Theory of Circular Partitioning Sequences [was: set theory & composition]

> Steven,

> If you haven't already seen it check out the thread on Polya's
> counting theorem from Dec. 2000/Jan 2001. I discussed how to
> count the number of sets.

Steve -- if you find this thread please post an article
number. I have difficulty with the yahoo search system.

I'm super interested in this problem -- the only thing
I've found before now is Ramanujan & ??'s difficult
solution to the problem of partitioning a whole number,
which is related, but not the same thing.

> If you are interested, I could cook up a short computer program
> to actually produce a table of sets for a given ET.

> Polya's theorem can give us the number of sets of various sizes,
> but the tables could be tricky to generate.

> David Bowen dmb@sgi.com

I've got one I made that might be useful to some people.
I'll append the manual to this email so that people can
look at it and see if would help them.

If any of my discoveries below are helpful to anyone
and not ancient news, I'd appreciate being referenced in
any mathematical journal articles or theses, or other
work derived.

Actually, I'd appreciate any feedback from the
mathematically inclined among you regarding if anything I
say here is new or old.

Oh, by the way, I use the example in the manual below of
88cET as 8th root of 3:2, but I do know that 88cET is
also 11th root of 7:4, both of which are ways I tune it.
(Gary tells me he inclines towards the 7:4 view himself.)

-----------------------
Mode Maker Manual
(c) 1997 by X. Jeff Scott

This program determines the number of pattern-repeating
scales possible from a generating set of n equal-sized
intervals, called srutis for convenience in the code.

For example, 12 tone equal temperament is a 12 srutis per
octave scale generating set. Various scales can be formed
by selecting a subset of the tones present in the set -
size 5 subsets are pentatonic scales, etc.

You can also form an 8 srutis per fifth generating set, or
4 srutis per third, or what-have-you.

The number of scales, including modes (which are rotations
of scales) that can be formed from a n-sruti generating
set is 2 raised to the (n-1) power. This can be seen by
imagining a scale as a n-digit binary number with the most
significant number place containing a 1 and holding the
tonic. Scales then can be represented as the various
patterns of bits where a 1 in a position means that that
tone is present in the scale.

Calculating the number of scales possible that does not
include rotations or modes of each other is a much more
difficult counting problem and I have not figured out a
general formula yet.

To get around this, I wrote this command-line driven
program to count and generate all the scales, not
including modes that can be formed from a n-sruti
generating set.

This is very useful way to generate all sets of scales
possible in equally spaced generating sets.

For example, say you are working in 88 cent equal
temperament, which divides the perfect fifth into nearly 8
equally sized divisions. You may want to play music using
scales that repeat every fifth.

Compile mode maker for your computer and call

modemaker -d -s 8

The results in the report:

A total of 1 different unitonic scales can be formed from
any 8 equal-sruti repeating set.
A total of 4 different duotonic scales can be formed from
any 8 equal-sruti repeating set.
A total of 7 different tritonic scales can be formed from
any 8 equal-sruti repeating set.
A total of 10 different quadratonic scales can be formed
from any 8 equal-sruti repeating set.
A total of 7 different pentatonic scales can be formed
from any 8 equal-sruti repeating set.
A total of 4 different hexatonic scales can be formed from
any 8 equal-sruti repeating set.
A total of 1 different heptatonic scales can be formed
from any 8 equal-sruti repeating set.
A total of 1 different octatonic scales can be formed from
any 8 equal-sruti repeating set.

A grand total of 35 8-sruti repeating scales are possible.

To see all the pentatonic scales that are possible in this
system, run modemaker as follows:

modemaker -u 88 -t 5 -s 8

This produces the result, given in terms of interval
distance in cents between keys. Each key is represented by
a period and the starting point is shown as an asterisk:

8 sruti repeating, pentatonic scales:
* 88 . 176 . 88 . 176 . 176 *
* 88 . 88 . 264 . 88 . 176 *
* 88 . 88 . 176 . 176 . 176 *
* 88 . 88 . 176 . 88 . 264 *
* 88 . 88 . 88 . 264 . 176 *
* 88 . 88 . 88 . 176 . 264 *
* 88 . 88 . 88 . 88 . 352 *
(a total of 7 pentatonic scales)

The source code for this program is presented here as a
starting point in the land of scale generation.

The total number of unique scales from 1 to 14 sruti sets
is {1, 2, 3, 5, 7, 13, 19, 35, 59, 107, 187, 351, 631,
1181 ...}. This is a very interesting sequence since it
starts out looking almost like the prime numbers -- a
sequence that is defined as having no pattern.