Language of Ratios, Partch

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


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.


3/10

GENESIS

OF

A

MUSIC

The2tolrelat ionship,basedonthefactorof2'appl iesonlytopro-

gr*ri""

L'pi,.tt

by

2/1's.îhere

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 vêry 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

âa

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

îr,,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


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


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


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


7/10

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


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.


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


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,

Documents

Language of Ratios, Partch