A problem involving a bit of group theory to solve. Actually it may well be thoroughly solved already, but I'm not sure how to go about finding the solution. I call it the Scale Problem, and it is essentially about determining which rational numbers give rise to feasible scales for composing music, particularly music using "Just" intonation, where all the tones are rational multiples of a fundamental frequency.

Specifically, I want a way to find a set of rationals j where for N an integer, N >= 2 { j : 1 <= j < N } which form a finite group under the tonal addition operation. Tonal addition is much like multiplication with a modulus: given two rationals a and b,

a tonal+ b = a * b * N^k

for some integer k such that the resulting value is returned to the original range 1 <= a tonal+ b < N.

Actually I don't even know how many (if any!) such groups there may be for a given N. If there is an algorithm for enumerating the scale groups, then those which have the "simplest" rationals (in the sense of having smaller numerators & denominators) would be preferred. Additionally the cardinality of the group needs to be less than 17 * N or so since scales with more tones turn out to be too fine grained for humans to be able to distinguish the tonal difference between individual notes.

The reason I have doubts about the existence of any scale group is due to the fact that humans have never yet worked out any really satisfactory Just intonation scheme (other mathematical music geeks may differ in their opinions on this :) Current musical practice uses a scale group which is isomorphic to a cyclical group whose generator is the 12th root of 2. This "equal temperament" scale provides a close approximation to an 11 element rational scale group with N = 2; however it fails to be a scale group due to the irrational nature of the generator (which produces notable dissonances to the trained ear).

So to phrase this as a direct question:

1) how many scale groups exist for a given N? 2) how would I go about finding one?

TIA :)

  • 2
    $\begingroup$ Any group of the kind you want will be cyclic, and in order for all its elements to be rational, $N$ must be a $g$-th power where $g$ is the cardinality of your group. So, no, there is no 100% rational scale except for the trivial ones. $\endgroup$ Aug 7 '10 at 21:49

Darij Grinberg answered the question in a comment, but I thought a more geometric interpretation might be helpful.

The group of positive rational numbers under multiplication is isomorphic to an infinite sum of copies of $\mathbb{Z}$, where the set of primes forms a natural basis. If you choose a positive integer $N$, you can think of the subgroup $N^\mathbb{Z}$ as a copy of the integers embedded as a line in an infinite dimensional lattice, with the generator $N$ given in coordinates by its prime factorization. Torsion elements of the quotient group $\mathbb{Q}_{>0}^\times/N^\mathbb{Z}$ must then lift to elements in the lattice that "lie between" elements of $N^\mathbf{Z}$. The torsion part of the quotient group is then generated by $N^{1/g}$, where $g$ is the largest integer such that $N$ is a perfect $g$th power.

  • $\begingroup$ Between your answer and Mr. Grinberg there are a number of useful and interesting results for algorithmically generated (or computer-assisted) musics, even though the result I was looking for is not available. Thank you all :) $\endgroup$
    – Kumoyuki
    Aug 8 '10 at 6:54

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