Synthetic Musical Scales

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Synthetic Musical ScalesAuthor(s): J. Murray BarbourSource: The American Mathematical Monthly, Vol. 36, No. 3 (Mar., 1929), pp. 155-160Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2299681Accessed: 21-12-2015 07:51 UTC

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2/7

1929]

SYNTHETIC MUSICAL SCALES

155

to the

first

uestion below.

For

several

particularpairs

of

values

of n

and

r,

we have

taken

b

=

m

=

1

in

(2) and obtained

an

affirmativenswer.

All

evidence

which we

have

collected also favors an

affirmative

nswer

to

the last

two

problems. We invite thereader to answerthefollowing uestions.

Problem1. Ifn > r, s

thewn of (15d)

the

maximumnumber hat

can

appear

in

a solutionof

2) ?

Problem

. If n>r,

is W1W2

. .

.

Wn

the

maximum roduct f

any

set

of

n

numberswhich

constitute solutionof

2)?

Problem

.

If

n>r,

is

Wl+W2+

-

* *

+Wn

the

maximum um

of

any

set

of

n

numberswhich

constitute solution

of

2)?

SYNTHETIC

MUSICAL SCALES

By

J.

MURRAY

BARBOUR, Wells

College

1.

Busoni's problem. In his little book, A New

Esthetic

j

Music,' Ferrucio

Busoni describes a method of formingmusical

scales by raising or lowering

various tones of the

scale

of

C major. By this

method

he

has obtained 113

scales, the majority

of which differ rom he ordinarymajor and minor scales

in having their ntervalsdifferentlyrranged.

Apparentlyhis results have not

been questioned since his book was published,fortwo rather recentworks2

accept

themas

authoritative.

A seriousobjection

to

Busoni's scheme s

that,

n

accordance

with theuisual

method

of

notation

and with the conceptionof a

seven-tone scale on successive

alphabetical degrees,

his octave contains

twenty-one

ifferent ones instead

of

the twelve that

belong

to our

system

of

enharmonic

emperament

n

the

piano.

This would seem to

be an unnecessarycomplication-and restriction-in a

proposal that is

otherwise

o

novel.

The question

also arises as to whether t is properto include all the cyclic

permutations

f

any

given scale

or whether

certain

arrangement

f

intervals

should be counted only once, irrespectiveof the point at which the series

begins.

From

the modern tonal

view-point,

ll of the

medieval church modes

are

variants

of

the scale

of

C

major.

Should one

look

at

these

new

scales

ac-

cording

to the

medieval

or

the modern

tandard?

2.

The

harp,

as

basis.

A

good way to avoid both the difficultiesmentioned

above

is

to restate the

problem

n terms

of

the

harp.

The

octave

of

the

harp

contains the

twenty-one

ones,

of

which

only

seven can be

used

at

any

one time.

On

the

harp

it is

literally impossible

to

form

a

scale

containing

the

tones

Ab, A,

A#;

on

the piano there s no practical reason

why

these tones should

not

1

Ferrucio Busoni, Entwurf einer neuen Aesthetikder Tonkunst, 1907), translated by Dr.

Th. Baker, (1911), pp. 29-30.

2

George Dyson,

The New

Music, 2nd edition (1926), pp. 94-5-7. Albert

A.

Stanley,

Greek

Themes

n

Modern

Musical Settings, 1924), introduction, . xiii.

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3/7

156

SYNTHETIC MUSICAL SCALES

[Mar.,

occur

in

a scale empiricallyformed, nstead of the

enharmonically quivalent

G#,

A, Bb that

would be written n one of Busoni's scales.

The difficulty

f

cyclic permutations s also

eliminated by confining ur

attention to theharp. One simply sks in how manyways a harp can be tuned.

The harp has seven

pedals, each controlling ll of

the stringsof a particular

letter. The

"natural" key of the harp is Cb major.

The C pedal will raise

all

of

the Cb strings

ithera semitone or a tone to C or

C#.

The other pedals

operate similarly.

Thus each stringmay produce one

of three sounds, and the

total

number

of

ways the harp may be tuned is

37

or

2187.

Although

this

answer

is

correct,

the

numbergiven above includes many

tunings that,

aurally considered, re

not

scales but chords. Often a composer

desires the

glissando

of

the

harp

to sound

like

a

seventh chord, arpeggiated

with extreme rapidity. Rimsky-Korsakow,for example, in his orchestral

suite "Scheherazade"

uses

the following unings: Cb,D,

E#,

F,

G#,

b, B; C,

D#,

Eb,

F#,G#,Ab,B#;

C,D#,Eb,

F,G#,Ab,B#;

C,Db,Eb,

F#,G,A,Bb.

Of these,

the

first wo

are

chords of

the diminished

eventh; the

third

s a chord

of

the

"added

sixth";

and

only

the

fourth,which, however,

s

typical

of

many

in

the

composition,might

properly

e

called a

scale.

In

accordance with the customary definition

f

the term "scale" we

must

rule out such

a

case

of

tonal

duplication

as

G#-Ab'

and the

rarer

ones

of

tonal

overlapping,

B#

b and

ES

-

Fb. These restrictions

re

sufficiently

bvious

and

reasonable.

The ordinary

major

and

minor scales contain

minor

seconds, (C-Db),

major seconds,

(C-D),

and

augmented seconds,

(C-D#).

One

of

Busoni's

examplescontains the

doubly augmented econd,

Fb'

G#.

In

this

paper,

there-

fore,

he interval

of

four

emitones s included with

the

morefamiliar

ypes.

3.

Method

and

results.

The method used

here to

compute

the number of

scale-tunings

s

exemplified

n

Table

1. In

it

is

found

first he

number

of

two-

tone scales

containing

"inflections"

#,

~,

and

b)

of

C and

B-either

1

or

0 for

each of

the 9

combinations,

as determined

by

the

given

restrictions.

For

example,

C#-B#,

C-B,

Cb'-B

'

are

possible,

but

C-B#, Cb-B#,

and

Cb'-B

are not.

Then inflections f

A

are added

to

form hree-tone

cales.

These

are

cumu-

lative.

For

example,

A#

may occur

in a scale with

B#

or

B,

and

the

numbers

opposite

A#

re the sums

of

thoseopposite

B#

and

B

in

each

of

the three

respec-

tive columns.

Generally peaking,

f

n

any

column the

numbers

opposite

the three

nflec-

tions of

a

letter re

a, b,

and

c,

those

opposite

the

inflections

f

the

letter

below,

if

it is

a semitone

ower, are a, a+b,

and

a+b+c.

If it

is a

tone

lower,

the

numbers are

a+b,

a+b+c,

and

a+b+c.

The

sole

principle

nvolved

is the

avoidance of tonal duplication and overlapping. The totals are the sums of

the last

three numbers

n

each

column.

The

sum

of

the three

columns, 363,

is

the result which

we

are seeking; i.e.,

the number

of

ways

the

harp may

be

tuned

to form

"scales."



4/7

1929]

SYNTHETIC

MUSICAL

SCALES

157

If we

start with

any

letter

other than

C, the totals

of the three columns

will

differ,

ut

their

um

will remain

constant.

Therefore

he number

of

scales

that

may be

formed

on

all

the twenty-one

ones

in the octave

will

be

7

X363

or 2541. One may wonderif thereis much enharmonicduplication of entire

scales

by

this nsistence

on the separate

identity

f tones

a

"diminished

econd"

apart.

Of

the363, however,

here

are only

four uch pairs

of

scales,

all of

them

well-known.

They

are

Db-C#

maj.,

Gb-F#

maj.,

Cb-B

maj.,

and

Db-C#

min.,melodic

form.

Table

1

Table

2

C#

c

Cb

Number

of scale-tunings

for each

tone

B

1

0

0

1

1

0

B#

57

Fl

1

1

1

E#

59

Gl

A

2

1

0

A#

81

Gl

3

2

1

3

2

1

D#

87

Dl

G

5

3

1

G#

105

Al,

8

5

2

C#

149

El

8

5

2

F

13

8

3

F#

153

Bb

21

13 S

B

153

F

21

13

5

E 155 C

E

13

8

3

34

21

8

A

177

G

55

34

13

D

189

D

D

47

29

11

102

63

24

0

63

24

Totals

149

155

59

By

a method

similar

to

that

illustrated

n Table

1,

the number

of

scales

that do not contain the doubly augmentedsecond can be foundforany tone.

For C

it

is

134. Thus

Busoni's

figure

f

113

is somewhat

ess than

the

correct

number

of

scales without

the uncommon

interval

and

much

less than

the

number

of scales

(155)

formed

in accordance

with

his declared

method.

Evidently

he

followed

no

scientific lan,

but was content

with writing

own

the

entire 113,-a

task

indeed. Oddly

enough,

this

number

s not

far

from

ne

third

the

sum

of the three nflections

f

any

letter:

363

.

3 =

121.

4.

Applications

to

number

theory.

If the numbers

representing

possible

scales

for he

different

egrees

are

put

in

order,

Table

2 results.

It

is

interesting

to observe that each number in it can be represented by the equation

s=lta+13b.

This table,

to

a musician,

s the

most

remarkable

nd

interesting

eature

of

the

entire

discussion.

Each

tone

is a

perfect

fifth istant

fromthe tone

next



5/7

158 SYNTHETIC MUSICAL SCALES [Mar.,

above

or below it in the table.

Considered

as

the tonic or key-note

f

a major

scale,

B

(at

the

top

of

the

left-hand

olumn) would have a key signature

of

12

sharps. For each tone below there would

be one less sharp

in

the signature.

D has two sharps, C none, and each tone above C in the right-hand olumn

has a

signature

of one additional flat. The

whole

is

often termed by

musical

theorists

"the circle of fifths" nd it is then

written

n

the

formof a circle

by

using the enharmonic

oincidence

of

Gb

and

F#

and omitting he tones beyond

them.

Table 3

B#

3 8

21

39

96

57

E#

3

8

15

37

96

59

A#

3 8

15 37

96

81

D#

3

6 15

39 102

87

G# 3 6 15 39 72 105

C#

3

8 21

55

102

149

F#

3

8

21

39

96

153

B

3 8

21

39

96 153

E

3 8 15

37

96

155

A

3 8

15 37

96

177

D

3 6

15

39

102

189

G

3 6 15

39 72 177

C

2 5 13

34

63

155

F

2

5

13

24 59

153

Bb

2 5

13 24

59

153

Eb 2 5 9 22 57 149

A

b

2

5

9

22

57

105

DI

2

3 7

18

47

87

Gb

2 3 7

18

33

81

Cb

1

2 5 13

24

59

FL

1

2 5

9

22

57

Table

4

Combinations

Number

of

of

intervals

permutations

7

1122222

=

21

5 2

7

1112223

=

140

3 3

7

1111233

4 2 =

105

7

1111224

4 2 =

105

7

42

1111134

5 =

Total

413

Table

3 is formed

y

adding

by

threes

he

numbers

nthe

columns

of

Table

1,

for

all

the

twenty-one

ones,

arranged

s

in

Table

2.

It will

be seen

that

Si

(the

first erm

of

a

series)

s

3 from

#

to C,

2 from

C

to

Cb,

and

1

from

Cb

to

the

end

of

the

table.

In

general,

ollowing

semitone

as

F

to

E), Sn-3(S,-,-S,-2);

following

whole

tone (as G to F),

Sn=3Sn_1-Sn2.

Also, forthe first tones

in the

table,S6=285-3S4,

after

semitone,

nd

S6=285-S4,

after

tone.

We

must

assume

SO

to be

1 inevery

case.

The

series

S.

=3

(S,-,

-

Sn-2)

is

not

of

great

interest,

mathematically,

except

for

ts

fluctuations

nd

changes

of

sign.

If

the

first

wo

terms

re

a

and

b,

the

series

s:

a,

b,

3b-3a,

6b-9a,

9b-18a,

9b-27a,

-27a,

-27b,

.

.

The

entire

series

will

consist

of

repetitions

f

the

6-term

portion

given

above,

multiplied ach timeby the constantfactor (-27). The expression -27)

n/6

is a sort

ofsymbol

ofthe

series.

The

terms

of

the

series

Sn=3Sn_1-S4-2

are

a,

b,

3b-a,

8b-3a,

21b

-8a,

*

-

. Since

a and

b occur

in everyterm,

t is

only necessary

to

derive



6/7

1929] SYNTHETIC MUSICAL SCALES

159

a formula for the coefficient eries 1,

3, 8,

21, . These terms may be

written as follows: 1=1; 3=3; 8=32_1; 21=33-2.3; 55=34_3.32+1;

144

=35-4. 33+3 3; etc. By

induction

Sn= 3n-1 (n-2)3n-3

+

2(n

-

3)(n

- 4)3n-5-

(n

-

4)(n

-

5)(n

-

6)3n +

The ath termof the above series s

(-l)a-1(n

-

a) 3n-2a+1

(a

-

1) (n

-

2a

+

1)

If

n is even the last term s (-n)(n-2)'2(n/2) 3. If n is odd the last term

is

t_

(n-1)

/2.

This series s ofthe greatest mportance n any generalization f the problem.

We have seen that in a single column t applies whenever here s a whole tone.

But when the totals of three columns are added it is necessaryto have at least

3 successive whole tones before he aw of the series becomes operative.

5. Permutation f ntervals. Similar to Busoni's problem s that of finding

in an octave of

12

semitones the possible combinations of the given intervals

with all theirpermutations.

This

is a simple algebraic problem and its solution

is clearly shown n Table 4, wherethe numbers n the column headed "combina-

tions of intervals" refer o the size

of

the intervals: 1, minor second; 2, major

second; 3, augmented second; 4, doubly augmented econd.

Since, under the conditions

of

the Busoni problem, no more than

4

semi-

tones may occur in succession, the 14 permutations n the fifth ow in which

all 5

semitones come together hould be subtracted

from

the total

in

order o

give the numberofdifferentcale-formsctually occurringn the scale-tunings.

With this

change,

it

is an

interesting act that the number

of

permutations

representing cales without

an

augmented second (21), those with

a

singly

augmented second (266),

and

those

with a

doubly augmented

second

(399)

are

exactly

the

same

as

the

cyclic permutations

f

the

scales

containing

B#

or

Fb,

using

these intervals.

The reason

therefor

as not

been

found.

Since the figures

n

Table

4

include cyclic permutations, ividing by

7 will

give numbers to be compared with the 363 scale-tunings. For example,there

are 15

major-scale-tuningsnatural,

7

sharps,

and 7

flats);

but

of

the

3

permuta-

tions of

major

and

minor

econds

in

the first ow

of

Table

4, only

one

(2212221)

is

called a major scale. At the other extreme s the 4th combination

of

this

table,

with the intervals

arranged

as

given. Only

one

such

scale can be

so

constructed;viz.,

C#,

D, Eb, Fb, Gb, Ab, Bj$.

This shows very clearly the limitations f the Busoni method. On the piano

this last-mentioned

permutation

of

intervals

might begin

on

any

white

or

black

key.

Of course

its notation would involve

the

use

either

of

one letter

twice and the omission ofanother,or else of double sharpsor flats.

6. Scales not heptatonic. In musical composition the whole-tone scale

of

6

tones

is

common,pentatonic

cales

(as

on

the

black

keys

of

the

piano)

are

used

in

the folk-songs

f

certain countries, nd the

minor

cale mightvery properly



7/7

160

SYNTHETIC MUSICAL SCALES [Mar.,

be said to contain 9 tones (in A minor, -A, B, C, D, E, F,

F#,

G,

G#).

There-

fore t is not making the question purely academic to pursue the line of nquiry

shown n Table 4 in respect to scales containing ess or more than 7 tones. The

method s identical; the results ppear in Table 5.

When we leave the heptatonic scale we leave also the troublesome uestion

of notation in alphabetical sequence. However, in the 2nd column of Table 5

the

four-semitonenterval

s

still the largest one used. Since this restriction

s

not in accordance with the freedom f this phase of the inquiry, he size of the

intervals n the 3rd column

s

unrestricted,making the problemthe simpler

one

of finding he numberof permutations f

11

things, aken (n -1) at a time.

Table

5

Number

of tones

Permutations

ncluding

Permutations

ncluding

in scale doubly augmented econd all intervals

2 0 11

3

1

55

4 31

165

5 155 330

6 336

462

7

413

462

8

322 330

9

165 165

10 55 55

11

11

11

One

may

well

ask

how

profitable

o the

composer

s

the

knowledge

that

he

is

free to select any

one

of

thousands

of

hitherto unknown scales,

as

the

foundation

orhis

creative

work.

Unfortunately, he whole problem s

of

greater

theoretical interest

than

of

practical

worth.

At

present, there

is the

most

astounding

license

in

composition;

there seems to be an

intuitive attempt

to

obtain euphony if

not

harmony

n

the

classical

sense) by the combination

f

the

most diversetonalities; tonality

tself s

applied

to

melodies

s almost

a

thing

of

thepast.

To

compose

on

the basis of

any artificiallyreated scale would

be

to

fasten

on

again

the

shackles that were

slipping

n

Wagner's day

and that were

thrownoff ntirely n the early years of this century. The composersof this

generation

seem

to

have attained the

ultimate

freedom

possible

under

the

system

of

duodecuple

division

of

the

octave.

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