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**.
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]].
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>.<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>.</body></html>