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7/25/2019 Language of Ratios, Partch
1/10
CHAPTER FOUR
The Language
of Ratios
Frequenciess Ratios
Or
rno unNy characteristics of a
musical
tone the most important for
musical
science s
the
number
of
pulsations
it
creates, n a
given length
of
time, n the air
in whi ch it is heard.
We call these
ulsations
ibrations,and
u'hen we use the
sixtieth part
of a
minute
as our measureof time, it
is
now
customary to call them cycles.
The
number of cycles-per second-de-
termines he
pitch
of the tone.r
A systemof music
is
an organization of relationships
of
pitches,
or
tones,
to one another,
and these relationships are inevitably
the
relationship
of
numbers. Tone is number, and sincea tone in music is always heard in re-
lation to one or
several
other tones-actually heard
or
implied-we have
at
least o numbers o deal with: the number
of the tone under considera-
tion and the number of the tone heard or implied in relation to the first
tone. Hence, the rat io.2
It is
u'ell
to
plunge
at once nto ratio nomenclature
nd
to disregard he
more familiar "A-B-C
"
terminologv by
which
the ratios in our con-
ventional
scales
re expressed.
he
advantages f doing so, n openingnew
tonal vistas,
n getting
to the analyzable root of music
and
the core
of
the
universe of tone, are
inestimable. If
time is taken out to translate each ratio
into what is assumed to be a synonymous word
value, these
vistas
are
dimmed or
lost
altogether, and the
values, u'hich
are not synonyms, are
nevertheless convicted
of
fraud
by alleged
synonyms. After
hearing a
"major
third"
on-the organ or
piano
or
some
other instrument
with
tem-
pered intonation,
this
interval
becomes ixed in the mind as a
pretty poor
consonance, t
leastby some
comparisons, certain modern composer, ne
Ifhe
use ofthe
word cycles
prevents
any
confusion as between
vibrations
in whole and
halves
of vibrations, thc latter
being
the French manncr of
indicating frcquencies.
'1In
Euclid's words,
"all
things which consist of
parts
numcrical, whcn compared
togethr,
arc subject to the
ratios
ofnumber; so that music al sounds or notes
[tones]
compared
together,
must
consequently be in some numerical ratio
to
each othcr."
Da\\, L.lk$,2t265.
76
of
numerous
is, of not be
just
interval
If ratios
language
so
of musical s
results are m
um
surround
ing and ted
any case.
Before pr
instruments
tones ested
this
precise
facile
hinki
and functio
The han
an exact ma
tave") abov
This additio
since
200 cy
cycles.
To g
the 2/1 belo
constitutea
with
the
sam
for
example
relationship
This2tol
why they
ca
result,
ust
a
"octave" at
and
its
doub
same
quality
(200
to 600
3Hindemith
4With
cents
which
represcnt
7/25/2019 Language of Ratios, Partch
2/10
for
of
vibrations,
an d
e, it is now
or tones,
relationship
of
re-
r considera-
the
our con-
opening
new
of
the
tio
ar e
hearing a
with tem-
pretty poor
one
in whole and
together,
together,
THE LANGUAGE
OF
RATIOS 77
of numerous
possible
examples, convicts
his
"thirds" of
inexactitude-that
is, of not being
exact
widths-for no real reason
except
his confounding of
just
intervals and
tempered intervals3
(see
page
i53).
If ratios seem a
new language, let it be said that it is in actual fact a
Ianguage so
old
that
its
beginnings as an expressionof
the essentialnature
of
musical sound can only
be conjectured. In learning any new
language,
results are more immediate
if
a
total
plunge
is made, so that the
new
medi-
um
surrounds and
permeates he thinking; and it is no more time-consum-
ing and tedious
than translation, which frequently cannot be exact
in
any case.
Before proceeding
to a study of the Monophonic intervals, experimental
instruments
will
be
described on which each ratio can be computed and
its
tones
tested. For the
present, what is required is a facility in thinking
with
this
precise
article-this
sine
qua
non of musical
structure-and for such
facile
thinking it is necessary o have a
thorough
understanding
of its nature
and functions.
The handling and
consideration of tones s, by virtue of their
vibrations,
an
exact mathematical
process. f
a
tone
makes
200 cycles, he 2/l
("oc-
tave")
above
t makes
400
cycles,a
doubling of 200, or 200 more cycles.
This additional
200 cycles s not, however, constant for the
2/1
measurer
since 200 cycles added to 400 do not give the 2/1 ("octave") above 400
cycles. To get the 2/l above
400 we multiply
by
2 and,,conversely, to
get
the 2/1 below
400 we
divide by
2. The cycles, or frequencies,of two tones
constitute
a ratio, and
it has long been established hat two
pairs
of tones
with the same
ratio-200
cycles
o 400 cycles and 400 cycles o 800 cycles,
for example, both
2/1's-are accepted by the ear as identical musical
relationships.
This 2
to
1 relationship is a constant one. Musicians
frequently wonder
rvhy
they cannot
add the ratio of one interval to another to
get
the correct
result,
just
as they add a
"perfect
fifth" and a
"perfect
fourth" to
get
an
"octave"
at the
piano. But the fact is that Nature does not offer one tone
and
its
doubling
(200
to 400) as a
given quality
of relationship, and
the
same quality of relationship in two tones which are not a ratio of doubling
(200
to 600, for example ).4
sHindemith,
CtuJt
oJ Mudeal
Conposition,
1t78,
rwith
cents,etplained
at
thc
end ofthis chap ter, t is possible
o
add and subtract
quantities
which
represcnt atios.
7/25/2019 Language of Ratios, Partch
3/10
GENESIS
OF
A
MUSIC
The2tolrelat ionship,basedonthefactorof2'appl iesonlytopro-
gr*ri""
L'pi,.tt
by
2/1's.^ihere
is
of
course
much
more
to
the
calculation
of
musical
intervals
than
successive
/1's'
But
if
the
factor
of
2 applies
to
the
2/1.
then
certain
decrements
of
the
factor
of
2 must
apply
to
intervals
;;.;;;;;;";u.J
fro*
u 2/1,
or-increments
of
the
factor
of
2
starting
;;i.
i; a;ust Intonation the comPutations are v^ery imple' sinceeach
,*ff-nrr-U.i
ratio
is itself
a
patticular
measure
of
the
factor
of
2'
The
i"[*"i]oolzoo,
or 312
fit'si
"perfect
fifth")'
for
examPle'
an
interval
,ru..o*e.
than
a
2/1,
represents
a certain
measure
of
the
factor
of
2'
and
the
,"i
ZSo
zOo,
or
5/4-(just
"major
third'')'
an
even
smaller
measure'
Ob.riorrrly,
then,
if
it
is
impossible
to add
fre
quencies
by
some
constant-
ZOO
"u.t..,
fo,
.*ample'
which
was
suggested
bove-to
get a
series
of
2/1's'
t;;ilil;..tbi.
to
^aa
f'equencies
by
constants
ogtt
any
series
of
in-
i.*Jt
.t
ri-
than
2/1.
Positively,
if a
sequence
of
2/1's
is
de
ermined
by
ii.
i"",-
.f
2, by
multiplication,
then
any
sequence
of
intervals
narrower
ir,""i/r
i.
a.t".-ined
by
their
respective
proPortions
of
the
factor
of
2-by
-,-ri,lii"u,io.t.
Therefore,
when
any
two
ratios-two
intervals-are
to
be
addei,
muttiPlY
heir
ratios'
--
frc
ttt"afi"g
of
small-number
ratios,
representing
the
intervals
to
*frl.tr
it.
ea,
i, irost
responsive,
nvolves
nothing
more
than
simple
multi-
ol icationanddivis ionofimProPerfractions.onlywhentheexpedientof
H;ia.'i;
;i.,trod.,..d
ao
tt't
computations
hcome
at all
complicated'
*i.'"
i.g"ti,ttt".
are
employed
to
produce.
deliberately
chosen
irrational
oercentJees
of
the
factor
of
2
(seepage 101)'
*^;;;:i;.
;i;
,.1*r,..
of
d"e
intJrvals,
somewhat
at
random
in
pitch'
rr".,ine
a
lo*et
consiant
of
1200
cycles,
the
five
ratios
being:
2400/1200'
iaoiiii
zoo,
ooo/1200,
500/1200
2000
1200'
n their
owest
erms
hese
;:;;:;:;.:;i-,
ili,
+it, stq,
s/:'
tris
same
eries.or
erationships
ourd
i".il *.rr b.'.o.t.id"..d *ith u" ttpptt
t9"t 1nt
9l'^YI'^2400
vcles;
he
i.",i*
*."rt
th.r'
u.
24oo
rzoo,
i+oo
1tooo,
2400/1800'240011920'
i-+oO7f
oO,
-rti"h,
reduced
o the
owest
ossible
erms
re'.again:
/1'3/2'
+tii'it+,
i/1.
rtt,r.
a
musical
atio
rePresents
he
relationship
etween
..r.i.r,
,id,r..d
to
its
owest
errns
t
is n
abstract
uantity
applicable
o
^ir,,l*..ioi,.h
in
the
otal
musical
cale,
nd
n
this
orm
ts
primacy-
il i;.ii;frJ;;nk
in
the
significant
esources
f
music-is
he
more
mani-
fest,
as
will
be
shown
in
the
next
chapter'
i i ,s in . . , in theconceptof theinterva l2/ | (the. .octave ', ) , the lower
number
is
e
numbcr
is
e
than
a
2/1;
such
as
3/2,
sequentlY,
w
halved or th
16/5
is brou
nithin
a
21
expressed
n
A
syste
cated
n ev
s,vmbols-r
211,
and
th
Nlusicians
given
"A"
i
any
given
9
cians
will
fi
should be e
But
such
a
seven
2/1's
its
own,
an
the
numbe
two
or
mo
up
and
do
2,i
1,"
"Hig
It
is co
built
up*'a
and
this
P
is
specifica
ities,and i
cation
is
s
reverse
wi
Itonocho
SupPo
a
mark
on
rhat
ndic
7/25/2019 Language of Ratios, Partch
4/10
only to pro-
calculation
to the
to intervals
of 2 starting
since
each
of 2. The
an
interval
of 2,
and the
measure.
constant-
series
f 2/1's,
of in-
determined
by
narrower
of
2-by
to be
to
multi-
of
complicated,
in pitch,
2400 1200,
terms
these
could
cycles; the
2400
1920,
2/1,3/2,
between
to
its primacy-
more mani-
the lower
THE
LANGUAGE
OF RATIOS
; )
number
is exactly half the upper
number,
any ratio in
which the lower
number s ess han
half the
upper,
such
as 5/2, represents
n
interval
wider
than a 217; and
one
in which
the
lou,er
number
exceedshalf the
upper,
such
as
3/2,
represents n interval smaller,
or
narrower,
han
a 2/1. Con-
sequently, when an
interval is wider than
a
2f 1,
the upper
number may
be
halved or the
lower number doubled to bring i t within
a
2/7.
The ratio
16/5 is brought
within a 2/1by
rvriting
t
8.,/5, nd the ratio
5,/2
s
brought
lithin a 2/1 by
rvriting it 514. Nearly
all
ratios
in this
exposition
ar e
cxpressedn the less han 2/l form.
A s)stemof music
s determined or
one
2/1;
the system s
then dupli-
cated n everyother
2/1, aboveor below, hat is
cmploycd.
Consequently,
svmbols-ratios in this exposition-are
used o denote the
degrees
f
one
2 1,
and
the
symbols
are
repeated in
every 2/1
of
the
musical gamut.
\lusicians are accustomed
o this
idea;
the
"octave"
above
or below
a
given
"A"
is sti l l
"A."
The situationhere
s dentical;
a
2/1
above
or below
any
given 9/8 is still 9/8. Onl y the
physicists
who
are not pract ising
musi-
cians will find
this
objectionable,
since,acoustically,
a 2/1
below a given
9
8
should be expressed9/16, and,a
2fl
above 9/8 should
be expressed
9/4.
But such a
procedure would mean
that every one
of the approximately
seven
2/1's of the
common musical gamut would
have
a
set
of symbols
of
its own, and when forty-three degrees-ratios-in a single 2/1 are involved
the number of total
symbols would
be unwieldy. The
relative positions
of
tuo or more 2/1's,
when tables
or diagrams or examples nvolve
ratios
both
up and down from
a given 1/1,
are
indicated
n this
exposition
by
"Lower
2,'1,"
"Higher
2/1,"
"Third
2/1," and.
Fourth
2/1,"
etc.
It is
common
practice
musically to
consider ratios
(intervals)
as being
built upwards
(with
the larger
numbers above)
from a lower
constant,
and this
practice is followed
throughout this
book except when
the reverse
is specifically ndicated.
Such ratio symbolism sjust
one
of
several possibil-
ities,
and
is a matter of arbitrary choice;
the
reverse
orm in practical
appli-
cation is synonymous, and in
order
that
this fact may not
be obscured
the
reversewill
be
indicated rom
time to time.
Monochord rocedures
Suppose
we have
a metal
string
stretched
across wo
bridges, and make
a mark
on
the wood beneath
that divides the string in
half, then
a mark
rhat
indicates
a third ofthe string, and finally
a mark that
divides the
third
7/25/2019 Language of Ratios, Partch
5/10
80 GENESISOF A
MUSIC
in half, or
into sixths of the whole string, as shown
in
Diagram
1. Suppose
the whole
string makes 100 cycles when in vibration;
if
a
third
bridge
is
placed at the halfway mark, either half of the
string would then give tones
of200 cycles,and
ifthe bridge is
placed
at the one-third mark and only one-
third of
the string
set n
vibration the
resulting
tone
would make 300 cycles.
Thus the relationship of the half to the whole is 200 to 100,or 2/1 ; and the
Drlcnlu
1.-Trre RuauoNsmp or Cvcr-rs o Pasrs or StnNc
relationship
of the third
part to
the
half is
300
to 200,
or
3/2. Each of these
ratios representsboth
a given tone and an interval between two tones.
The
rario 3f2, via
the
agent
3, represents the higher tone of that
relation-
ship; at
the same ime
it represerits he interval from 3 to 2, or 3
2.
For lack
ofa
better
term this concept
might
be called
"upward
ratio thinking" from
a
lower cotslant,
the number 2,
it
3/2
(200
cycles),
representing the
con-
stant.
Now,
without
regard to
cycles,
et us think in terms of
parts
of a
string
length,
the ancient monochord
proceduie.
If one-half the string represents
three equal parts (each a sixth of the whole string) one-third of the string
represents
wo equal
parts.
When
sounded, one-third represents
2 parts
and
one-half
represents3
parts,
or
the ratio 2/3. The ratio 2/3, via the agent 3,
represents
he
lower tone, while at the same time
it
represents
he interval
from
2 to 3, or 2/1, exactly
the same nterval as 3
2-
lr{e
thus
see
hat the
numbers
of vibration are
inversely proportional to the string
lengths
(see
page 99 for
reservations regarding this
generalization). And the mental
processof considering
ratios as
parts
of a
sounding
body,
rather
than as
vi-
brations, or
cycles, as
here presented, s essentially
"
downuard atio think-
inq" lrom
rrnc of
thc
These
s
ncl ing
o
i.
arb i t rar
: r r rar s thc
r
-1.
2. 8
.r ci 5
-l
rc
r , ,
11).4
rn l ( Ival
to
i r l icc l
ntcr
rnr l
an
im
Exprcs
l -nqths,
th
: i r lc the o
r :Lro-rvou
icnc. but
rr 1 1. Be
The sc
rhe upper
asccnds n
rhe
same
r
aiucs, h
"Tttinkin
Upon
rts
comPr
numbers
In the
ar rhc
up
l:cnce
he
:ntcrval
f
* i th in
th
51
44
54
7/25/2019 Language of Ratios, Partch
6/10
L Suppose
bridge is
then give
tones
and only
one-
300
cycles.
or 2/1
;
and
the
of
these
two tones.The
relation-
3
2.
For lack
thinking"
from
the
con-
of a string
represents
of
the string
2 parts
and
via
the agent
3,
he interval
see hat
the
lengths
(see
the
mental
than
as vi-
ratio
think-
TIl
E LANGUAGE OF RAT]OS
BI
ing"
from an upperconstant, the number ? in
2 3-representing
the upper
tone
of the
re ationship-being
the constant.
These
very elementary examples
are essential to
a thorough under-
standing
of the
Monophonic
procedure,
where he
under number
of a
ratio
is arbitrarily
chosen o represent1/1, unity,
or the Prime
Unity, and
is
aluar
s
the ower of thc t\,\ 'o atio
tones,heard
or impli cd. In the ratios
5/4,
1.
3, 3
2,
8/5 the under numbers4,
3, 2, and
5
represent
he constant1
1,
and
5/4
rcpresents single one
upuard from 1/1
(and,
n
implied
interval
ro lil),4/3 represents single tone uprvard from 1/1 (and an implied
inlcrval
to 1/1),3/2 represents single
one upward
from 1/1
(and
an im-
plicd interval
to 1/1), and 8/5 rcpresents
single one
upward from 1/l
tand
an implied nterval o 1/ l) .
Expressed n the
"dounuard thinking" manner,
as
parts
of
string
lcngths,
hesc atios .r 'ould e 4/5,3/4,2i3,
5 i8t and 4,3,2,
and 5-this
time the
over numbers and the higJher ones,
heard
or
implicd,
of each
ratio-would
again reprcsent1
11
And
each ratio rvould represcnt
a single
tone,
but dou nuard nstead
of upward from 111,
and an implied interval
to 1/1. Below s
a
schema
of theseexamples:
]
8
(higher
ones
f
intervals
pward rom 1/l)
? 2: t
t t '
the Prirne
UnitY
3 8 (lower ones f intervals ownward rom 1/1)
The scale
of four
tones s
designed
o be identical in the two
processes;
rhe
upper
scale
ascends rom 5 to
8
(from
left
to right); the lower scale
ascends
nly
ifread in reverse,
rom 8 to
5
(right
to left). To
achieve exactly
the same
pilches
n
both
scales,
without regard
to synonymous
interval
values,
he lower
scale
u'ould be written:
5/8,
2/3,3/4, 4/5.
"Thinking" in Ratios
Upon
further investigation
of the nature
of ratios we find that each has
its
complement within
the
2/1,
and if
the ratio is composed
of
small
numbers its
complementary ratio is
also composed
of
small numbers.
In
the
ratio
3/2, 2 represents /1,
the lower limit
of the
2/1. The
tone
at tlre
upper
limit
of the 2/1 may
be
represented
y
4
(a
doubling of
2) ;
hcnce
he
interval
from the
3 of 3/2 to this
upper
limit
of the 2/1
is
th e
interval
from
3 to
4,
or
4/3,
which is
therefore he
complement of 3/2
*itlrin
the 2/1i the two intervals
might
be expressedhus: 2:3:4. In the
54
43
54
7/25/2019 Language of Ratios, Partch
7/10
GENESIS OF A MUSIC
ratio
5/4,4
represents 1/l; the 2/1 above
4
is 8; the complement of 5/4,
therefore,
s the
interval
from 5 to 8, or 8/5, and the two
intervals might be
expressed,
:5:8. In
the
ratio
6/5, 5
represents /1; the 2/l above
5
is 10,
and the complement of 6/5
is
therefore the
interval
from
6 to 10,
or
70/6,
or, in
its lowest
terms, 5/3; the two
intervals might
be
shown in this form,
5:6:10.
To find the sum of two
intervals
multiply the two
ratios. The sum
of
5/4
and 6/5, f or example, 5/4x6/5:30/20 ,
which, reduced o its lowest
terms, is 3/2
(a
"major
third" and a
"minor
third" make a
"perfect
fifth").
To
find
the interval
between
t\ /o tones invert the smaller or narrower
ratio and
multiply. For example, to find
the
interval
between
3/2 and
4/3
invert
the
smaller,4/3,
and use
t
as a
multiplier:
3/2X3/4:9/8,which
is therefore the
interval representing
he tonal distance between
the 4 of
4
3
and the of 3
2.
(The
difference between a
"perfect
fourth" and a
"perfect
.fifth"
is a
"major
second.")
To find a
given interval
above a
given
tone is of course
simply
a matter
of
multiplying the two ratios involved; to frnd the same
interval
distance
downward from the same tone, the procedure is inversion and multiplica-
tion. Forexample,a6f5 above3/2
is
arrived at r}Lvs3/2X6
/5:9/5;
and
a
6/5 below 3/2 t hrs: 3/2X5/6:5/4.
Were
Do RatiosFall
on
hePiano?
It is inadvisable to think
of
these ratios in terms
of
piano
keys except
with the most
precise eservations.To
do so without reservations
s
a triple
abuse-of
the ratios, of the
piano,
and
of oneself.One
can
go
crazy trying
to
reconcile
irreconcilables,
but
given
an appreciation of the essentiality of
ratios in understanding musical resourcessome knowledge of the
piano's
discrepancies
may
prove
enlightening.
If in the teaching of simple arithme tic the number 1 was called Sun, 2
called Moon, 3
called
Jupiter,
and
4
called Venus, and
if this procedure
were carried to the
point where
the teachers hemselves o
longer knew that
Sun:1, Moon:2,
Jupiter:3,
and Venus:4, and
forced upon students
the euphemistic
proposition
that Moon*Moon: Venus, because hey
had
learned
it
that
way, we would
have in simple
arithmetic
a
fairly
exact
parallel to the
"Tonic-Supertonic-Mediant"
or the
"C-D-E"
nomencla-
ture
in the teaching
of
the science
of musical vibrations.
And the idea that
Moon*Moon
:
Venus could accurately represent2*2:4
\s
no more awk-
ward,
to
p
vals
"C-F*
In
res
adopt
the
would
say
tone flat i
think
of th
and
the m
tion
and
i
rtould
sa
equal
sem
the pursu
recommc
crepanci
Ellis'
Me
One
m
presente
principle
ander
J.
E
tions J
T
scmitone
ables
he
magnitud
sches.
Th
lish his
n
number
1
subscque
l-'oundar
If
"G"
cents
as
s
rains
400
ccnts.
Th
Thcrefor
irc cxpre
lonc.
sPage
7/25/2019 Language of Ratios, Partch
8/10
of 5/4,
might
be
above
5
is 10,
or 70/6,
in
this form,
The
sum
of
to its
lowest
"perfect
fifth").
or narrower
3f 2
and 4f 3
/4:9
/8,
which
he 4
of 4/3
a
"perfect
a matte
distance
and multiplica-
/5
:9
/5;
and
keys
except
s
a triple
crazy
trying
to
essentiality
of
of the piano's
called
Sun, 2
knew
that
had
a fairly
exact
the idea
that
no
more
awk-
THE
LANGUAGE OF RATIOS
*'ard, to
put
it charitably,
than the idea that the ascending
musical inter-
vals
"C-F *F-C
:
C-C" can accurately epresent 3
X3
/
Z 2 1,
In resorting to the
piano two
procedures
are
possible. First, one can
adopt the negative
procedure
of regarding
ratios as altered
piano tones.
One
would say that 16
9,
for example, is
"F"
one tu'enty-fifth
of an equal
semi-
tone flat
in
the
"key
of G." On the
other hand, it
is quite
as
possible
o
think of the
piano tonesas altered
atios.This is the constructive pproach,
and the more fruitful
one,
since t predicatesan understanding of the
func-
tion and indispensability f ratios. In accordance 'ith this procedureone
rlould say
that "F"
in the
"key
of
G" is 1679
plus
one t$'enty-fift h
of an
equal semitone. f translation
nto conventional
valuesseems esirable n
the pursuit
of
the Monophonic theory,
the second
s
certainly
the
procedure
recommended.
Follorting
the explanation
of cents a
table
of
piano
dis-
crepancieswith the
nearest
mall-number ati os will be
given.
Ellis' Measure
f
Cents
One mori
step n the simple
mechanics f dealing
with ratios must be
prcsented
n preparation or the exposition f
the Monophonic
concepts
nd
principles, namely,
the measure of
musical intervals established
by Alex-
anderJ. Ellis
n an appendix
o hi s translationof Helmholtz's
On theSensa-
tians J Tone.6 his measure s the cent, the hundredth part of an equal
semitone-1200 to rhe
2/1. Cents
provide a logarith mic device
which en-
ables he theorist
o add and
subtract numbers epresenting
he respective
magnitudes f the
various atios,
which he
cannot
do
with
the
ratios hem-
selves. he y give the adventurerhis
longitudeand
latitude and thus estab-
lish
his
u'hereaboutsn that
vast.
barely
explored
sea which
lies
from the
number 1 to the araway
shores f the
number 2. The
ratios
on
previous
nd
subscquent
ages,
hen, are
the familiar or exotic
slands hat
lie within the
boundariesof this
little-knoransea.
If
"G"
is the starting
point,
the
intcrvalsof the
piano keyboardcontain
centsas
shown n Diagram 2.
The tempered
major
third,"
"G
to
B,"
con-
rains400 cents.
The true
"major
third," 57'4, ontains
only a
tr iflc
over 386
cents.The diference, nearly 14 cents, sapproximatelyone-seventh f 100.
Thcrefore
B"
in the
"key
of G,"
which is 14 cents
harPer han
5/4.
may
be cxpressed s 5/4
plus 14 cents,or approximately
one-seventh
f a semi-
tone.
sPaces
446-451.
7/25/2019 Language of Ratios, Partch
9/10
B4
GENESIS
OF
A MUSIC
The tempered
minor
third,"
"G
to
Bl,"
contains300 cents,whereas
the true
"minor
third," 6/5, has nearly
316.
The
difference, a trifle
less
than 16 cents, s approximately
one-sixth
f 100.Therefore
Bb"
in the
"key
of G," which is 16 cents latter than 6/5, may be stated as 6/5 minus 16
cents, or approximately
one-sixth of a
semitone.
Below are translations
of all so-called diatonic intervals to the nearest
small-number
ratios,
the discrepanciesbeing expressedn approximate
plus
or minus number of cents:
scale
and
placcs.
or
suflicient.
lrc-.ides,r
rable of
A
( nlv to tha
r oncl he
r
pl:rce
oga
r
i Llclrnh
l l l is cxpla
.nrPlc.
an
r ' i rh thc
t
'
rnrrncnts
r : rusical
c
For
pu
i , ,nat ion
th
:1. i .
r rork
:( lP{ ramc
(
: l . l l ) l ish in
INTERVAL
RATIO
"G"
1/1
(the
unison)
"G
to Ab" 16/15 minus 12
cents
9/8 minus 4
cents,ot
10/9
plus
18
cents
, ,G
to
Bb',
6/5 minus 16 cents,or
7 6 plus 33 cents
5/4
plus
14
cents
4/3
plus
2 cents
RATIO
7/5
plus
17.5 cents,or
10/7 minus 17.5
cents
3/2
minus
2 cents
8/5 minus 14 cents
5/3
plus
16 cents,or
12/7 minus 33 cents
16/9
plus 4
cents,
or
9/5 minus 18 cents
15/8 plus 12 cents
INTERVAL
"G
to C*"
"G
to Eb"
"G
to
F* "
This
table represents,
of course, the falsities
that are
found not
only
in
the
"key
of G" but in any
"key"
of Equal Temperament. If the
"key
of
C"
is
chosen,
C
to Dt"-the
smallest nterval-is
16/15 minus 12 cents,etc.
For finding
the number of cents n
a
given
ratio Ellis
provides
a simple
arithmetical
method-not
adequate for investigation
of a
many-toned
Drncnalr
2.-Celrrs oN THE PreNo KeysoaRD
7/25/2019 Language of Ratios, Partch
10/10
cents,
whereas
a
trifle less
Bb"
in
the
"key
minus
16
to
the
nearest
plus
RATIO
I / .)
cents,
ot
mrnus
I
/.)
cents
14 ccnts
16
cents,
or
minus
33 cents
4
cents,
o/
18
cent s
plus
12
cents
not
only in
the
"key
of C"
12
cents,
etc.
a simple
a many-toned
THE LANGUAGE
OF RATIOS
scale-and alsomethods
by
logarithms
hat give
results
up
to
threedecimal
placcs.For presentpurposes omputations
o a tenth of a ccnt
are
generally
sufficient.All
the
Monophonic ratios n
this exposition,and manv
others
besidcs, re givcn n cent s o one dccimal point, either n
the text or in
th e
table
of
Appendix I. Knowledgc as to com putation
of ccnts s important
only to that adventurous
oul rlho rvishes
o organize
a
scale
or
systcm
bc -
vond
the
ratios
expounded
n
this volumc. For t his purposc
a tablc of five-
place ogarithms,
obtainableat almost an; bookstore, nd the ibrary loan
of Hclmholtz's On heSensationsJZonrarc the esscntials. n pagcs448-449
Ellis cxplainshis procedure or
obtaininq rcsults o a tenth of
a cent, by ex-
ample,
ancl
on pages450-451hc supplit's
ablcs o be uscd n conjunct ion
uith
the
tablc
of
{ivc-place ogarithms. Ratios
and ccn ts are thc nvo in-
s(rumcnts
bl
rrhich thc investi {ator xaurines
nd organizcs is t}rcorctical
musical csourccs.
For
purposcs
of an
immediatc papcr
ccrnparisonof ratios n
Just
In -
tonation the logarithm
is
no bcttcr th an rhe ratio,
and
is therefore
uscd n
this rvork onlf in computing ccnts and in examination
of the numcrous
tempcraments.
or exactitude
uc havc
thc ratio itsclf:lbr the purpose
of
cstablishing hcreaboutsby-
prina
Jacit
comparison \'c
havc cents,