Scale products and scale powers
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Original Wikitext content:
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 and utonality|otonality, utonality or ambitonalisty]], 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 balls|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 [[Otonality and utonality|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.Original HTML content:
<html><head><title>Scale products and scale powers</title></head><body>Given two <a class="wiki_link" href="/periodic%20scales">periodic scales</a> S and T with the same repetition interval <strong>O</strong>, the <em>scale product</em> 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 <em>scale power</em> is the iterated scale product; S^2 is S∗S, S^3 is S∗S∗S, and so forth.<br />
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In terms of S and T as reduced to the repetition interval <strong>O</strong>, the product can be defined via a finite sum over S[i] and T[j] in 0 ≤ S[i], T[j] < <strong>O</strong> where the sums S[i]+T[j] are reduced modulo <strong>O</strong> to the interval 0 ≤ I < <strong>O</strong>. If S and T are written multiplicatively, of course, the scale product is over products S[i]*T[j] reduced modulo <strong>O</strong>. 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 <a class="wiki_link" href="/Otonality%20and%20utonality">otonality, utonality or ambitonalisty</a>, as the reduction of the scale product is simply the set of products of the two reductions.<br />
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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 <a class="wiki_link" href="/Crystal%20balls">crystal ball</a>, 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.<br />
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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 <a class="wiki_link" href="/Otonality%20and%20utonality">otonal, utonal or ambitonal</a>; 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.</body></html>