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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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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."
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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
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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
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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
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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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