Yahoo Tuning Groups Ultimate Backup tuning FW: [MusicTheory] 351 possible scales?

Anyone have a neat answer for this 12-tET question?

-----Original Message-----
From: Michael Edwards [mailto:mje@foxall.com.au]
Sent: Saturday, December 09, 2000 4:04 PM
To: MusicTheory@egroups.com
Subject: [MusicTheory] 351 possible scales? - but I can't prove it.

Michael Edwards.

I have a musical/mathematical question, to which I think I know the
answer,
but I cannot confirm it mathematically. I once raised this on the Finale
list,
and a couple of people there seemed to think my answer was correct - but I
thought it would be fun to see what anyone on this list thinks. (If anyone
here
is also on the Finale list, I apologize to you because I'm about to subject
you
to this again - but it was a year ago, after all, and it will mostly be new
people who read it now.)
It's a permutational question concerning the number of possible scales
you
can invent in the equal-tempered 12-note octave. Perhaps it can also be
expressed in terms of set theory (not by me, though). I must emphasize that
I
am *not* talking about Pythagorean tuning for this question, but a closed
system
of just 12 different pitches in equal temperament, plus all their octave
transpositions.
I have to define my terms exactly; especially, I have to define
precisely
what I mean by a scale.
If you start with the 12 chromatic notes of the equal-tempered octave,
you
invent a scale by selecting any number of these notes from 1 to 12 notes,
and
arrange them in order of pitch. The pitches and intervals are considered
circular, so that, for the C-major scale, for instance, you don't get a new
scale just by starting on another note, as in the old church modes. By my
definition, these are 7 different modes; but it's all the same scale. More
generally, I would define a mode as any scale structure considered as
beginning
with one particular note and one particular interval. So my definition of
the
diatonic scale, for instance would simply be the sequence of 5 whole-tones
and
two semitones, considered to wrap around in a circle, with the semitones
being
separated by two whole-tones in their closest separation - and this says
nothing
about whether any particular note or interval is considered the start of the
scale.
Also, if you have a similar scale structure in B major or Ab major or
whatever, I am counting this as the same scale: in other words,
transpostions
don't count. I am solely concerned with distinct circular sequences of
intervals which make up one octave, with no distinction about which interval
you
start with (which mode of the scale you are considering), and with no
concern
about what pitch you cast the entire structure in (the key it's in). In
analyzing problems relating to this, it is easier to focus on the intervals
between the notes rather than the notes themselves.
(My definition might be a bit wordy; but it's very important to the
problem
to be absolutely clear on what makes scales different by this definition,
and
what doesn't.)

The number of possible scales, as just defined, is far less than I
originally would have expected. Before considering the problem, I might
have
naively expected thousands, if not millions, of scales, considering all the
possible permutations of notes and intervals you can arrive at. But I am
now
fairly sure that there are only 351 scales. It's amazing how many
permutations
are ruled out because they are simply different modes of the same scale,
and/or
different transpositions.
I found out the number of scales by the brute-force method of listing
all
possible scales, expressing them by circles of numbers representing the size
of
intervals counted in semitones: first listing all 1-, 12-, and 11-note
scales
(only one of each), then 2- and 10-notes scales (only 6 of each), and so on.
Then I simply counted them, and it came to 351 scales.
(It's easiest to list each scale on paper as a row of numbers beginning
with the largest interval in each, but regarding the line as circular; the
horizontal arrangement and the beginning with the largest interval have no
permutational significance, but just make it easier to see if you've missed
or
duplicated any permutations. Notionally you regard each sequence of numbers
as
a circle of numbers - that's vital to the problem I'm posing.)
Here are the numbers of scales I found with each number of notes:

No. notes No. of
in scale scales

12 1
1, 11 1, 1
2, 10 6, 6
3, 9 19, 19
4, 8 43, 43
5, 7 66, 66
6 80

TOTAL 351

(I debated whether to include a scale of 0 notes in it, but rejected
this
as having no musical meaning. (I suppose John Cage's 4' 33" is the only
piece
written in a 0-note scale. :-) ) But certain analysis I have done with the
properties of various scales, regarding them purely as mathematical
entities,
only makes sense and retains perfect symmetry of structure if you do accept
a
notional 0-note scale. This stems from the fact that scales form
complementary
pairs such that, when you select notes for a scale, the notes left in the
chromatic scale you *didn't* select form another scale complementary to the
first, so that every 5-note scale can be paired uniquely with a particular
7-note scale, every 4-note scale with a particular 8-note scale, and so on.
6-note scales are either their own complement (at a different pitch of
course,
such as the whole-tone scale), or pair up with another 6-note scale. The
justification for allowing a 0-note scale is that it is the only possible
complement for the 12-note scale (whose existence surely no-one would deny).
Seen in set-theory terms, I suppose the 12-note scale would be the universal
set, and the 0-note scale the empty set. But I have not included the 0-note
scale in the table of figures above.)
I catalogued the 351 scales on paper several times separated by some
interval of time, and arrived at the same result each time, and this
convinces
me it is the correct answer. Any mistakes I might make which cause that
number
to vary would be unlikely to be repeated on different occasions so as to
make
the number of scales arrived at inaccurate by exactly the same amount each
time;
in other words, if I made mistakes at all, I would almost certainly make
them
*differently* each time I did the exercise.
That's why I think 351 is in fact the correct number; but the fact
remains
that I cannot prove mathematically that it is the correct answer.
Can anyone prove it (and explain it in terms I can understand)?

There is the question of how many modes there are of these scales. Now
we
*do* take into account starting-points (but not transpositions to different
keys). Most scales will have the same number of modes as they have notes,
because each note chosen as a starting point will give a different pattern
of
intervals relative to that starting note (even though they are all the same
considered as a circular pattern). But some scales will have fewer modes.
For
instance, the scales C D# E G Ab B C and C Db D# E F# G A Bb C have only two
modes each, because some of the modes duplicate each others' interval
patterns.
(Enharmonic differences in spelling are ignored, and can't be avoided no
matter
how you arrange things.) The first of these two scales has the two modes
3 1 3 1 3 1 and 1 3 1 3 1 3 (with the numbers expressing successive
intervals
counted in semitones); the second scale's two modes are 1 2 1 2 1 2 1 2 and
2 1 2 1 2 1 2 1. The scale C Db E F# G Bb C (the "Petrushka" scale) has *3*
different modes (1 3 2 1 3 2, 3 2 1 3 2 1, and 2 1 3 2 1 3); the whole-tone
scale has only *1* mode (2 2 2 2 2 2), because, whatever note you start on,
the
intervals are always identical.
It would be easy in principle, if time-consuming, to just calculate all
modes of all scales (1*1 + 6*2 + 19*3, etc.), then eliminate the ones that
duplicate their interval patterns. I estimate that the number would be in
the
region of 1,700 or thereabouts. But is there a neat mathematical way of
working
out the number of modes from a consideration of the earlier problem about
the
number of scales (viewing it as a permutational problem)? If so, it is way
beyond my mathematical ability.

I've established that there are exactly 16 periodic scales. I define a
periodic scale as one which divides into two or more equal portions of
identical
intervallic make-up. (For instance, the three scales I cited above; the
whole-tone scale; the chromatic scale.) Another way of expressing this is
scales with fewer distinct modes than notes per octave. Messiaen's
so-called
"modes" (which I think should be called scales) of limited transposition are
7
of these periodic scales; but his list of 7 omits a few interesting scales,
as
well as trivially simple examples (2-, 3-, or 4-note scales, of limited
practical use but nonetheless undeniably still being periodic scales as I
defined them).
(I think such scales of the sort I am now talking about are more
commonly
termed "symmetrical scales"; but I abandoned this term when I noticed that
there
were two different ways can be symmetrical, and thus the term is ambiguous.
I
now refer to periodic scales and self-reflective scales for the two types.)
I've examined the properties of these periodic scales in considerable
detail, because they kind of fascinate me, and have done so for many years;
and
I can state definitely that the number of these periodic scales breaks down
thus:

No. notes No. of
in scale scales

12 1
1, 11 0, 0
2, 10 1, 1
3, 9 1, 1
4, 8 3, 3
5, 7 0, 0
6 5

TOTAL 16

I have seen some of these scales in actual scores, if only briefly in
most
cases - but not all of them.

Has anyone systematically studied the properties and relationships of
scales and modes? A musician I have corresponded with in England told me
that
the English composer John Foulds had done some analysis along these lines,
and
he sent me a photocopy of a list of several dozen scales than Foulds had
apparently arranged in some way. But the information I was given was
tantalizingly too little for me to infer what Foulds was doing, or how he
was
classifying the scales.
I wonder if anyone knows more about this, or knows of anyone else who
has
studied scale patterns in depth. (I once saw a thick book of hundreds of
scales
compiled by Nicholas Slonimsky; however, to me it looked comprehensive, but
not
all that systematic.)

Michael Edwards.

MusicTheory onelist.

This posting is in response to Michael Edwards posting which is quite long,
so I will not include the original hear...

To determine the number of possible scales under Mr. Edwards criteria, we
are doing basic number theory... this topic has been extensively covered in
music theory literature...

Allen Forte: The Structure of Atonal Music
Robert D. Morris: Composition with Pitch-Classes
John Rahn: Basic Atonal Theory
Joseph Struass: Introduction to Post-Tonal Theory

Unfortunately there is no simple mathematical answer to this question that
I know of... The total number of possible sets in 12tet is 12^2 or 4096,
this includes the full chromatic scale and the null set... within that 4096
many of these sets will be transpositionally equivalent, for instance
C-Major = D-Major and so on... after we take out transpositional
equivalence there are 352 (including sets with 0 elements)... set theory
further reduces this number through inversional equivalence... for instance
{C-E-G} and {A-C-E} are both considered the same set they are related
through inversion (basically reading the intervals backwards) C-E-G is a
M3rd + m3rd A-C-E is a m3 + M3. After subtracting inversionally equivalent
sets there are only 158 unique sets in 12tet (including sets with 0
elements)...

So, it seems that you are correct Mr. Edwards that there are only 351
scales (not counting the 0 element set). I am rather impressed that you did
this all by hand... Allen Forte used computers to compile his set tables,
but you did everything by hand and got everything right...

Your approach towards the possible number of scales is different than most
set theoretic literature since you don't consider sets related by inversion
as equal... for instance if you look at the intervals of the harmonic minor
scale are as follows

C, D, E-flat, F, G, A-flat, B, C (2,1,2,2,1,3,1)

inverted you get this...

C, D-flat, E, F, G, A, B-flat, C (1,3,1,2,2,1,2)

these are obviously two different musical creatures... atonal set-theory
doesn't distinguish between these since there interval content is the
same... but the principles of set-theory are very valuable for looking at
how musical sets will interact... you can analyze different set in many
different ways... one valuable way is to look at the common notes between
two different musical sets and determine ways to modulate between sets...
for instance... not only can you look at the similarities between different
sets, but the similarities between different transpositions of the same
set... for instance C-Major and G-Major have 6 notes in common... C Major
and F Whole-tone have four notes in common... you can generalize conepts of
modulation from these ideas... also, if you look at the make up of a set...
for instance the Major Scale has 12 different forms, and a given scale
contains a single tritone and three major triads... simply by knowing this
you know that there is one other scalar form that contains that tri-tone and
that a single major triad will be found in 3 other scales... e.g. the B-F
tri-tone is found in C-Major and F#-Major... the C-Major Triad is found in
C, F, G major... this is also a very powerful compositional tool concerning
harmonic ambiguity in abstract sets and again in modulation.

A book that is very valuable, even though it considers only the 158 sets
with inversional equivalence is Jeffrey Johnson's Thesaurus of Abstract
Musical Properties... unfortunately it is a very expensive book...

I am basically writing a small text book concerning 'scale theory' and set
theory for an independent study in Music Theory... if there are any more
questions or if clarifications are desired feel free to ask...

Steven Kallstrom
Illinois Wesleyan University
Senior Composition/Theory Major
Harpsichordist
skallstr@sun.iwu.edu

FW: [MusicTheory] 351 possible scales? - but I can't prove it.

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