable in the vibration of resonant bodIes while he
was also responsible for the use of the term fundamental ll
fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy
parisons as well as for the term Itnodes for those
pOints at which no motion occurred--terms which like
the concepts they represent are still in use in the
discussion of the phenomena of sound
CHAPTER IV
THE HEIRS OF SAUVEUR
In his report on Sauveurs method of determining
a fixed pitch Fontene11e speculated that the number of
beats present in an interval might be directly related
to its degree of consonance or dissonance and expected
that were this hypothesis to prove true it would tr1ay
bare the true source of the Rules of Composition unknown
until the present to Philosophy which relies almost enshy
tirely on the judgment of the earn1 In the years that
followed Sauveur made discoveries concerning the vibrashy
tion of strings and the overtone series--the expression
for example of the ratios of sounds as integral multip1es-shy
which Fontenelle estimated made the representation of
musical intervals
not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2
lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille
2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette
III
112
Sauveur had been the geometer in fashion when he was not
yet twenty-three years old and had numbered among his
accomplis~~ents tables for the flow of jets of water the
maps of the shores of France and treatises on the relationshy
ships of the weights of ~nrious c0untries3 besides his
development of the sCience of acoustics a discipline
which he has been credited with both naming and founding
It might have surprised Fontenelle had he been ahle to
foresee that several centuries later none of SallVeUT S
works wrnlld he available in translation to students of the
science of sound and that his name would be so unfamiliar
to those students that not only does Groves Dictionary
of Muslc and Musicians include no article devoted exclusshy
ively to his achievements but also that the same encyshy
clopedia offers an article on sound4 in which a brief
history of the science of acoustics is presented without
even a mention of the name of one of its most influential
founders
rrhe later heirs of Sauvenr then in large part
enjoy the bequest without acknowledging or perhaps even
nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)
3bontenelle Eloge II p 104
4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd
113
recognizing the benefactor In the eighteenth century
however there were both acousticians and musical theorshy
ists who consciously made use of his methods in developing
the theories of both the science of sound in general and
music in particular
Sauveurs Chronometer divided into twelfth and
further into sixtieth parts of a second was a refinement
of the Chronometer of Louli~ divided more simply into
universal inches The refinements of Sauveur weTe incorshy
porated into the Pendulum of Michel LAffilard who folshy
lowed him closely in this matter in his book Principes
tr~s-faciles pour bien apprendre la musique
A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time
The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System
5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55
Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque
114
LAffilards description or Sauveur1s first
Memoire of 1701 as new principles for leDrning to sing
from the ordinary notes hy means of his General Systemu6
suggests that LAffilard did not t1o-rollphly understand one
of the authors upon whose works he hasAd his P-rincinlea shy
rrhe Metrometer proposed by Loui 3-Leon Pai ot
Chevalier comte DOns-en-Bray7 intended by its inventor
improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed
the 01 vislon into t--tirds constructed hy ([luvenr
Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose
You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines
In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions
~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins
La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l
r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J
0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al
7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U
M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195
8 Hardin~ Ori~ins p 12
115
of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums
llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9
Pajot then specifies a rule by the use of which
the lengths of a pendulum can be calculated for a given
number of thirds and subJoins a table lO in which the
lengths of a pendulum are given for vibrations of durations
of 1 to 180 half-thirds as well as a table of durations
of the measures of various compositions by I~lly Colasse
Campra des Touches and NIato
9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons
On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi
POlrr ~
connoi tre
les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3
Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules
De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna
lOIbid pp 193-195
116
Erich Schwandt who has discussed the Chronometer
of Sauveur and the Pendulum of LAffilard in a monograph
on the tempos of various French court dances has argued
that while LAffilard employs for the measurement of his
pendulum the scale devised by Sauveur he nonetheless
mistakenly applied the periods of his pendulum to a rule
divided for half periods ll According to Schwandt then
the vibration of a pendulum is considered by LAffilard
to comprise a period--both excursion and return Pajot
however obviously did not consider the vibration to be
equal to the period for in his description of the
M~trom~tr~ cited above he specified that one vibration
of a pendulum 3 feet 8t lines long lasts one second and
it can easily he determined that I second gives the half-
period of a pendulum of this length It is difficult to
ascertain whether Sauveur meant by a vibration a period
or a half-period In his Memoire of 1713 Sauveur disshy
cussing vibrating strings admitted that discoveries he
had made compelled him to talee ua passage and a return for
a vibration of sound and if this implies that he had
previously taken both excursions and returns as vibrashy
tions it can be conjectured further that he considered
the vibration of a pendulum to consist analogously of
only an excursion or a return So while the evidence
does seem to suggest that Sauveur understood a ~ibration
to be a half-period and while experiment does show that
llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400
117
Pajot understood a vibration to be a half-period it may
still be true as Schwannt su~pests--it is beyond the purshy
view of this study to enter into an examination of his
argument--that LIAffilnrd construed the term vibration
as referring to a period and misapplied the perions of
his pendulum to the half-periods of Sauveurs Chronometer
thus giving rise to mlsunderstandinr-s as a consequence of
which all modern translations of LAffilards tempo
indications are exactly twice too fast12
In the procession of devices of musical chronometry
Sauveurs Chronometer apnears behind that of Loulie over
which it represents a great imnrovement in accuracy rhe
more sophisticated instrument of Paiot added little In
the way of mathematical refinement and its superiority
lay simply in its greater mechanical complexity and thus
while Paiots improvement represented an advance in execushy
tion Sauve11r s improvement represented an ac1vance in conshy
cept The contribution of LAffilard if he is to he
considered as having made one lies chiefly in the ~rAnter
flexibility which his system of parentheses lent to the
indication of tempo by means of numbers
Sauveurs contribution to the preci se measurement
of musical time was thus significant and if the inst~lment
he proposed is no lon~er in use it nonetheless won the
12Ibid p 395
118
respect of those who coming later incorporateci itA
scale into their own devic e s bull
Despite Sauveurs attempts to estabJish the AystArT
of 43 m~ridians there is no record of its ~eneral nCConshy
tance even for a short time among musicians As an
nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy
stern of Just Intonation it was perhans mo-re sucCO~t1fl]l
than wore the systems of 55 31 19 or 12--tho altnrnntlvo8
proposed by Sauveur before the selection of the system of
43 was rnade--but the suggestion is nowhere made the t those
systems were put forward with the intention of dupl1catinp
that of just intonation The cycle of 31 as has been
noted was observed by Huygens who calculated the system
logarithmically to differ only imperceptibly from that
J 13of 4-comma temperament and thus would have been superior
to the system of 43 meridians had the i-comma temperament
been selected as a standard Sauveur proposed the system
of 43 meridians with the intention that it should be useful
in showing clearly the number of small parts--heptamprldians
13Barbour Tuning and Temperament p 118 The
vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J
A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation
119
or decameridians--in the elements as well as the larrer
units of all conceivable systems of intonation and devoted
the fifth section of his M~moire of 1701 to the illustration
of its udaptnbil ity for this purpose [he nystom willeh
approximated mOst closely the just system--the one which
[rave the intervals in their simplest form--thus seemed
more appropriate to Sauveur as an instrument of comparison
which was to be useful in scientific investigations as well
as in purely practical employments and the system which
meeting Sauveurs other requirements--that the comma for
example should bear to the semitone a relationship the
li~its of which we~e rigidly fixed--did in fact
approximate the just system most closely was recommended
as well by the relationship borne by the number of its
parts (43 or 301 or 3010) to the logarithm of 2 which
simplified its application in the scientific measurement
of intervals It will be remembered that the cycle of 301
as well as that of 3010 were included by Ellis amonp the
paper cycles14 _-presumnbly those which not well suited
to tuning were nevertheless usefUl in measurement and
calculation Sauveur was the first to snppest the llse of
small logarithmic parts of any size for these tasks and
was t~le father of the paper cycles based on 3010) or the
15logaritmn of 2 in particular although the divisIon of
14 lis Appendix XX to Helmholtz Sensations of Tone p 43
l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in
120
the octave into 301 (or for simplicity 300) logarithmic
units was later reintroduced by Felix Sava~t as a system
of intervallic measurement 16 The unmodified lo~a~lthmic
systems have been in large part superseded by the syntem
of 1200 cents proposed and developed by Alexande~ EllisI7
which has the advantage of making clear at a glance the
relationship of the number of units of an interval to the
number of semi tones of equal temperament it contains--as
for example 1125 cents corresponds to lIt equal semi-
tones and this advantage is decisive since the system
of equal temperament is in common use
From observations found throughout his published
~ I bulllemOlres it may easily be inferred that Sauveur did not
put forth his system of 43 meridians solely as a scale of
musical measurement In the Ivrt3moi 1e of 1711 for exampl e
he noted that
setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18
It seem Clear then that Sauveur believed that his system
his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur
16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II
l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451
18Sauveur uTable GeneraletI p 416 see vol II p 165 below
121
so accurately reflected contemporary modes of tuning tLat
it could be substituted for them and that such substitushy
tion would confer great advantages
It may be noted in the cou~se of evalllatlnp this
cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy
luted by llily~ens is intimately re1ate~ to a meantone
temperament 19 Table 17 gives in its first column the
names of the intervals of Sauveurs system the vn] nos of shy
these intervals ate given in cents in the second column
the third column contains the differences between the
systems of Sauveur and the ~-comma temperament obtained
by subtracting the fourth column from the second the
fourth column gives the values in cents of the intervals
of the ~-comma meantone temperament as they are given)
by Barbour20 and the fifth column contains the names of
1the intervals of the 5-comma meantone temperament the exshy
ponents denoting the fractions of a comma by which the
given intervals deviate from Pythagorean tuning
19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere
12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts
t-comrriU parts ~-comma 91 parts ~-comma 13d ports
L-comrr~a 247 parts r8--comma 499 parts n Barbour
Tuni n9 and remnerament p 126
20Ibid p 36
9
122
TABLE 17
CYCLE OF 43 -COMMA
NAMES CENTS DIFFERENCE CENTS NAMES
1)Vll lOuU 0 lOUU l
b~57 1005 0 1005 B _JloA ltjVI 893 0 893
V( ) 781 0 781 G-
_l V 698 0 698 G 5
F-~IV 586 0 586
F+~4 502 0 502
E-~III 391 +1 390
Eb~l0 53 307 307
1
II 195 0 195 D-~
C-~s 84 +1 83
It will be noticed that the differences between
the system of Sauveur and the ~-comma meantone temperament
amounting to only one cent in the case of only two intershy
vals are even smaller than those between the cycle of 31
and the -comma meantone temperament noted above
Table 18 gives in its five columns the names
of the intervals of Sauveurs system the values of his
intervals in cents the values of the corresponding just
intervals in cen ts the values of the correspondi ng intershy
vals 01 the system of ~-comma meantone temperament the
differences obtained by subtracting the third column fron
123
the second and finally the differences obtained by subshy
tracting the fourth column from the second
TABLE 18
1 2 3 4
SAUVEUHS JUST l-GOriI~ 5
INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS
VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0
VI 893 884 893 + 9 0 vUI) 781 781 0 V
IV 698 586
702 590
698 586
--
4 4
0 0
4 502 498 502 + 4 0 III 391 386 390 + 5 tl
3 307 316 307 - 9 0 II 195 182 195 t13 0
s 84 83 tl
It can be seen that the differences between Sauveurs
system and the just system are far ~reater than the differshy
1 ences between his system and the 5-comma mAantone temperashy
ment This wide discrepancy together with fact that when
in propounding his method of reCiprocal intervals in the
Memoire of 170121 he took C of 84 cents rather than the
Db of 112 cents of the just system and Gil (which he
labeled 6 or Ab but which is nevertheless the chromatic
semitone above G) of 781 cents rather than the Ab of 814
cents of just intonation sugpests that if Sauve~r waD both
utterly frank and scrupulously accurate when he stat that
the harpsichord tunings fell precisely on t1e meridional
21SalJVAur Systeme General pp 484-488 see vol II p 82 below
124
divisions of his monochord set for the system of 43 then
those harpsichords with which he performed his experiments
1were tuned in 5-comma meantone temperament This conclusion
would not be inconsonant with the conclusion of Barbour
that the suites of Frangois Couperin a contemnorary of
SU1JVfHlr were performed on an instrument set wt th a m0nnshy
22tone temperamnnt which could be vUYied from piece to pieco
Sauveur proposed his system then as one by which
musical instruments particularly the nroblematic keyboard
instruments could be tuned and it has been seen that his
intervals would have matched almost perfectly those of the
1 15-comma meantone temperament so that if the 5-comma system
of tuning was indeed popular among musicians of the ti~e
then his proposal was not at all unreasonable
It may have been this correspondence of the system
of 43 to one in popular use which along with its other
merits--the simplicity of its calculations based on 301
for example or the fact that within the limitations
Souveur imposed it approximated most closely to iust
intonation--which led Sauveur to accept it and not to con-
tinue his search for a cycle like that of 53 commas
which while not satisfying all of his re(1uirements for
the relatIonship between the slzes of the comma and the
minor semitone nevertheless expressed the just scale
more closely
22J3arbour Tuning and Temperament p 193
125
The sys t em of 43 as it is given by Sa11vcll is
not of course readily adaptihle as is thn system of
equal semi tones to the performance of h1 pJIJy chrorLi t ic
musIc or remote moduJntions wlthollt the conjtYneLlon or
an elahorate keyboard which wOlJld make avai] a hI e nIl of
1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v
menntone temperament which has been shown to be prHcshy
43 meridians was slJbject to the same restrictions and
the oerformer found it necessary to make adjustments in
the tunlnp of his instrument when he vlshed to strike
in the piece he was about to perform a note which was
not avnilahle on his keyboard24 and thus Sallveurs system
was not less flexible encounterert on a keyboard than
the meantone temperaments or just intonation
An attempt to illustrate the chromatic ran~e of
the system of Sauveur when all ot the 43 meridians are
onployed appears in rrable 19 The prlnclples app] led in
()3( EXperimental keyhoard comprisinp vltldn (~eh
octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx
XX to HelMholtz Sensations of Tone pp 466-483
24It has been m~ntionerl for exa71 e tha t JJ
Jt boar~ San vellr describ es had the notes C C-r D EO 1~
li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~
are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result
126
its construction are two the fifth of 7s + 4c where
s bull 3 and c = 1 is equal to 25 meridians and the accishy
dentals bearing sharps are obtained by an upward projection
by fifths from C while the accidentals bearing flats are
obtained by a downward proiection from C The first and
rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In
f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~
bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh
C at its head and ends with F6b at its hase (the exponents
1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy
tion of multiple sharps and flats) The second anrl fourth
columns show the number of fifths in the ~roioct1()n for tho
corresponding name as well as the number of octaves which
must be subtracted in the second column or added in the
fourth to reduce the intervals to the compass of one octave
Jlhe numbers in the tbi1d column M Vi ve the numbers of
meridians of the notes corresponding to the names given
in both the first and fifth columns 25 (Table 19)
It will thus be SAen that A is the equivalent of
D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians
rphrOl1fhout t1 is series of proi ections it will be noted
25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)
The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r
Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain
127
tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)
1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (
18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )
1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)
- 0) -860 40 -175 (7) +215 (~))
G7S (3~) -8()O 15 (E) +1J (~
4 (31) -1317 33 ( I) t ) ~) ) (()
(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)
(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)
(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)
(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)
525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )
~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb
(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())
Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )
JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17
(10) -215 (5) 35 -825 (33) + (3() I )
( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j
200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I
(6) -129 (3) 21 -925 ( )7) + r1 tJ
- )
( ~~ (~) (6 (2) 3()
+( t( ) -
()_GU 14 -(y(~ ()) )
7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )
G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075
128
that the relationships between the intervals of one type
of accidental remain intact thus the numher of meridians
separating F(21) and F(24) are three as might have been
expected since 3 meridians are allotted to the minor
sernitone rIhe consistency extends to lonFer series of
accidcntals as well F(21) F(24) F2(28) F3(~O)
p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt
li chrornitic scmltono ie formed hy addlnp ~gt morldHn1
The table illustrates the general principle that
the number of fIfth projections possihle befoTe closure
in a cyclical system like that of Sauveur is eQ11 al to
the number of steps in the system and that one of two
sets of fifth projections the sharps will he equivalent
to the other the flats In the system of equal temperashy
ment the projections do not extend the range of accidenshy
tals beyond one sharp or two flats befor~ closure--B is
equal to C and Dbb is egual to C
It wOl11d have been however futile to extend the
ranrre of the flats and sharps in Sauveurs system in this
way for it seems likely that al though he wi sbed to
devise a cycle which would be of use in performance while
also providinp a fairly accurate reflection of the just
scale fo~ purposes of measurement he was satisfied that
the system was adequate for performance on account of the
IYrJationship it bore to the 5-comma temperament Sauveur
was perhaps not aware of the difficulties involved in
more or less remote modulations--the keyhoard he presents
129
in the third plate subjoined to the M~moire of 170126 is
provided with the names of lfthe chromatic system of
musicians--names of the notes in B natural with their
sharps and flats tl2--and perhaps not even aware thnt the
range of sIlarps and flats of his keyboard was not ucleqUtlt)
to perform the music of for example Couperin of whose
suites for c1avecin only 6 have no more than 12 different
scale c1egrees 1I28 Throughout his fJlemoires howeve-r
Sauveur makes very few references to music as it is pershy
formed and virtually none to its harmonic or melodic
characteristics and so it is not surprising that he makes
no comment on the appropriateness of any of the systems
of tuning or temperament that come under his scrutiny to
the performance of any particular type of music whatsoever
The convenience of the method he nrovirled for findshy
inr tho number of heptamorldians of an interval by direct
computation without tbe use of tables of logarithms is
just one of many indications throughout the M~moires that
Sauveur did design his system for use by musicians as well
as by methemRticians Ellis who as has been noted exshy
panded the method of bimodular computat ion of logari thms 29
credited to Sauveurs Memoire of 1701 the first instance
I26Sauveur tlSysteme General p 498 see vol II p 97 below
~ I27Sauvel1r ffSyst~me General rt p 450 see vol
II p 47 b ow
28Barbol1r Tuning and Temperament p 193
29Ellls Improved Method
130
of its use Nonetheless Ellis who may be considerect a
sort of heir of an unpublicized part of Sauveus lep-acy
did not read the will carefully he reports tha t Sallv0ur
Ugives a rule for findln~ the number of hoptamerides in
any interval under 67 = 267 cents ~SO while it is clear
from tho cnlculntions performed earlier in thIs stllOY
which determined the limit implied by Sauveurs directions
that intervals under 57 or 583 cents may be found by his
bimodular method and Ellis need not have done mo~e than
read Sauveurs first example in which the number of
heptameridians of the fourth with a ratio of 43 and a
31value of 498 cents is calculated as 125 heptameridians
to discover that he had erred in fixing the limits of the
32efficacy of Sauveur1s method at 67 or 267 cents
If Sauveur had among his followers none who were
willing to champion as ho hud tho system of 4~gt mcridians-shy
although as has been seen that of 301 heptameridians
was reintroduced by Savart as a scale of musical
30Ellis Appendix XX to Helmholtz Sensations of Tone p 437
31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below
32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa
131
measurement--there were nonetheless those who followed
his theory of the correct formation of cycles 33
The investigations of multiple division of the
octave undertaken by Snuveur were accordin to Barbour ~)4
the inspiration for a similar study in which Homieu proshy
posed Uto perfect the theory and practlce of temporunent
on which the systems of music and the division of instrushy
ments with keys depends35 and the plan of which is
strikingly similar to that followed by Sauveur in his
of 1707 announcin~ thatMemolre Romieu
After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36
Aft0r sumwarizing the method employed by Sauveur--the
division of the tone into two minor semitones and a
comma which Ro~ieu calls a quarter tone37 and the
33Barbou r Ttlning and Temperame nt p 128
~j4Blrhollr ttHlstorytI p 21lB
~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I
SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of
la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR
36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons
3Ibld p 488 bull quart de ton
132
determination of the ratio between them--Romieu obiects
that the necessity is not demonstrated of makinr an
equal distribution to correct the sCale of the just
nY1 tnm n~)8
11e prosents nevortheless a formuJt1 for tile cllvlshy
sions of the octave permissible within the restrictions
set by Sauveur lIit is always eoual to the number 6
multiplied by the number of parts dividing the tone plus Lg
unitytl O which gives the series 1 7 13 bull bull bull incJuding
19 31 43 and 55 which were the numbers of parts of
systems examined by Sauveur The correctness of Romieus
formula is easy to demonstrate the octave is expressed
by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s
or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the
quotient 6 gives the number of tones in the octave while
c remalns Thus if c is an aliquot paTt of the octave
then 6 mult-tplied by the numher of commas in the tone
plus 1 will pive the numher of parts in the octave
Romieu dec1ines to follow Sauveur however and
examines instead a series of meantone tempernments in which
the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r
38 Tb i d bull It bull
bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste
39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite
133
gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy
ment to which howeve~ it is not so close as to the
1 414-conma temperament Romieu expresses a preference for
1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u
recommends the ~-comma temperament which is e~uiv31ent
to division into 55 parts--a division which Sauveur had
10 iec ted 42
40Barbour Tuning and Temperament n 126
41mh1 e values in cents of the system of Huygens
of 1 4-comma temperament as given by Barbour and of
2 gcomma as also given by Barbour are shown below
rJd~~S CHjl
D Eb E F F G Gft A Bb B
Huygens 77 194 310 387 503 581 697 774 890 1006 1084
l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4
~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9
The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and
the anaJogous total for the system of Huygens and that
of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37
42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel
134
The system of 43 was discussed by Robert Smlth43
according to Barbour44 and Sauveurs method of dividing
the octave tone was included in Bosanquets more compreshy
hensive discussion which took account of positive systems-shy
those that is which form their thirds by the downward
projection of 8 fifths--and classified the systems accord-
Ing to tile order of difference between the minor and
major semi tones
In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system
According to this definition Sauveurs cycles of 31 43
and 55 parts are primary nepatlve systems that of
Benfling with its s of 3 its S of 5 and its c of 2
is a secondary ne~ative system while for example the
system of 53 with as perhaps was heyond vlhat Sauveur
would have considered rational an s of 5 an S of 4 and
a c of _146 is a primary negative system It may be
noted that j[lUVe1Jr did consider the system of 53 as well
as the system of 17 which Bosanquet gives as examples
of primary positive systems but only in the M~moire of
1711 in which c is no longer represented as an element
43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)
44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees
45BosanquetTemperamentrr p 10
46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53
135
as it was in the Memoire of 1707 but is merely piven the
47algebraic definition 2s - t Sauveur gave as his reason
for including them that they ha ve th eir partisans 11 48
he did not however as has already been seen form the
intervals of these systems in the way which has come to
be customary but rather proiected four fifths upward
in fact as Pytharorean thirds It may also he noted that
Romieus formula 6P - 1 where P represents the number of
parts into which the tone is divided is not applicable
to systems other than the primary negative for it is only
in these that c = 1 it can however be easily adapted
6P + c where P represents the number of parts in a tone
and 0 the value of the comma gives the number of parts
in the octave 49
It has been seen that the system of 43 as it was
applied to the keyboard by Sauveur rendered some remote
modulat~ons difficl1l t and some impossible His discussions
of the system of equal temperament throughout the Memoires
show him to be as Barbour has noted a reactionary50
47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below
48Sauvellr Table Geneale1r 416middot vol IIl p see
p 159 below
49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive
50Barbour History n p 247
12
136
In this cycle S = sand c = 0 and it thus in a sense
falls outside BosanqlJet s system of classification In
the Memoire of 1707 SauveuT recognized that the cycle of
has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51
He objected however that the differences between the
intervals of equal temperament and those of the diatonic
system were t00 g-rea t and tha t the capabl e instr1Jmentshy
alists have rejected it52 In the Memolre of 1711 he
reiterated that besides the fact that the system of 12
lay outside the limits he had prescribed--that the ratio
of the minor semi tone to the comma fall between 1~ and
4~ to l--it was defective because the differences of its
intervals were much too unequal some being greater than
a half-corrJ11a bull 53 Sauveurs judgment that the system of
equal temperament has its use among the least capable
instrumentalists seems harsh in view of the fact that
Bach only a generation younger than Sauveur included
in his works for organ ua host of examples of triads in
remote keys that would have been dreadfully dissonant in
any sort of tuning except equal temperament54
51Sauveur Methode Generale p 272 see vo] II p 140 below
52 Ibid bull
53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below
54Barbour Tuning and Temperament p 196
137
If Sauveur was not the first to discuss the phenshy
55 omenon of beats he was the first to make use of them
in determining the number of vibrations of a resonant body
in a second The methon which for long was recorrni7ed us
6the surest method of nssessinp vibratory freqlonc 10 ~l )
wnn importnnt as well for the Jiht it shed on tho nntlH()
of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich
is available only in Fontenelles report of 1700 57 The
limits established by Sauveur according to Fontenelle
for the perception of beats have not been generally
accepte~ for while Sauveur had rema~ked that when the
vibrations dve to beats ape encountered only 6 times in
a second they are easily di stinguished and that in
harmonies in which the vibrations are encountered more
than six times per second the beats are not perceived
at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny
as 132 beats in a second aTe audihle--an assertion which
he supposed would appear very strange and incredible to
acol1sticians59 Nevertheless Helmholtz insisted that
55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7
56Scherchen Nature of Music p 29
57 If IfFontenelle Determination
58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I
59Helmholtz Sensations of Tone p 171
138
his claim could be verified experimentally
bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l
of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc
~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60
If as seems likely Sauveur intended his limit to be
understood as one beyond which beats could not be pershy
ceived rather than simply as one beyond which they could
not be counted then Helmholtzs findings contradict his
conjecture61 but the verdict on his estimate of the
number of beats perceivable in one second will hardly
affect the apnlicability of his method andmoreovAr
the liMit of six beats in one second seems to have heen
e~tahJ iRhed despite the way in which it was descrlheo
a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy
ing the pitCh of the pipes or strings emJ)loyed by octavos
Thus pipes which made 400 and 384 vibrations or 16 beats
in one second would make two octaves lower 100 and V6
vtbrations or 4 heats in one second and those four beats
woulrl be if not actually more clearly perceptible than
middot ~60lb lO
61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168
139
the 16 beats of the pipes at a higher octave certainly
more easily countable
Fontenelle predicted that the beats described by
Sauveur could be incorporated into a theory of consonance
and dissonance which would lay bare the true source of
the rules of composition unknown at the present to
Philosophy which relies almost entirely on the judgment
of the ear62 The envisioned theory from which so much
was to be expected was to be based upon the observation
that
the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3
Iontenelles prediction was fulfilled in the theory
of consonance propounded by Helmholtz in which he proposed
that the degree of consonance or dissonance could be preshy
cis ely determined by an ascertainment of the number of
beats between the partials of two tones
When two musical tones are sounded at the same time their united sound is generally disturbed by
62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille
63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre
140
the beate of the upper partials so that a ~re3teI
or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~
But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64
Fontenelle or perhaps Sauvellr had also it soema
n()tteod Inntnnces of whnt hns come to be accepted n8 a
general rule that beats sound unpleasant when the
number of heats Del second is comparable with the freshy65
quencyof the main tonerr and that thus an interval may
beat more unpleasantly in a lower octave in which the freshy
quency of the main tone is itself lower than in a hirher
octave The phenomenon subsumed under this general rule
constitutes a disadvantape to the kind of theory Helmholtz
proposed only if an attenpt is made to establish the
absolute consonance or dissonance of a type of interval
and presents no problem if it is conceded that the degree
of consonance of a type of interval vuries with the octave
in which it is found
If ~ontenelle and Sauveur we~e of the opinion howshy
ever that beats more frequent than six per second become
actually imperceptible rather than uncountable then they
cannot be deemed to have approached so closely to Helmholtzs
theory Indeed the maximum of unpleasantness is
64Helmholtz Sensations of Tone p 194
65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49
141
reached according to various accounts at about 25 beats
par second 66
Perhaps the most influential theorist to hase his
worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to
have heen in an important sense his follower nt nll
tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)
67de 1 Iarmonie in which he had attempted to make music
a deductive science hased on natural postu1ates mvch
in the same way that Newton approaches the physical
sci ences in hi s Prineipia rr 68 before he l)ecame famll iar
with Sauveurs discoveries concerning the overtone series
Girdlestone Hameaus biographer69 notes that Sauveur had
demonstrated the existence of harmonics in nature but had
failed to explain how and why they passed into us70
66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance
67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)
68Gossett Ramea1J Trentise p xxii
6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)
70Ibid p 516
11-2
It was in this respect Girdlestone concludes that
Rameau began bullbullbull where Sauveur left off71
The two claims which are implied in these remarks
and which may be consider-ed separa tely are that Hamenn
was influenced by Sauveur and tho t Rameau s work somehow
constitutes a continuation of that of Sauveur The first
that Hamonus work was influenced by Sauvollr is cOTtalnly
t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the
Nouveau systeme of 1726 Hameau had begun to appreciate
the importance of a physical justification for his matheshy
rna tical manipulations he had read and begun to understand
72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7
he had 1Idiscllssed in detail the relatlonship between his
73rules and strictly physical phenomena Nonetheless
accordinv to Gossett the main tenets of his musical theory
did n0t lAndergo a change complementary to that whtch had
been effected in the basis of their justification
But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of
the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74
If Gossetts estimation is correct as it seems to be
71 Ibid bull
72Gossett Ramerul Trait~ p xxi
73 Ibid bull
74 Ibi d
143
then Sauveurs influence on Rameau while important WHS
not sO ~reat that it disturbed any of his conc]usions
nor so beneficial that it offered him a means by which
he could rid himself of all the problems which bGset them
Gossett observes that in fact Rameaus difficulty in
oxplHininr~ the minor third was duo at loast partly to his
uttempt to force into a natural framework principles of
comnosition which although not unrelated to acoustlcs
are not wholly dependent on it75 Since the inadequacies
of these attempts to found his conclusions on principles
e1ther dlscoverable by teason or observabJe in nature does
not of conrse militate against the acceptance of his
theories or even their truth and since the importance
of Sauveurs di scoveries to Rameau s work 1ay as has been
noted mere1y in the basis they provided for the iustifi shy
cation of the theories rather than in any direct influence
they exerted in the formulation of the theories themse1ves
then it follows that the influence of Sauveur on Rameau
is more important from a philosophical than from a practi shy
cal point of view
lhe second cIa im that Rameau was SOl-11 ehow a
continuator of the work of Sauvel~ can be assessed in the
light of the findings concerning the imnortance of
Sauveurs discoveries to Hameaus work It has been seen
that the chief use to which Rameau put Sauveurs discovershy
ies was that of justifying his theory of harmony and
75 Ibid p xxii
144
while it is true that Fontenelle in his report on Sauveur1s
M~moire of 1702 had judged that the discovery of the harshy
monics and their integral ratios to unity had exposed the
only music that nature has piven us without the help of
artG and that Hamenu us hHs boen seen had taken up
the discussion of the prinCiples of nature it is nevershy
theless not clear that Sauveur had any inclination whatevor
to infer from his discoveries principles of nature llpon
which a theory of harmony could be constructed If an
analogy can be drawn between acoustics as that science
was envisioned by Sauve1rr and Optics--and it has been
noted that Sauveur himself often discussed the similarities
of the two sciences--then perhaps another analogy can be
drawn between theories of harmony and theories of painting
As a painter thus might profit from a study of the prinshy
ciples of the diffusion of light so might a composer
profit from a study of the overtone series But the
painter qua painter is not a SCientist and neither is
the musical theorist or composer qua musical theorist
or composer an acoustician Rameau built an edifioe
on the foundations Sauveur hampd laid but he neither
broadened nor deepened those foundations his adaptation
of Sauveurs work belonged not to acoustics nor pe~haps
even to musical theory but constituted an attempt judged
by posterity not entirely successful to base the one upon
the other Soherchens claims that Sauveur pointed out
76Fontenelle Application p 120
145
the reciprocal powers 01 inverted interva1su77 and that
Sauveur and Hameau together introduced ideas of the
fundamental flas a tonic centerU the major chord as a
natural phenomenon the inversion lias a variant of a
chordU and constrllcti0n by thiTds as the law of chord
formationff78 are thus seAn to be exaggerations of
~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy
hnps from an overestim1 t on of the extent of Snuvcllr s
interest in harmony and the theories that explain its
origin
Phe importance of Sauveurs theories to acol1stics
in general must not however be minimized It has been
seen that much of his terminology was adopted--the terms
nodes ftharmonics1I and IIftJndamental for example are
fonnd both in his M~moire of 1701 and in common use today
and his observation that a vibratinp string could produce
the sounds corresponding to several harmonics at the same
time 79 provided the subiect for the investigations of
1)aniel darnoulli who in 1755 provided a dynamical exshy
planation of the phenomenon showing that
it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant
77Scherchen Nature of llusic p b2
8Ib1d bull J p 53
9Lindsay Introduction to Raleigh Sound p xv
146
being the algebraic sum of the displacements for each simple harmonic node SO
This is the fa1jloUS principle of the coexistence of small
OSCillations also referred to as the superposition
prlnclple ll which has Tlproved of the utmost lmportnnce in
tho development of the theory 0 f oscillations u81
In Sauveurs apolication of the system of harmonIcs
to the cornpo)ition of orrHl stops he lnld down prtnc1plos
that were to be reiterated more than a century und a half
later by Helmholtz who held as had Sauveur that every
key of compound stops is connected with a larger or
smaller seles of pipes which it opens simultaneously
and which give the nrime tone and a certain number of the
lower upper partials of the compound tone of the note in
question 82
Charles Culver observes that the establishment of
philosophical pitch with G having numbers of vibrations
per second corresponding to powers of 2 in the work of
the aconstician Koenig vvas probably based on a suggestion
said to have been originally made by the acoustician
Sauveuy tf 83 This pi tch which as has been seen was
nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy
cally simple approximation of the pitch then in use-shy
Culver notes that it would flgive to A a value of 4266
80Ibid bull
81 Ibid bull
L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull
83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86
147
which is close to the A of Handel84_- came into widespread
use in scientific laboratories as the highly accurate forks
made by Koenig were accepted as standards although the A
of 440 is now lIin common use throughout the musical world 1I 85
If Sauveur 1 s calcu]ation by a somewhat (lllhious
method of lithe frequency of a given stretched strlnf from
the measl~red sag of the coo tra1 l)oint 86 was eclipsed by
the publication in 1713 of the first dynamical solution
of the problem of the vibrating string in which from the
equation of an assumed curve for the shape of the string
of such a character that every point would reach the recti shy
linear position in the same timeft and the Newtonian equashy
tion of motion Brook Taylor (1685-1731) was able to
derive a formula for the frequency of vibration agreeing
87with the experimental law of Galileo and Mersenne
it must be remembered not only that Sauveur was described
by Fontenelle as having little use for what he called
IIInfinitaires88 but also that the Memoire of 1713 in
which these calculations appeared was printed after the
death of MY Sauveur and that the reader is requested
to excuse the errors whlch may be found in it flag
84 Ibid bull
85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of
Sound p xiv
87 Ibid bull
88Font enell e 1tEloge II p 104
89Sauveur Rapport It p 469 see vol II p201 below
148
Sauveurs system of notes and names which was not
of course adopted by the musicians of his time was nevershy
theless carefully designed to represent intervals as minute
- as decameridians accurately and 8ystemnticalJy In this
hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind
notes to represent all conceivable musical sounds rather
than simply to facilitate the solmization of a meJody
Sauveur transcended in his work the systems of Hubert
Waelrant (c 1517-95) father of Bocedization (bo ce di
ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy
zation (la be ce de me fe gel and Karl Heinrich
Graun (1704-59) father of Damenization (da me ni po
tu la be) 90 to which his own bore a superfici al resemshy
blance The Tonwort system devised by KaYl A Eitz (1848shy
1924) for Bosanquets 53-tone scale91 is perhaps the
closest nineteenth-centl1ry equivalent of Sauveur t s system
In conclusion it may be stated that although both
Mersenne and Sauveur have been descrihed as the father of
acoustics92 the claims of each are not di fficul t to arbishy
trate Sauveurs work was based in part upon observashy
tions of Mersenne whose Harmonie Universelle he cites
here and there but the difference between their works is
90Harvard Dictionary 2nd ed sv Solmization 1I
9l Ibid bull
92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ
COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)
149
more striking than their similarities Versenne had
attempted to make a more or less comprehensive survey of
music and included an informative and comprehensive antholshy
ogy embracing all the most important mllsical theoreticians
93from Euclid and Glarean to the treatise of Cerone
and if his treatment can tlU1S be described as extensive
Sa1lvellrs method can be described as intensive--he attempted
to rllncove~ the ln~icnl order inhnrent in the rolntlvoly
smaller number of phenomena he investiFated as well as
to establish systems of meRsurement nomAnclature and
symbols which Would make accurate observnt1on of acoustical
phenomena describable In what would virtually be a universal
language of sounds
Fontenelle noted that Sauveur in his analysis of
basset and other games of chance converted them to
algebraic equations where the players did not recognize
94them any more 11 and sirrLilarly that the new system of
musical intervals proposed by Sauveur in 1701 would
proh[tbJ y appBar astonishing to performers
It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95
llatl1Ye of Music p 18
94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus
95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle
150
These two instances of Sauveurs method however illustrate
his general Pythagorean approach--to determine by means
of numhers the logical structure 0 f t he phenomenon under
investi~ation and to give it the simplest expression
consistent with precision
rlg1d methods of research and tlprecisj_on in confining
himself to a few important subiects96 from Rouhault but
it can be seen from a list of the topics he considered
tha t the ranf1~e of his acoustical interests i~ practically
coterminous with those of modern acoustical texts (with
the elimination from the modern texts of course of those
subjects which Sauveur could not have considered such
as for example electronic music) a glance at the table
of contents of Music Physics Rnd Engineering by Harry
f Olson reveals that the sl1b5ects covered in the ten
chapters are 1 Sound Vvaves 2 Musical rerminology
3 Music)l Scales 4 Resonators and RanlatoYs
t) Ml)sicnl Instruments 6 Characteri sties of Musical
Instruments 7 Properties of Music 8 Thenter Studio
and Room Acoustics 9 Sound-reproduclng Systems
10 Electronic Music 97
Of these Sauveur treated tho first or tho pro~ai~a-
tion of sound waves only in passing the second through
96Scherchen Nature of ~lsic p 26
97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi
151
the seventh in great detail and the ninth and tenth
not at all rrhe eighth topic--theater studio and room
acoustic s vIas perhaps based too much on the first to
attract his attention
Most striking perh8ps is the exclusion of topics
relatinr to musical aesthetics and the foundations of sysshy
t ems of harr-aony Sauveur as has been seen took pains to
show that the system of musical nomenclature he employed
could be easily applied to all existing systems of music-shy
to the ordinary systems of musicians to the exot 1c systems
of the East and to the ancient systems of the Greeks-shy
without providing a basis for selecting from among them the
one which is best Only those syster1s are reiectec1 which
he considers proposals fo~ temperaments apnroximating the
iust system of intervals ana which he shows do not come
so close to that ideal as the ODe he himself Dut forward
a~ an a] terflR ti ve to them But these systems are after
all not ~)sical systems in the strictest sense Only
occasionally then is an aesthetic judgment given weight
in t~le deliberations which lead to the acceptance 0( reshy
jection of some corollary of the system
rrho rl ifference between the lnnges of the wHlu1 0 t
jiersenne and Sauveur suggests a dIs tinction which will be
of assistance in determining the paternity of aCollstics
Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de
Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy
eries concernin~ the phenomena which were later to be
s tlJdied by Sauveur and while among these T~ersenne had
152
attempted to present a compendium of all the information
avniJable to scholars of his generation Sauveur hnd in
contrast peeled away the layers of spectl1a tion which enshy
crusted the study of sound brourht to that core of facts
a systematic order which would lay bare tleir 10gicHI reshy
In tions and invented for further in-estir-uti ons systoms
of nomenclutufte and instruments of measurement Tlnlike
Rameau he was not a musical theorist and his system
general by design could express with equal ease the
occidental harraonies of Hameau or the exotic harmonies of
tho Far East It was in the generality of his system
that hIs ~ystem conld c]aLrn an extensIon equal to that of
Mersenne If then Mersennes labors preceded his
Sauveur nonetheless restricted the field of acoustics to
the study of roughly the same phenomena as a~e now studied
by acoustic~ans Whether the fat~erhood of a scIence
should be a ttrihllted to a seminal thinker or to an
organizer vvho gave form to its inquiries is not one
however vlhich Can be settled in the course of such a
study as this one
It must be pointed out that however scrllpulo1)sly
Sauveur avoided aesthetic judgments and however stal shy
wurtly hn re8isted the temptation to rronnd the theory of
haytrlony in hIs study of the laws of nature he n()nethelt~ss
ho-)ed that his system vlOuld be deemed useflll not only to
scholfjrs htJt to musicians as well and it i~ -pprhftnD one
of the most remarkahle cha~actAristics of h~ sv~tem that
an obvionsly great effort has been made to hrinp it into
153
har-mony wi th practice The ingenious bimodllJ ar method
of computing musical lo~~rtthms for example is at once
a we] come addition to the theorists repertoire of
tochniquQs and an emInent] y oractical means of fl n(1J nEr
heptameridians which could be employed by anyone with the
ability to perform simple aritbmeticHl operations
Had 0auveur lived longer he might have pursued
further the investigations of resonatinG bodies for which
- he had already provided a basis Indeed in th e 1e10 1 re
of 1713 Sauveur proposed that having established the
principal foundations of Acoustics in the Histoire de
J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702
107 and 1711 he had chosen to examine each resonant
body in particu1aru98 the first fruits of which lnbor
he was then offering to the reader
As it was he left hebind a great number of imporshy
tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch
tne overtone series the echometer and the formulas for
tne constrvctlon and classificatlon of terperarnents--as
well as a language of sovnd which if not finally accepted
was nevertheless as Fontenelle described it a
philosophical languare in vk1ich each word carries its
srngo vvi th it 99 But here where Sauvenr fai] ed it may
b ( not ed 0 ther s hav e no t s u c c e e ded bull
98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w
99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull
Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T
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156
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