The Theories of Tartini

8/2/2019 The Theories of Tartini

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A S T U D Y O F T H E T H E O R I E SO F G I U S E P P E T A R T I N I

Alejandro Enrique Planch art Ca ra ca s, Venezuela

Befo re beginning a sy ste ma tic study of the th eo rie s of GiuseppeTar tin i (1692-1770), i t is nec ess ary to examine briefly his position andhis goals a s a t heo rist and to give a sum ma ry of the contents of theworks to be discussed. Of a l l the th eo ri st s of his time no one wasmo re singlemindedly con cerned with throwing light upon the m ost bas icprin cipl es of harmony; in consequence, anything that in hi s opinion didnot s erv e this purpos e was not brought into consideratio n [T,cf . le t terto Count Decio Agostino Tr en to , without page numbe r, a t the beginningof the book]. Thus we find little o r nothing sai d about pra ctic al ru le sfor chord progression o r about the s tru ctura l o r affect ive uses of h ar-mony. To him, th e bas ic principles of harmony a r e an exact science,derived fro m nature to be su re , but from a nature r igidly ruled bymathematical proportions in a C artesian sense. In his work, mathe-matical relatio n and proportio n take priority, and he derid es those whothink that the princip les of harmony may depend upon feeling o ra r t i s t ry .

He is quite aware of the difficulties into which other theoristshave been led in attem ptin g to explain consonant min or harm ony anddissonance, and he tries throughout his work to re lat e the th ree ap-parently different foundations of majo r, min or, and dissona nt harmonyto an a pr ior i principle which will account for all three. This unive rsalprinciple, he believes, is to be found in the relationship of the circum-ference of a cir cle to i t s diame ter projected a s the side of a circum-scr ibed square (Figure 1).

Figure 1.

Th is leads him to devote much spac e in his work to long and ab -st ru se mathematical and geo metric proofs in which he see ks to demon-st ra te the validity of thi s a pri ori principle and the neces sar y inclusionof the principle s of maj or, min or, and dis sonant harmony in thi s uni-ve rs al principle to thus produce a complete and closed syste m.

To most of his contemporar ies these proofs were incomprehen-


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THEORIES OF GIUSEPPE TARTINIsible, and those who understood them found e r r o r s in Ta rti ni 's calcu-lations and attacked his system violently. Even Ta rti ni 1s follower,Benjamin Stillingfleet say s: "One cannot without s om e imp res sio n ofcompassion, s ee him wandering in the perplexing labyrinths of ab st ra ctideas, alm os t without guide, o r a t best with one which it is most likelywould mislead him [, 16]!' And later on: "What I have already saidwill be sufficient fo r my not enteri ng into a detail on this long chap ter[x,Ch. 111 a s such a de tail would be e xtr em ely tedious to som e, unin-telligible to oth ers and would appear (at lea st) stran ge to the only menwho a r e qualified to for m any judgement on this ma tte r, I mean themathematicians [6, 17]!' This se em s to explain why the re has neverbeen an attempt to account fully for this p art of Ta rt in i1 swork, to showwhere the e r r o r s a r e and in what way they affect his conclusions. Suchan a t tempt is made in this essay .

Tart ini ' s theoretical works a re : the Trattato di Musica [TI thatappeared in 1754, De' Principj del l lArmonia [PI that appea red in 1767,and a Risposta di G. ..T. . .al la cr i t t ica del di lui Trat ta to [ a ] that ap-peared in 1767. His o ther works a r e two sm all tr ac ts on ornamen ta-tion [A] and vio lin playing [L,]. He h as been cr ed it ed with the au th or -s hi p of a n anonymous ~ i s ~ o s t a l th at a pp ea re d ino J . J . Rousseau1769.

This essay is a co mparative study of the Trat tato , the P rinci pjand the R isposta of 1767; al though i t would have b m i r a b l eto-pa re th ese works with the anonymo us Rispos ta, no copy of th is workha s been available to me.The Trattato is a long and abtruse work which contains the mostcomprehen sive account of Ta rt in i ' s theo retic al thought. After an in-troduction, where Tart ini exposes the ma th em ati ca l pr em is es Tor thedemonstration s in the second and third chap ters, the work begins withan account of s ev er al phenomena which point to the possi bility of aphysical bas is fo r harmony. Of par ticu lar importance here is his ac-count ofth e difference-tones whi char e the m ost se cu re ph ysical founda-tion for his system ; in the Principj he clai ms to have discovered thisphenomenon in 17 14 [P, 361. In the second and thi rd ch ap te rs we findthe bulk of his mathematical calculations and g eom etric proofs; he t ri esto prove the harmonic nature of the cir cle and to derive f ro m the re la -tion ofthe circum ference to the diam eter the ma jor and mino r s yst ems ,the diatonic and chro mati c dissonances, and even an enharmon icsys tem . The la st two ch ap ter s a r e devoted to the derivation of thescale, the establishment of the basic ba ss progression, and a discus -sion of s ev era l inte rv als found in the mu sic of hi s day. The work con-cludes by answering som e possible argu men ts that could be rai se dagainst his system.

The Principj is a much cl ea re r work than the Trattato; the mathe-matical proofs of the ea r l ie r work a r e for the most par t absent here .There is a gen eral re-exposition of the prin cipl es of the Trat tato , butlimited to the diatonic system. The work is divided into four parts.The fi rs t de als with the physical foundations of harm ony , and conta insaccounts sim ila r to those in Chapter I of the Tra tta to . The second

1. Risposta d i un anonimo a 1 celebre signor J . ..J . . . RousseauVenice: Antonio Decastro, 1769.


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TABLE ITARTINI'S THIRD SOUNDS

for the fifth:

for the fourth:

I -for the m ajo r thi rd:

for the minor thi rd:

for the ma jor tone:

for the mino r tone:

for the major semitone:

for the mino r semitone:


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THEORIES O F GIUSEPPE TARTINIdea ls with the systematic foundation of harmony and is concerned withthe derivation of intervals. The third tre at s what Tartin i calls "themu sic al foundation of ha rmon yu and co ns ist s of the d eriva tion of thesc ale and the establishment of the basi c bas s progression. The fourthrela tes the mater ia l ofth e th ree previous ch ap ter s into a unified whole.The Risposta is a sm all polemic work in which Tart ini ans wer sthe objectibns rai se d to his T ra tt at o by J e a n A dam ~ e r r e ? t containsve ry l i t t le that has not been dealt with mo re extensively in the othertwo books. The mo st imp ortan t propositionme tri c explanation fo r the formula of the "third sound

in the Ris ostaP is a geo-(i. e. difference-tones).Tar t in i ' s sys tem is a clo sely woven one. Unfortunately i t s au th oris "quite ig noran t of geo me try and algeb ra [B, 51" and his mistakencalculations lead him astray . He firmly believes that he has found theanswer to all the prob lem s of har mon ic theory and has discovered anew s et of pr op ert ie s for the geo met ric figu res with which his proofsdeal. In the Risposta he mentions that he has developed a new sciencewhich however he will not ye t publi sh sinc e the few glim pses of it thatappeared in the Trattato we re so i l l-received 13, 101.

The attack on Tar t in i ' s sy stem by the French theor is ts led Fdt isto consider Ta rti ni 's theories a s directly opposed to the theor ies ofRameau [z, viii, 1871. This is denied by Shirlaw [a, 2931; and i t willbe evident to anyone who co mp are s the works of the two th eo ris ts that,in spite of Tartini 's polemic attitude towards Rameau and the attack onTart ini 's s yst em by Ram eau ts followers, the basic theoretical conclu-sions of both men a r e quite si mi la r3

In the Trattato, Tart ini ob serv es that when a monochord str in gis plucked, even though i t is one str ing, i t produces thre e distinctsounds: the fundamental, the twelfth, and the sevente enth , which a r e inthe harmonic proportion of 1:1/3:1/5 [T,101. Likewise, he noticesthat when an org an key (with a m ixtu re stop) is pres sed many pipes ofdifferent pitch sound; but since the sounds produced by the pipes a r e inharmonic proportion, the au ral imp ress ion is that of a single sound[T , 121. F r o m thi s he dedu ces that the es sen ce of harmony is unity, aunity that divides itself into multiplicity only to re tu rn again to unity a si ts basic pr inciple [T,131. He finds fu rth er supp ort for thi s in thefact that instrum ents such a s the tromba mar ina produce no sound un-les s the s t r ing is divided into an aliquot part4 of the total length (unity)of the string [T , 11-12]. In this sense the sounds of the monochordstr ing a r e "harmonic monads" and the te rm aliquot pa r ts does notproperly describe them [T,121 .

Rameau st at es that musical sound is not one but three 15, 2881.2. In the Obsemtions sur les principes de ltha.rmonie, occa-

siondes par quelqu&s gcrits modernes sur ce sujet, et particuli&rementpar.. . le traiix! de th6orie musicale de M. W i n i . . . [~eneva: enriAlbert Gosse et Jean Gosse, 17631. This work was not available to me.3. There is an excellent account of Rameauys theories bu Joan Ferrls [8].4. By an aliquot part of any quantity, line, or surface is meant such a part as will measure the whole wlthout a remainder. That is, an integral divisor.


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ALEJANDRO ENRIQUE PLANCHARTWhen Tar tin i st at es that the thr ee sounds of the monochord str i ng a r eharmonic monads, he does not place himself in di rec t opposition toRameau sin ce he con si de rs multiplicity a function of unity and re ga rd sthe division of unity into mul tipli city and the resol ution of mul tipli cityinto unity a s pa rt s of a complete cycle [T,131. Rameau ' s emphas is onmultiplicity would not contradict-directly Ta rt in i' s assu mp tion of unitya s the basic harmonic principle.

Whether o r not multiplicity should be co nsid ered beyond theth re e sounds (1, 113, 115) which a r e audibly produced by the mono-chord st r in g i s a ma tter of extensive discussion in both the Tr atta toand the Prin cipj . In the conclusion of the Trat tato , Ta rti ni ad mi tsthat other theor i s ts c la im to hea r the sounds 1/ 2, 1/ 4, and even 1/6 inaddition to the three sounds he has alrea dy reported a s being producedby a plucked string. Tar t ini admits tha t this may be true but that i twould not invalidate his th eori es since these additional sounds ar e als oin harmonic proportion [T_, 1'701. He open s the Pr in ci pi with a disc us -sion of the specific as se rt io ns of Rameau [& 1941 and d lA lem be rt[_P, 43ffl with re sp ec t to this question. Both cl aim to have he ard th esounds 112 and 114 produced by a plucked str in g.

Tar tin i points out that although they make th is cl aim, they ignorethe ir f indings in setting up harm onic sy st em s based solely on 1, 113,and 115 [P , 31. He ad mi ts that he ha s been unsuccessful in his atte mp tsto determ ine the pre se nc e of the sounds 112 and 114 in the vibrati on ofthe plucked st rin g because, a s octave doublings, they a r e swallowed upby the fundamental sound of the string [_P, 41. He cl ai m s tha t if Ram eauand dlAlem bert have indeed hea rd these sounds, the ir sys te ms invali-date themselve s by not making use of th eir em piri cal f indings [I?, 31.In his se ar ch for a m or e com prehensive physical foundation for aharm onic sy ste m he tu rn s to his discovery of the difference-tones,which he c al ls the "phenomenon of the thir d sound [E, 511'

As repor ted in the Tra t ta to , he determ ines the ir exis tencethrough a se r ie s of experiments in which two instrum ents play seve ra linte rval s. If the inte rval s a r e played with just intonation and loudenough, a third sound, gene ral ly low er than e it he r of the two soun dsbeing played, will be heard [ T , 13-14]. This account is accompaniedin the Tra ttat o by a l is t of the third sounds for the different int erva ls(Table I, p. 34): fo r the octave and the unison no thir d sound is givensince Tart ini clai ms that the se intervals produce none. The thirdsounds of an interv al and i t s invers ion a r e the same, some times withan octave displacement [T,14ffl.

Any in ter va l will produce a thir d sound. If two violins play thefollowing: it will be noticed that the third sound glidesfro m that of one in te rva l to that of the other, ne ve r ceas ing to soundeven though a n incomm ensurable num ber of intervening in te rva ls haveoccur red [X, 171. To him, the third sound is the real physical funda-men tal b as s of any given i nter val and of any given pa ir of m elodic lines;the succ essi ve t hi rd sound s produced by th ei r combination constitutethe tr ue fundamental b as s of the melodies. Any oth er b as s would beabsurd or , a t bes t , a r t i f ic ia l [T, 171.Ta rti ni ob se rve s that any combination of in ter va ls out of the


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THEORIES OF GIUSEPPE TARTINIse na rio up to the sounding of the whole s en ar io produces a single thirdsound, which in the Tr att at o, he mistakenly identifies a s 112, that is,an octave above the fundamental rT, 181. However, in the conclusionof the Tra ttat o he mentions that othe r t heo rists have identified the thir dsound a s 1, o r the fundamental of the se ri es . Thi s, he says , does notupset his as se rt ion of the third sound a s the fundamental ba ss andphysical root of any interval o r harmonic s er ie s [T_, 170-11.

As a consequence of the identi ficatio n of the thi rd sound of theharmonic se r i es a s constant in 112, Tart in i bel ieves that he has founda novel property of harm onic rat i os. Given I / 2: 113, the re sul ta nt(thir d sound) is 112; given 1 /3: 1/ 4, the resu ltant is 112 and s o on [T,18-19]. This goes agai nst the ba sic pr em is e of the system , which con-si de rs harmony a s a projection of unity. Ta rt i ni overlooks this con tra-diction.

In the Principj , Tar t ini re vis es his view and identifies the thirdsound, s t i l l mistakenly, a s constant in 1 o r in unison with the funda-mental of the harmonic series [_P, 221; in addition, he gives a mathe-ma tical form ula f or the deduction of the t hir d sound of any inter val.Th e int erv al of the fifth, 1/ 2: 1 3, can be exp ress ed ari thmetically a s3:2. Tar t i n i a s se r t s t ha t t he product of the values that rep re se nt thes imple s t ar i thmet ic express ion of an in te rval wi ll g ive the cor rec t th i rdsound. In the ca se of the fifth, 3 x 2 = 6; thus 6 will re pr es en t theth ird sound of the in te rv a l 3:2. If 6 is exp ressed a s 1 of the harmonicse ri e s, 3 will be 112 and 2 will be 113 [_P, 5-61. When an interval isexpres sed in a m ore complex form , i . e. the octave a s 2:4, the productof the te r m s of the rat io mu st be divided by the common fac tor in o r -der to a r r i ve a t the co r re c t thi rd sound [_P, 61.This for mula enab les Tar t ini to determine the third sounds fo rthe octave and the unison. He explains that they a r e inaudible beca usethey a r e in unison with one of the g ener ating tones [_P, 51. Ta rt in i fai lsto notice that he ha s invalidated his as se rt io n that the third sound ofthe senario is constant in 1. The thi rd sound of the octave 1/2:1 /4,which is included in the senario, is, by his formula, 112.In the Risposta, t rying to defend the Tratt ato a t al l cos ts , heattempts to justify both 1 and 112 a s third sounds. This is done byme ans of an involved mathem atical proof that at te mp ts to demonstra tethat 112 is the ge ome tric ex pre ssi on of the thi rd sound, while 1 is thear i thm et ic one k, 25ffl. It is important to note that Tar t in i doe s notconsider the thir d sound a s a foreign element added to the generatinginterval; the inter val, the third sound, and a ll the harm onic divisionsin between, fo rm a harmo nic whole [E,6-71.The formula for the th ird sound and the operation i t re pr es en tsa r e s im ila r to a formula proposed by Leonhard Euler . Eu le r ' s formulais base d on the m ultiplication of the t e r m s of the ra tio a t which twost rin gs vibr ate when producing a n interval; the product, together withal l i t s in tegral d ivisors, wil l fo rm a complex s im ilar to the harmoniccomplex of T ar t i ni ls sy ste m of the third sound [L39ffl.Tar t in i arg ues that Eu le r l s formula fai ls to produce a harmonicproportion. The vibratio n of two s tr in gs producing the int er va l of afifth a r e in the rat i o 3:2, the product of these t e r m s is 6. The integra ldiv iso rs of 6 a r e 3, 2, and 1, and the complex formed by E ul er ls lawis 6:3:2:1, which is not in harm onic proportion. The complex 6:3:2,


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Figure 2.

Example 2 .

Example 3.

Example 4.

Example 5.


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THEORIES OF GIUSEPPE TARTINIproduced by Ta rt in i 's system , i s [E, 8-91. Moreover, in the case ofthe minor third, 6:5, the complex arr iv ed a t by me ans of Eu le r ' sformula is not even consonant, while the one ar ri ve d a t by Ta rt in it sformula is .

F r o m this, Ta rti ni concludes that the law of the third sound isthe law of harm onic proport ion and has i ts basis in unity; in conse-quence, the ra tio s contained within the se na rio should be considered a snecess ar i ly bas ic [_P, 91.

To Tar tini, the natu re of the third sound cons ists in thr ee prop-er t i es : f i r s t , i t is intrinsically physical harmonic unity a s a principle;second, it produces harmonic proportion with the generating sounds;third, it s simultaneous union with the two generating sounds and a ll theharmonic divisions in between gives r is e to the harmonic sy st em ofwhich the third sound is the fundamental bass [_P, 111. However, thethird sound of an inter val which ha s 1 a s one of the t e r m s of its rati ocannot produce a proportion with that int erv al because it will be identi-ca l with one of the sounds of the gene ratin g interv al. In consequence,any inter val which has the unit a s one of i t s te rm s is only potentiallyharmonic; the third sound will st a r t producing har mon ic proportionwith the generating sounds f ro m the second inter val on in any or de re ds e r i e s [_P, 151. Th is ma kes the second inte rval of any harm onic se r ie sthe interval that determines the harmo nic nature of the s e ri e s [_P, 151.

Within the sen ario th er e ar e only two possible s er ie s, one bydifference 1 (superpart icular) a s 1 :1/ 2 : 1/ 3 and ano ther by difference 2(superpar t iens) a s 1:1/3:1/5. Any at tempt to construct a s er ie s withinthe sen ari o using a la rg er difference will never rea ch the second in-terval which is necessary to determine the nature of the series [P, 121.

Returning to the phenomenon of the monochord str ing , Ta rt in ipoints out that, while the s ys te m of the thi rd sound by differe nce 1 pro-duces the whole of the se na rio , the sy st em by difference 2 produces theesse nce of the se nar io in 1:1/3 :1/5, the sounds that he has he ard in themonochord strin g. The phenomenon of the monochord s tr in g is thenonly a pa rt ia l one while th at of the th ird sound is the general one [E,31.

So fa r, Ta rt in i ha s worked only with the harm onic division of thestrin g which produces the maj or syst em . As soon a s he at tempts toexplain the minor system, trouble ar is es . In the Trattato he gives thethird sounds fo r the m inor tr iad, which a r e not one, a s in the ma jortri ad, but two (Ex. 2). The resu ltan t is a dissonant combination whichcau ses Tart ini to comment that , if the third sounds were m ore audible,only music with m aj or t ria ds would be possible [T,671.

Minor harmony, derived from the arithmetic division of thestring, had been considered by other theoris ts a s "borrowed" fr omarithme tic science and essential ly foreign to harmony. Thi s amountsto an admission of mo re than one principle for any suc cessful harmonicsystem, and Tart ini f inds this assump tion absur d and fundamentally op-posed to the idea of sy st em itself [T,681. Tart in i adm its the " imper-fection" of minor harmony a s compared to m ajo r - major harmony has"priority of nature " ove r min or - but he denies that min or harmonymay s te m from a di fferent pr inciple and considers i t a s an in t r ins ical lyinseparable consequence of major harmony [T , 681.The argum ent in both tre at ise s is esse ntia lly the sam e. Givena str ing of determ ined length, e. g. 60, he divides the stri ng harmoni-


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ALEJANDRO ENRIQUE PLANCHARTcally (F ig . 2, p. 38). Ea ch of the segm ents taken fro m A to the pointof division produces one of the sounds of the series shown in Ex. 3(p. 38). He call s this se ri es the syst em of fractions and i t is nothingother than the senario.

Th ese divisions, when taken to B, produce another se ri es ofstr ing lengths which produce .the s e r i e s of sounds shown in EX. 4. Hecall s thi s the sys tem of advances and cons ider s i t the physical proofthat min or harmony and the subdominant a r e an inseparable conse-quence of major harmony [T, 69; 2, 25ffl. He over looks the fa ct th atthe sy ste m of advances con sists of frac tions such a s 516 which a r e notaliquot parts of the string and could not be produced by such instru-ments a s the t romba marina.Rameau re lat es minor harmony to the co-vibration of multiplesof the fundamental string [3, 2215 ~ a r t i n i o ns id er s R a me au 's pointwhen he supe rim pos es the sy st em s of frac tio ns and of advanc es upon aconstant fundamental sound (Ex. 5). The r esultant third sounds (Fand~b in part icular) ar e the sa m e a s the sounds of Rameau's sympathetics t r ings [_P, 26-71. He s e e m s not to notice that grea t-C is not the funda-mental of that se ri es and that the successiv e third sounds a r e the funda-me ntals of eac h interval. His s er ie s, a s Shirlaw points out , is actuallymade of interv als belonging to different harmonic s e r ie s and a t best i tproves only that min or harmony is some so rt of inverted ma jor h ar -mony E, 295-61.

Tart in i is aware that mere physical proofs are incomplete andse ts out to se ar ch f or a mathematical foundation for his system. How-ever , mathema tical principles, in or de r to be valid, must be evident inthe observed physical fact s. "wh ere the point is to establish a s ystem,it i s nece ss ar y to unite the two cate gori es, physical and mathem atical,in such a way that they would be in sepa rable and f or m a single p rin -ciple [T_, 201:'The unity which he believes to have found in the physical founda-tion of harmony is best exp ress ed by the ci rc le which is in itself one.Eac h of the infinite num be r of ra dii that ent er into i ts construction canser ve a s unity [T,211. This is not enough. The circ le must be provento be a harm onic unity, and to thi s end T ar tin i cons ide rs it in connec-tion with the stra ight line repr esen ted in the squa re. In Tar t i n i t sopinion, t his p re mi se has a definite physical basi s. The strai ght lineof the monochord str ing , when it vib ra tes to produce the harm onicproportion 1:1/3:1/5, produces so me so rt of c irc ular figure, and thea i r ma ss es se t in mot ion by the s t r in g take a spheric shape. These

5. Rameau had s ta ted , mistakenly, t h a t a vib rat ing s t r i ng w i l lcause t o v ib r a t e o the r s t r i n gs which a r e i t s r i u lt i pl e s i n l en g th a tt h e i r resp ec ti ve fundamentals. This would imply a se r i e s of "undertones"i n invers e proport io n t o th e s e r ie s of "overtones!' Accordingly, a funda-mental C would generate F , A ~ ,C . I n t h e D & o n s t r a ti ~ a [J], Rameauadmits th a t t h i s observa t ion was e r roneous; th e longer s t r i ng s v i bra tei n segments corresponding t o t h e unison of the e x c i t i n g s t r i n g . S t i l l ,he continue s t o der ive minor harmony from t h e pro por tio n 6:5:4 andmaintains that while major harmony i s a d ir e c t product of nature, minorharmony i s " indica ted by na ture" [B, 2361.


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THEORIES O F GIUSEPPE TARTINIphenomena can be bes t exp ress ed by the circ le which is the most per-fect of a l l curv es [_T, 211. The c irc ula r l ine presupposes the exis tenceof a stra ig ht line in the radi us , which, upon completion of the cir cu m -ference produces the diam eter [?_,211. The dia me ter , placed tangentto the circ um fere nce , becom es the sid e of a squa re which ci rc um -sc ri be s the ci rcl e (s ee Fig. 1, p. 32).

F r om this premise , Tar t in i a t tempts to demonstra te the ha r -monic nature of the c irc le and to der i ve f ro m i t the basic pr inciples ofmaj or, minor, and dissonant harmony, and chromat icism; and to provethat the lim it of consonant harm ony is to be found in the senario.

Here i t is necessa ry to in t roduce some of Ta r t i n i t s a r i thmet icp r e m i s e s . In the Tra t ta to pr em ess o to the Tra t ta to he s ta tes tha t,given the duple ra ti o 60:120, the harm oni c mea n will be 80, the ar it h-met ic mean will be 90, and the coun ter-h arm oni c mea n will be 100.T h e r e is a mean miss ing, the geo met ric mean, which Tart ini est ab-l i she s a s 84 [T, 11.

The nat ure of the geo met ric mean is that the squ are of the meanis equal to the product of the ext re me s o r that the differences betweenthe ex t rem es and the mean a r e in the sa me ra t io a s the two success ivera t ios which fo rm the propor t ion. F o r example , 2 is the geometr icmea n between 1 and 4. In consequence, the square of 2 is equal to thepro duct of 1 x 4, and the differ ence s between 1 and 2 and 2 and 4 a r e inthe ratio 1:2 o r 2:4, the two ra ti os that for m the proportio n 1:2:4.In Ta rt in i' s proportion 120:84:60, th is does not hold. This pro-port ion, reduced to i t s s imp lest te rm s is 10:7:5. It is obvious that thedif ferences between the m ean and the ex trem es a r e not in the s am eratio since the mean does not produce the same ratio with both ex-tre me s; besides , the s quare of 7 is 49 while the product ofth e ext re me sis 50. Ta rti ni adm its thi s when he sa ys that we cannot have a notion ofthis mean a s geometr ic because i t cannot be express ed by integers butonly by a line [T , 1-21? Thus he ma ke s a distinction between what he- - .-te rm s the complete i r ra t ion al geom etr ic mean and the incompleterational[z, 21. 1he f or m er would b e mthe la t ter ism (7) . Tar tin i ope rate s with the second one. Thi s isnot a valid prem ise , but Tartini, through this and oth er such approxi-mations, a t tem pts to warp geometry to sui t his sys tem.Another te rm which we encounter in Ta r t in i ' s theory is the d 2 -cre te geome tric proportion. By this Tar tin i m ean s a proportion inwhich the product of the harm oni c and the ari th me tic m ea ns is equal tothe product of the extremes. In this way, the irr atio nal geo met ricmean that l ie s between the harmonic and ar i thmetic means can be ex-press ed d isc re te ly in te rm s of ra tiona l in tegers . Fo r example , theprop or tion 12:9:8:6, of which 9 is the ar i thm etic me an and 8 is the ha r -monic mean, is a dis cret e geo met ric proportion sin ce both 12 x 6 and9 x 8 equ al 72.

T a r t i n i t s f i r s t g oa l is to prove the harmonic n atur e of the circle .Taking the ci rc le AB (Fig. 3, p. 42) with the dia me ter di vi de dr at io n-ally at any point, le t us say x, Tar t in i a t te mpts to show that xy will bethe harm onic mea n and xz the ari th me tic mean of the rati o Ax:xB into

6. The l i n e t o which Tar t in i re fe rs is th e diagonal of a squarewhich i s always i r ra t i on al s ince it i nvolve s a l e ng th equa l t o w .


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Figure 3.


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THEORIES OF GIUSEPPE TARTINIwhich the di am ete r AB is divided.

If, Ax:xB = 3:7then: Ax2 = 9 -B ~ = 49G2 = 21 (se e footnote 7 )

G2 = 25 ( beca use it i s r 2 ) .Expanding the ratio 3:7 (by multiplying by 5) we ar r i v e a t the dis 3r etegeometric proportion 15:21:25:35, of which 21 is the harmonic meanand 25 the ari thm eti c mea n [T , 22, proposition 111. Th is s e e m s to in-dicate that the length of the perpe ndicu lar dropped f ro m any point onthe diameter to the circumference of a circle is the harmonic mean ofthe ratio of the line segments into which the diameter is divided by thepoint on the diameter [T,391.Tar t ini ' s e r r o r in this proof l ie s in the expression ofthe discretegeom etric proportion of the ra tio 3:7. He mult iplies al l the te r m s byfive except the harmon ic mean. If we redu ce the dis cr et e geo metr icproportion to it s orig inal f orm , then 15:21:25:35 be co me s 3:21/5:5:7.However, 2115 does not rep re se nt the segm ent xy which is actuallya for which 2 I V 3 i s a close enough approximation to make thee r r o r le ss evident . Tart ini se em s to have been vaguely aware thatsomething was out of o rd er in his ex pre ssi on of the ra tio 3:7 a s theproportion 15:2 1:25:35 since he a tte mp ts to dem onstra te, la te r in theTrat tato , that the propor tion 15 :2 1:25:35 is a num erical express ion forthe lengths of the lines in the proportion Ax:xy:xz:xB [T, 241.

The second proof of the harm onic natur e of the c i r g e is throughthe sa m e figu re with the chord Ay and the hypotenuse Az added to it(F ig . 4, p. 42).Given the ci rc le AB, with the diam eter divid edra tiona lly at x.Ta rti ni sa ys that Ay will be the harmo nic mea n and Az the counter-harmonic mean of the ratio Ax :xz.

If, Ax:xB = 5:9then: Ax2 = 25-B ~ = 81

-2 = 45XYz2 = 49 (b ecaus e it is r 2 )By the theorem of pythagoras? Ax2 + F2= Ay2(70) and z2+ Ax2 =-Z ~74). z 2 : z 2 70:74, which, reduced to the simplest t e rms , is35:37. If AX:^ i s expressed a s a discrete geometr ic proportion, we

7. 5 i s t h e p e r p e nd i c u la r dropped from th e v e r t e x of t h e r i g h ttr ia ng le AyB,>ence th e square of 5 i s e qu al t o Ax t imes xB. Subs t i --u t i n g 3 f o r Ax and 7 f o r xB we have 3 x 7 = 21 which i s the square ofa. 8. The sum of th e squa res of t h e si d es of a r i g h t t r i a ng l e i sequal to the square of the hypotenuse.


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ALEJANDRO ENRIQUE PLANCHARThave 30:35:37:42, of which 35 is the harmonic mean and 37 the counter-harmon ic mean. Th is again s ee m s to indicate the harmon ic nature ofthe circ le; and furt her , that the nature of the squ are se em s to be bothar i thm etic and counter-harmonic [_T, 22ff, proposition 1111,

Again he re , a s in the prev ious proof , Tar t i n i1s e r r o r l ie s in theexpression of his t e r m s a s a propor tion. If we reduce the proportion30:35:37:42 to i t s or iginal fo rm we have 5:35/6:37/6:7. H ~ r e het e r m s 35 / 6 and 371 6 do not r ep re se nt the line s Ax (m )nd Az (mbut only approxi mate them. Tar tini nev er revi sed his calculations.The Princ ipj contains no calculations of th is so rt , but in the Risposta,published the sa m e ye ar a s the Pr incipj , he s ta tes " the c irc le is a n in -f inite num ber of harmo nic means [&, 141:' Th is placed him in oppo si-tion to al l the mathemat icians who had demon strated that the ci rcl e isthe locus of the geometric mean of the infinite number of ratios intowhich the di am et er can be divided. He is a wa r e o f t h i s when he a t -tem pts to justify his view by s aying that the si ne (xy), a s ge ome tr icmean, contains both the arit hm etic and harmo nic means, the arith met icat the point where the sin e touches the dia me ter , the harmon ic a t thepoint where the sin e touches the circumferen ce [T, 22ff, propositionIV].

Having arr iv ed a t the conclusion that the ci rcl e is in tr insica l lyharmo nic while the sq ua re is in tr insica l ly ar i thm etic and counter-harmonic , Tar t in i says tha t the diame ter (ar i thmet ic per s ~ )hould bedivided harmonically [x,491. But fr o m a di am ete r divided harm oni -cally, the only sy st em that can be derived is that of consonant harmonyand a s such a pa r t icu la r sys tem. The universa l sys tem, Tar t in ic la im s, must be der ived f r om the chords, complements , s ines , andprotrac ted s in es a s well , which ar e derived f ro m the diameter dividedharmonically u,531 .

Theoretic ally, the diam eter can be divided harmonically gd infi-h m , but in ord er to produce a c losed syst em Tar t ini is force d to finda lim it to the divisions of the dia me ter . As Zarlino does, he s e ts thisl imi t a t the f i r s t s ix d iv isions of the d iamete r ( the s e n a r i ~ )[T,56-7,proposition VI].

Tar tin i atte mp ts to prove the necessity of thi s l im it a s follows.Given the ci rc le and the sq ua re (Fig. 5, p. 42) with the dia met er se tequal to 120 and divided harmon ically through the si mp le st s ix divi-sion s, the sq ua re s of the cho rds and of the hypotenuses in the fig ure(which according to Tar t ini a r e the harmonic and counter-harmonicmea ns of the pro por tion s considered in the preceding proof) will be:

Omitting the squ ar es of Aa' and of Aa" since both a r e the sam e,Ta rti ni c ons truc ts the following proportions using the sq ua re s of theremaining segments:


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- -- -THEORIES OF GIUSEPPE TARTINI

lower extrem e harmonic mean counter-harmonic mean upper ext rem e4000 Ab12 4800 Ab1I2 5200 60002700 A C ' ~3600 Ac1I2 4500 5400

The fi rs t proportion is a discrete geometr ic sesquial tera (as2:3); the second, a discrete geo met ric duple (a s 1:2); the third , a di s-cr et e geo me tric proportion of the ra tio 2:5; the las t, a dis cr ete geo-metric tr iple (as 1:3) [T,581.

Tart in i adds the ext reme s:

and ex tra ct s the square root of eac h total:

Ta rti ni ob se rv es tha t 100, 90, 84, and 80 a r e the m ea ns of theduple ratio 120:60 (which a r e the valu es assigned to the diame ter andthe rad ius of the given ci rc le) ? He has managed to base the entire cal-culation on a cir cle with the diam ete r divided in the ba sic ra ti os of thegenario and has obtained what he cons ider s a unified exp ress ion fo r al lthe mea ns of the duple ratio . Tar tini r eg ar ds this a s proof that the in-te gr al extension and lim it of the ha rmo nic s y st e m a r e in the s e n a r i ~[TI 601.His calculations fai l because two of his pre mis es a r e incorrect .Fi rs t , the tr ue geom etric mean of the rat io 120:60 is not 84b u t m . Second, the squ ar es of the chords, a s we saw in the p re -vious proof, a r e not the ha rmo nic mea n of the ra ti os produced by thesin es and the segm ents of the diam eter that fo rm a right triangle withthe si ne s and the chord.

The limitation of the harmonic sy ste m to the senario c rea t e s an -othe r problem. All accepted consonant inter vals a r e included in these na rio except the mi nor sixth which occ ur s between 115 and 118. Thisexclusion of the minor sixth brought cri t icis m fro m S er re . In both thePrincipi and the R isposta, Tart ini ans we rs that the minor sixth is in-capable of reduction to a ha rm on ic proportio n oth er than the continuous

9. A s we have seen , T ar t i n i i s us ing th e f a l s e premise th a t 84 i sa mean of t h e r a t i o 120:60, which i n f a c t it i s not .

(6


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THEORIES O F GIUSEPPE TARTINIharmonic proportion 1/5 :1/6 :1/7 :1/8 , which req uire s the use of thenon-diatonic seventh aliquot par t, an absurdity since the harmonicsys tem is essentially diatonic [E,461. However, he adds an attemptto derive the minor sixth from the se nario. The harmonic triple, whichis composed of a n octave and a fifth can be ex pres sed ar ithm etica lly a s1:2:3. In this form, the proport ion is indivisible, but the octave ex-pr es se d a s 2:4 ca nbe divided arithme tica lly 2:3:4 and the fifth, exp re s-sed a s 4:6 can be divided ari thme tically a s 4:5:6. The sm al le r rat i osof e ac h propor tion (3:4 and 5:6), when put one above the o ther ( a s mu si -cal intervals), for m the minor sixth (Ex. 6). To Tar t ini i t is obvious

Example 6.

that an inte rval produced entir ely by arith met ic division cannot be in-cluded in the harmonic system [E, 481.

Once Tar tini ha s established the univers al principle of his sy ste min the circ le, he undertakes the derivation of the sy st em from the re -lationships contained within the figures of the circle and the square.The basic interval of any sys tem is the octave since it is the first in-terval of any super part icu lar ser i es that s t ar ts f r om unity. The octaveitself does not form any sys tem since i t is only a rat io, and to have asys tem i t is ne ce ssa ry to have a proportion. Because of this, the oc-tave is considered the a prio ri rat i o of the syst em [x,24; 2, 221. Thesecond rati o of any sys tem is then the rat i o that determines the natureof the system ; in the cas e of the har mon ic syste m, this r ati o is the fifth(1/2 :1/3) . The octave and the fifth ar e then united in the fi rs t harmonicproportion of the system , namely, the triple proportion 1:1/2: 11312, 241.All this is exemplified, accord ing to Tar tini, in the constructionof the circ le ; the rad iu s being unity which, upon completion of t heci rc le, has been doubled in the diam eter and (approx imately) tripl ed inthe circum ference. Thi s produces the ari t hm etic se ri es 1:2:3, which,inverted, produces the harmo nic tr iple 1:1/2:1/3 [T , 27ffl. In the in-version, the ci rcle then repres ents harmonic a s well a s ar i thmet icunity. Tar tini attempt s to approximate this proportion through longcalculations using Archimede s1 me asu rem ents of 7 for the radius, 14for the diam eter, and 22 fo r the circum ference . He rep eat s the cal-culat ion two other t imes, the last t ime using such values a s1,000, 000, 000 fo r the radius , 2,000,000,000 fo r the diam eter , and6,283,185,507 fo r the circu mfe renc e[T, 33ffl.

The de rivati on of the inte rva ls of the senario in the Trattato pre -se nts no difficulty since they can be derived fr om the diam eter dividedharmonically. In the Principi, where Ta rti ni ha s avoided proofs con-cerning the circle, the explanation is m or e involved. He st at es thatthe five inter vals of the se na rio mus t not be derived fr om the divisionof an alre ady existent ratio. He finds the sy ste m based on the th re esounds (1:113: 115) of the monochord st ri ng unsa tisfa ctor y since the


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ALEJANDRO ENRIQUE PLANCHARTbasi c ra tio of the harmonic syste m, the octave (1: 1/21, mus t be ob-tained by dividing harmon ically the ra tio 1:1/3, and the las t inter val ofthe sena rip, the mino r third, cannot be obtained unle ss the fraction 1 /7i s included [_P, 58ffl. He goes then to the sy st em of the third sound andfinds that by the superpart icular se ri es al l the intervals of the sen ari ocan be obtained without derivin g any int erv al fr om anothe r and that theoctave finds its p rope r place a t the beginning of the se r ie s [E 59-60].The derivation of minor harmony follows in bo tht re at ise s a si m i-l a r c ourse which co nsists in deriving i t from the advances produced bythe harmonic division of a line (the diam eter ) a s was explained ea rl ie rin th is essay .However, in the Trat tat o, Ta rti ni explains the relatio n of such asy st em of division, which is essentially arit hme tic, to the cir cle by hisre m ar k that the ci rc le mus t be considered in conjunction with thesquare [T,361 - a projection of the dia me ter of the cir cl e - which isessent ial ly ar i thmet ic .In the Tr atta tg, af ter the explanation of the se nari o, Ta rtini givesthe following rule : "Of the par ts that form the s e n a r i ~ : , 112, 113, 114,115, 11 6, the terms , 1,1/2,1 /4,mu st not be put to8eth er [by themselves]although they are contained within the senarip!' O The explanation ofthe rule is that 1:1/ 2 :1/ 4 fo rm a continuous geom etric proportion anda s such they ar e the potential principle for dissonant harmony [T,621.

In the T rattato, dissonances a r e derived fro m the c ircl e in thefollowing manner:

Given the cir cle AB with the diam ete r divided up to the fir st si xdivisions and having the s in es and the chord s drawn (Fig . 6, p. 46),T ar ti ni deduces the s ys te m s given b g o w _ ~ B ~--equals the pitch C.Fro m the squ ares of KB,A Ab', Acl , Ad' , r e t : (See Ex. 7a).

Fr om the sq uar es of Ba' , El,K 1 , - E ' , Be': (See Ex. 7b).Fr om the squares of aa l , bbt , c c ' , ddl , eel : (See Ex. 7c).

Tar tini then places the f ir st (Ex. 7a) unde r the third (Ex. 7c), and underboth the succ essi ve formations of the sena rio (Ex. 8). From theseTa rti ni ob se rv es that, in the second position, the dissonance of theninth oc cu rs , form ed by the two fifths c -g and g-d' which produce acontinuous ge om etric sesq uial tera [T, 73ffl. F ro m the sa me example,he dra ws the following law: "Generally, any cho rd will be- dissonantthat contains two sim ila r int erva ls of different s pec ies except (m or e bycustom than by reas on) the octave; 'l1 Two fourths, two ma jor thirds,etc. will each produce a continuous geom etric proportion and, a s such,a dissonant combination.

Tart ini 's next concern is to determine which interval is the dis-sonant one when two s im ila r in terv als ap pear in the sa me chord and hesays : "Of two s im il ar int er va ls of different sp eci es , that one will beconsonant which intrinsically belongs to the harmonic [ma jor] o r ar ith -meti c [min or] syste m. The dissonant one will be that which does not[belong] and cannot possibly belong to ei ther of the two mentioned

10. Che nel le pe r t i i n t e e a l i del la sestupla armonica, 1, 112, 113,1/4, 1/5, 1/6, non s i pongano insieme qu est i t r e termini, 1, 1/2, 1/4,. . . . .benche contenuti nella sestupla [_T, 621.11. Che in genere qualunque accord0 m s ic a le sa ra dissonante , ses i seranno nell 'accordo due in te rv al li s im ili d i specie diversa eccetu-a t a (piu per uso che per ragione ) l a ottava [T, 741.


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THEORIES OF GIUSEPPE TARTINIsystems1112

In the sa me m ann er, Ta rti ni deals with the remaining notes ofEx. 7c) , ar riv ing a t the following l is t of dissonances: ninth, eleventh(fou rth), augmented twelfth (augmented fifth), and fourte enth (seventh) .Tar t ini obs erv es that the thirteenth is not present in this syst em , yethe inclu des i t by definition since it contains two four ths and is t he re -fore dissonant by the law given pre viously [T,761. In the case of thesecond he concludes that the dissonant note is not the upper note of theinterval and this he considers a reason against cal l ing the interval a"dissonance of the second" E, 771. He pro pos es no nam e fo r thi scombination.

Any dissonance, he states, should be prepared by a melodic uni-son; the dissonant note and the dissonant in terv al should be pr e-pared by a similar consonant interv al, with the ba ss moving likew iseby the same interval [x,99ffI a s in Ex. 9. The resol ution of di s-sonances should proceed downwards by st ep o r half ste p [_T, 82ffl. It

Exam ple 9. Exam ple 10.

i s curious that Tart ini even reso lve s the augmented twelfth in this man-ne r [_T, 82 and 1301, p roduc ing the r eso luti on shown in Ex. 10.In the Principi, Tartini is conc erne d with the diatonic d iss o-nances. F ir s t he con sider s the possibility of a s yst em in which thedissonant tones a r e arr ive d a t f ro m the rat ios outside the senario whichhave odd numb ers a s denominators; i. e. 117, 119, l / 11 etc. These a r ethe tones produced naturally by the tromb a ma rina and other si mi la rins t ruments .

If the fir st thr ee a liqu otpa rts with odd denominators a r e added tothe senario, we have a sy st em in which the dis sonan ces produced bythe rat ios with odd numb ers a s denominators outside the se na rio wouldhave a consonant basi s in the thre e basic sounds of the monochords t r ing [P, 901. This would explain the sp eci al tre atm en t of the mi norseven th (117) which is not always p repa red, since it stand s in thissyst em a s some so rt of mean between the consonant part i als 1:1/ 3: 115and the openly dissonant ones 1/9 :1 /11 :1/ 13 [E,901. Th is Ta rt in ifinds inconclusive since the odd-numb ered aliquot pa rt s produce al l theout-of-tune tones which cannot be con side red d iatonic and which a r e a l -

12. ...Che de due interval l i s i m i l i d i s p e ci e d i v e r s a sar& il con-sonante qu el l o che int r insecamente appar t iene a 1 s is tema armonico. Sarbil diss ona nte que llo, che i n niun modo appaxt iene, n8 pub appar tenerea ' due sudde t t i s i s t emi [T,751.


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Example 12.

6 :\a ' .3 : iL

Example 13.E=llExample 14. IEaExample 15.EsE3

Example 17.mExample


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THEORIES OF GIUSEPPE TARTINIte re d by skilled playe rs to fit the diatonic sys tem E, 951.

He retu rn s again to the sys te m of the third sound and const ruct supon the fundamental ba ss a diato-ic sca le. Those intervals between

the fundamental ba ss and the notes in theExample 11. sc al e which a r e not included in the s en ar lp(Ex. 11) a r e considered dissonant [z, 88;T , 1121. This re su lts in the sa m e diss o-

nances a s the ones derived in the Trattatoexcept for the augmented twelfth which doesnot app ear he re since i t cannot be considereda diatonic dissonance.

After a discu ssion of the u se of theword "harmony" by Greek theo rist s , Tar t ini

concludes that since the Greeks did not have simultaneous harmony,thei r meaning for the word must have been that of success ive c o n s o uharmony [x,143; 2,49ffl. Th is suc ce ssi ve consonant harmony he de -r ives f rom the bas ic rat io of the sys tem , the duple, with it s h arm onicand ari thmetic means. Thi s disc rete g eometric duple can be expressedmath ema tically a s 6:8:9: 12 and m usica lly a s in Ex. 12 (p. 50). The to nesc and c 1 a r e the extr em es of the duple, g is the harmonic mean, and fthe ari thmetic mean e,70ffl.

Inthe music of his time, su cce ssive consonant harmony is r ep re -sented by the t hre e cadences that can be derived fro m the duple. Thefi rs t and more perfect is the harmo nic cadence fro m the harmonicmean to the extr em e (Ex. 13). The second and le ss perfect is theari thmetic cadence from the a ri th me tic mean to the extr em e (Ex. 14).The third and leas t perfec t combines both mea ns and, in s o doing, bothnatures and is called a mixed cadence by Ta rti ni (Ex. 15). By order-ing the thr ee ca dences according to their d eg re es of perfection and en-closing them with the extr em es of the duple, Tart ini a rr iv es a t thebasic ba ss progression for the establishment of a key [T, 107; 2,741(Ex. 16).

F ro m the union of successive and simultaneous harmon ies, T a r -tini derives the scale, which is the ba sic princ iple of melody. Theunion of both harmonies is acco mpli shed through the us e of the"organic" formula, which is in turn derived from the senario. Thesen ario a s we have seen, can be expressed in i t s essence in the thre esounds of the monochord stri ng, but it m ay be exp ress ed in yet anotherway by taking only the thr ee upper tone s (Ex. 17). It is from this formof the that Tart ini der ives his "organic" formula: 1, 3, 5 withthe 8 added on top a s a duplication of the ba ss- ton e [E,28; T, 101-21.However, this accounts fo r ma jor harmony only. Minor harmony isderived fr om the thr ee lower notes of the arithme tic sextuple [B, 61ffla s in Ex. 18. The union of both ma jor and minor formulae fo rm s ad i scre te geome tric sesquial ter a C-G with i t s harmonic mean E and i tsar i thmet ic mean E~ E, 66ffl.

This orga nic form ula can be found in t hr ee different positions,and he re Ta rtini recognizes the theory of inversion although the wordinversion is nev er mentioned. The fi rs t position is 1, 3, 5, 8; thesecond, 1, 3, 6; and the third, 1, 4, 6 [T,1051. Of the th ree pos itions,


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Example 1 9 . l.EEaExample 20.

Example 21.

Example 2 2 .aExample 2 3 .

Example 25.

Example 26 .


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TH EO RE S O F GIUSE P PE TARTINIhe considers the f i rs t the most perfect in most cases . However, headm!ts that the second posit ion is mo re important in the minor t r iadthah in the ma jor tr i ad [T, 110-111.

Tar t ini examines the natural sc ale produced by such instrumentsa s the tromba marina , horns, t rum pet s etc. and finds that the ra t iosfro m one tone to another a r e not those used in m usical practice. The4th and 7th not es a r e "out of tune" [T , 95ffl. This leads to his der iva-tion of th e sc al e, o r ba sic princ iple of melody, fr om the union of thetwo kinds of consonant harm ony: si multa neou s and suc ce ssi ve.

The esse nce of succes sive consonant harmony is the discretegeo met ric duple exp ress ed a s in Ex. 19 (p. 52). Since the note C isduplicated, ther e a r e only thre e notes that form the esse nce of su cce s-sive consonant harmony: C, F, and G. Upon e ver y one of the se thr eenotes a chord is built ac cording to the orga nic formula, and the thr eeresultant chords a r e the bas is f or the formation of the sc ale [T, 98;-, 71ff] a s in Ex. 20. The re sulta nt sc ale is the Ptolemaic syntonicsc al e (svntonic ditoniaion) E,771.

h he T ra t t ab , Tart in i obse rves that such a scale needs to betemper ed since th ree ma jor thir ds will not rea ch the octave and fourminor th ird s will exceed it . He is quite di str es se d by the fact that mosttempera ments se em to him quite arbitr ary ; he mentions that Vallotti 'stemperament13 is the safest, but offers no solution of his own u, 1001.

The relative minor is derived fro m the la st note of Ex. 7b) (s eep. 46) which is E ~ ,nd which Ta rti ni con side rs to contain inherentlyboth harm onie s (ma jor and min or) because , when placed in it s prop erplace a s ar i thmet ic mean between C and G i t fo rms a major thi rd withG and a mi no r thi rd with C. Thus the distan ce between the .first notesof relative scales is proven to be a minor th ird [ T , 110- 111.

Tart ini gives thr ee fundamental bas ses for the sca le and conse-quently thr ee ways of adding figures to the sc al e when used a s bass .However, in both tr ea ti se s he w arns aga inst the confusion ofth e melodywith the successive consonant harmony ofthe cadences which results inthe misuse of the sca le a s a ba ss melody [T, 1061.

The fi rs t fundamental bas s for the sca le consists in underliningit with the prog ress ion of the or de re d cadences (Ex. 21). This funda-mental bass is completely harmonic in the se nse that it is formedtotally of ma jor harm onie s. The corresponding figuration for the scalewould be a s shown in Ex. 22 [x,1071.

The second fundamental ba ss is dire ctly derived fro m the conceptof the discrete geometric duple C, F, G, c. Tart ini considers the

1 3. T a r t i n i r e f e r s t o t h e Padre Francescantonio V al lo tt i (1697-1780) who was org ani st a t th e Franciscan seminary i n Padua a t th e t ime.Va l lo t t i had s t a r t e d wr i t i ng i n 1735 a t r e a t i s e i n fou r books, =uowever, Vallottif in i s hed only the f i r s t book. The o the r t h r ee were wr i t t en by Va l lo t t i ' sd i s c i p l e and successor, Lu igi Antonio Sa bb at in i (1739-1809). The wholework was publ ished a s Va l l o t t i ' s (under the t i t l e T ra t ta to d e l la modern8musica) i n 1950. I n th e Libro second^, Chapter IV, we fi nd Sab bat ini l sexpo si t ion of V a l l o t t i l s the or i es on temperament.


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ALEJANDRO ENRIQUE PLANCHARTsca le a s two sim ilar tetr ach ord s and the fundamental bas s underl inesthi s fa cto r (Ex. 23). The corr espon ding figuration is a s shown in Ex.24. This fundamental ba ss is harmonic in the sense that i t consists oftwo harm onic c aden ces (Ex. 25), but i t unde rlines , through a str on gcadence, the arith met ic m ean of the disc re te geom etric duple C, F, G,c [T I 109ffI. Another char ac ter ist ic of this ba ss which distinguishesi t f rom the f i r s t is the appeara nce of minor tr iads. The minor tr i ad sa r e on D and A, which a r e presently those tones on which the relativeminors of the tones underlined by the two cadences would be basedIT, 1101.The third fundamental ba ss is produced by the obser vation that,in the fi rs t one given, the re is a tritone relationship between the har-mon ies of the sixth and seventh st ep s (Ex. 26). This relat ion is con-sidered by Tart ini a s tolerable when the sca le is ascending since i t issoftened by the passing of the s ca le fro m a le ss perfect mean (ari thm e-tic) to a mo re perfect one (harmonic). However, when the scale is de-scending, the relation becomes intolerable since th er e is nothing tosoften it . This is inherent in the harmonic natu re which tends to a s -cend and not to descend D, 1311. Ta rt in i ls solut ion is qui te surpr i s -ing since her e he re so rt s to the seventh aliquot pa rt which he calls the"consonant seventhU(Ex.27). This produces a fundamental ba ss that isa re t rograd e of i t sel f , or , in Ta rt in i f s words, a com pleteci rcle [T , 1321.

Tart ini considers the use of the sca le in the ba ss unwise. Theba ss motion should be determined by the succ essi ve consonant harmonywhile the re al m of the sc al e is in the upper voice s.

The derivation of the chr oma tic and enharm onic ge nera is con-tained entirely within the Tratat to. I t app ear s afte r Tart ini gives abrief account of the Gre ek t etr ac ho rds and adds , "1 have rea d a ll thi sin Zarlino, a reasonable man and diligent collector of old things. Whatmay have been the rea son fo r dividing the sc al e in so many te tr a-chords . . . I do not know!'14

The possibility of inter polat ing chr om ati c tones in the diatonicsca le in or de r to provide for melodic chrom aticism finds i t s basicprinciple in Ex. 7c) (se e p. 46), which includes the notes G# and B~with the help of which the c hr om ati c genus can be cons tructe d [x,122ffl(s ee Ex. 28, p. 55).

Through the us e of G#, Tart in i a rr iv es at the following chromat ictetra chor d (Ex. 29). This tetrachord takes place in A minor, relat iveminor of C major,which is the tonality produced by the (given) se na ri o[T,1261. Thus, Tart ini rela te s chrom atic ism to the min or mode.

To deri ve the enharmonic tetra chor d the seventh aliquot par t isinterpolated in the sc ale f ro m a to d f which includes the bb (Ex. 30)[T,1271. Ta rti ni ad mi ts that som e theo rists will complain that thisnew sound is not pre pa red and not resolv ed by descending. To thi s heanswers that the seventh p artia l, having the sa me third sound a s thefir st s ix and being moreove r the true harmonic mean of the fourth a - d f

14. Tutto cib ho letto nel Zarlino, uomo ragionevole, e diligente raccolittore delle cose antiche. Quia sia stata la cagione di divider la scala in tanti Tetracordi... 10 non la so [x, 1211.


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Example 28.

Example 29 .

Example 30.

Example 31.

Example 32.

Example 3 3 .

Example 34 .

I


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Example 3 5 .


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THEORIES OF GIUSEPPE TARTINIis consonant [T, 1281. He cla ims that the m inor seventh before ca-dences need not be pr ep ar ed since the difference between the con-sonant seventh and the tr ue minor seventh is so sma l l that the minorseventh is thought of a s an alm ost consonant inte rva l E, 1291. HishArmonization of the sc al e using the s eve nth pa rt ia l (Ex. 27, p. 52) be -longs the re fo re to the enharmonic genus.

Tar t in i cons iders the augmented twelfth a chromatic dissonance,s ince the G occ urs in the chromat ic te t rachord, and resolve s i t a sshown in Ex. 10 (p. 49) [T, 1301. The mi no r seventh, in co ntr as t withthe minor fourteenth, which is considered a diatonic dissonance, is r e -garded a s an enharmonic dissonance and given the resolution shown inEx. 31 [x,1311.

Tart in i considers three other in tervals and the i r invers ions: thediminished third, inverted a s the augmented sixth; the augmented se c-ond, inverted a s the diminished seventh; and the diminished fourth, in-ver ted a s the augmented fifth. Al l these in terva l s a r e der ived f r om theinterp olated diatonic sc al e of Ex. 28 [T , 1581. In ac co rd an ce with thederivation, Tar t ini cl assi fies the augmented second and the diminishedfourth a s chromat ic in tervals and the diminished thi rd a s an enhar -monic interval [T, 1591. He s ta te s that the diminished seventh is t rea teda s a dissonance al though i t is not always prepared. The augmentedfifth(in contrast to the augmented twelfth) is t rea ted a s a non-essen t ia lcombination sinc e the tone that produces it is not prepar ed and i t a s -cends (Tart ini us es the word discordanza for i t ) a s in Ex. 32. Thediminished fourth is t rea ted a s a consonance (Ex. 33). The augmentedsixth is als o t reated a s a consonance [T, 157-581.The use of such inte rva ls can be d emonstrate d in the sc ale shownin Ex. 34. This scal e is base d on Ex. 7c) (p. 46). Ta rti ni obse rv esthat the sc al e c on sis ts of two equivalent tet ra ch or ds , d-a and a-dl andgives a n example of a sma ll piece composed using the sca le (Ex. 35)IT, 159ffl. Ta rti ni cla im s tha t in Ex. 35 the only disso nanc es a r e the4-3 suspensions a t the cadences. Th e only ot he r chor d that could beconsidered dissonant cannot be dissonant in the inve rsion in which it iswrit ten since the D# cannot be distinguished, eve n in the generation ofthe t hi rd sound, f rom the t rue ha rmon ic no te ~ b 'Ex. 36). Th e funda-men tal fo r thi s chord should then be a s shovl?l in Ex. 37. Ta rti ni con-cludes by pointing out the po ssibil ities of su ch harm onic m at er ia ls indramat ic mus ic [_T, 162ffl.

Rhythm and me ter , l ike harmony, a r e der ived by T art in i f r omthe bas ic r at io of the syste m, the duple. Th is rat io provides dupleme ter. The combination of the duple and the ses quia l ter a for m s thebas i s fo r t r i p l e me te r [x,1141. In the p roc ess of applying met er tothe s ucce ssive consonant harmony a nd to the me lodic voices, rhythmis produced in long and s ho rt ac ce nts (long and s ho rt note-values) whichcorr espo nd to the long and sh or t sy llabl es of prosody [T, 1151. How-ever , Tart in i observ es that mu sical rhythm, par t icularly in t r ip lem e t e r , is not a st ri c t application of syllabic valu es to the pitches, buta stylization of the syllabic value s which he illus tra tes a s shown in Ex.38. The fir st of the two settin gs is a s tr ic t application of the syllablevalues that r es ul ts in confused m ete r, while the second, a styl ization


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Example 39. ,Example 40.

Example 4 1

Example 42.


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THEORIES O F GIUSEPPE TARTINIof the sy llab ic values is clear rhythmically and metrically [T, 1391.

Example 38.

Tartin i mentions in passing that harmony se em s to have a defi-nite influence in determ ining the long and sho rt acc ent s in mus ic, andhe notices the association of downbeats with consonance [T_, 117ffl.

In the di scr ete geom etric duple, ex press ed a s C, F, G, c, Tar-tini se es a bas is for the three clefs [z, 1201 and the ba sic tonal re al mof tonic, dominant, and subdominant D, 146ff; 2, 98ffl.Tartini treats modulation very briefly. A ma jor key, becau se ofi ts harmonic nature will tend to modulate toward s the dominant, amino r key, beca use of it s arit hm etic nat ure will tend to modu latetowards the subdominant. Ta rti ni wa rns agains t "leaps" in modulationsuch a s going fro m a key in flats to a key in sharp s and requ ires thatthe principal key be made c le ar a t the opening and closing of any piece[T , 147ffl.

Ta rt in i 's notions about history, and about Greek theory, a r esc att ere d throughout the two trea tis es. Greek theory, for him, con sistsin a statem ent of the different divisio ns of the tetr ach ord [T, 1211 andan explanation of the adaptation of the tetr acho rd s to the st ri ngs of thekithara L_P, 531.

Ta rtin i believed that the Greek s had an octave-species base d onthe arith me tic and the harmonic divisions of the octave thus formin gtwo tetr ach ord s (Ex. 39). Because of this, he con sid ers his diatonicsystem a s essent ia lly the same a s the Greek [P, 49ff; T 143ffl. Sincethe Greek s did not have simultaneous harmony, the consonant in te rval swere the leap s of the fourth, fifth, and octave, while any in te rvalsm all er than the fourth was considered dissonant [T, 1431.

Tartini con siders the ecclesiastical modes a s combinations offou rths and fifths. The authentic modes, in which the fifth is at thebottom, he cons iders a s harmon ic modes, while the plagal ones withthe fourth on the bottom a r e cons idered a rith me tic modes. However,he finds the organization of the modes unsatisfactory since the har-monic and arithmetic divisions of the octave do not fall within the sameoctave (Ex. 40). Tartin i then proposes what he consid ers an improvedand "natural" organization of the modes based on the harmo nic andarithmetic divisions of the octave c-c', f-f \ and g -g', which a r e de-rived from the disc rete geom etric duple (Ex. 41). To these s ix modes,Tar t ini adds another series of modes based on the arithmetic and har-monic divisions of the octave s A-a, d-d ', and e- e' [T , 1371 (Ex. 42).Ta rt in i had little understanding of the organization of the mod esand his new modal sy ste m i s nothing but the m ajo r and minor sca le sthat fo rm the common tonal re al m of the music of his time.In the Tratth to he use s hexachord nomenclature to r ef er to anygiven note; this i s dropped in the Princiui. However, in the R is m st a,published the s am e ye ar a s the Principj, Tartini attacks Se rr e ' s use of


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ALEJANDRO ENRIQUE PLANCHARTthe syllable gi which, according to Tarti ni, shows S e r r e l s ignorance of"the masterful Italian solfeggio:' that is, hexachord solmisation [R,70-711.

Ta rti ni ' s opinions on the affections a r e found in the fifth chapterof t he T ra t t a t~ . Rega rding the affective power of music, Ta rti ni isquite skeptical. He doubts that music, a s practiced in his t ime, canexpress any precise affection. Gree k music could do so since it wasmonophonic and since pa rtic ula r p assions had th ei r own ran ges , speed,dynamic qualities, etc. A single voice, with perh aps a single ins tru -ment, can most effectively communicate an affection [T, 139ff, 149ffl.Ther efor e, som e of the old Italian dr am as which consisted almo st en-tire ly of recitative could mo st cle arl y communicate an affection [T,1351. However, in mo st mu sic of his day which employed many voicesthi s would hardly be possib le because the different pitche s, motion,etc., of the different voices would obs cur e any attem pt to communicateany given passion clea rly. The effect will be a ve ry gene ral one lean-ing toward one affection o r another [z, 1411.

The two bes t me ans by which to induce an affection in a he ar e r,Tarti ni concludes, a r e melodic orname nts and harmony itself [x,148-91. Tar t ini rem ar ks on the relat ion of majo r harmony to fi re andjoy on one hand, and minor harmony to melancholy, languor , andsweetness on the other. He cl aim s that a harmonic cadence c ar ri esthe ch ar ac ter ist ics of the ma jor harmony while an arith met ic cadencehas the cha ra cte ris tic s of mino r harmony. Ta rti ni feels that a studyof the affec tive value of the i nt er va ls could begin with th is and continueto account for those inter vals that a r e bette r suited to each voice. Heobs erv es tha t while oct aves and fifths sound quite sat isf act ory in thebass , even if use d constantly, they a r e quite repugnant in the uppervoices m,153-41. Such con side rat ions, ac cor ding to Ta rt in i, wouldre qu ir e anothe r tr ea ti se writte n by one who, like the Gree ks, would bea music ian, philosopher, and poet.

BibliographyP. Del principi dell'a rm oni a music ale contenuta nel diatonic0 genere,disse rtaz ione di Giuseppe Tart ini. Padua: Stamperia del Seminario,1767.L. Le tt er a del defonto Sig. Giuseppe Tart ini al la Signora MaddalenaLombardini inserviente ad una importante Lezione Der i Suonatori d iViolino. Venice: Colombani, 1770. Tr an sla ted into Eng lish by Dr.Burney. London: Br em m er , 1779.R . Risposta di Giuseppe Tar tin i alla crit tica del di lui Trattato di mu- sics di Monsieur l e Se rr e di Ginevra. Venice: Antonio Decastr o, 1767. A. Traitk des agrkments de la musique. . . composd par le cdlebre Giuzeppe Tartini.. . et traduit par le sigr. P. Denis. Pa ri s: Chez llauteur, [1782]. Also a vailable in English in the Jou rna l of Rese ar ch in Music Education, IV: 75ff.


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THEORIES OF GIUSE PP E TARTINIT. Trat tato di musica second0 la Vera scienza de lllar mo nia. Padua:Stam peria del Seminario, 1754.1. Eul er, Leonhard. Tentamen novae theoriae musicae ex ce rti ssi m isharmoniae principis dilucide expositae. . . Petropoli [St. Pet er sbu rg? 1:Typ. Academiae, 1739.2 . Fktis, F. J . Biographie universelle des musiciens et bibliographiegdndsale de la musique. Volume VIII. B ru ss el s: Leroux , 1835-44.3. Rameau, Je an Philippe. Ddmonstration du principe de l'h arm on ieservant de base 3 tout l l a r t musical thedrique et pratique. Par i s :Durand, 1750.4. - - - - - Nouvelles rdflexions s u r le princ ipe sono re [bound with theCode de musique pratique]. P a ri s : Im primeri e Royale, 1760.5. Shirlaw , Matthew. The Th eory of Harmony. London: Novello and Co.[1939].6. Stillingfleet, Benjamin. The Pr incipl es and Power of Harmonx.London: J. and H. Hughs, 1771.7 . Vallotti, Pa dr e Franc escan tonio [and Sabbatini, Luigi Antonio].Trattato della moderna rnusica. Padua: I1 me ssa gg ier o di S. Antonio,1950.8. F e r r i s , Joan. "The Evolution of Ram eau's Harmonic Theories:'Jour na l of Music Theory, III(1959):231-56.

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The Theories of Tartini