Scale products and scale powers

Given two periodic scales S and T with the same repetition interval O, the scale product S∗T is the set of notes {S[i]+T[j]|i, j ∊ Z} over all pairs if integers i and j, ordered by increasing size so as to constitute a new monotone periodic scale. The scale power is the iterated scale product; S^2 is S∗S, S^3 is S∗S∗S, and so forth.

In terms of S and T as reduced to the repetition interval O, the product can be defined via a finite sum over S[i] and T[j] in 0 ≤ S[i], T[j] < O where the sums S[i]+T[j] are reduced modulo O to the interval 0 ≤ I < O. If S and T are written multiplicatively, of course, the scale product is over products S[i]*T[j] reduced modulo O. If S and T are scales in rational intonation, we can use the reduction of the scale to a set of odd positive integers used to determine otonality, utonality or ambitonality, as the reduction of the scale product is simply the set of products of the two reductions.

Suppose, for example, that S is the 5-limit tonality diamond, 1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3. Products of pairs of these, reduced to the octave, gives the second 5-limit crystal ball, 1, 25/24, 16/15, 10/9, 9/8, 6/5, 5/4, 32/25, 4/3, 25/18, 36/25, 3/2, 25/16, 8/5, 5/3, 16/9, 9/5, 15/8, 48/25. In general, the nth 5-limit crystal ball is the scale power of S, S^n. Similarly, starting from the 7-limit tonality diamond, the nth 7-limit crystal ball is the nth scale power of the 7-limit tonality diamond.

Viewed as a subset of a lattice, the scale powers of a scale look like a blown-up version of the scale, but holes tend to be filled in, creating convexly closed scales. Scale powers also preserve the properties of being otonal, utonal or ambitonal; this follows from the fact that the highest number in the nth scale power is the nth power of the highest number in the reduction, and the same is true of the reduction of the inverse.