Generator complexity
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- This revision was by author genewardsmith and made on 2012-01-25 14:01:50 UTC.
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Original Wikitext content:
Suppose <0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and <0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, <0 1 -2 -2| is the generator mapping val for seven limit [[pajara]], and <0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ <0 0.631 -0.831 -0.712| is the val in weighted coordinates. Then the //generator complexity// of the temperament is P*(max(0 B₃ B₅ B₇ ... Bp) - min(0 B₃ B₅ B₇ ... Bp)). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C OH(I), where C is the generator complexity of the temperament, G(I) is the number of generator steps, times P, required to reach the tempered version of I, and OH(I) is the //odd height// of I, that is the [[Tenney height]] of K where K has the factorization of I without any factors of 2. Hence for any MOS of size N, floor(N/(C OH(I))) intervals with pitch class corresponding to I are guaranteed to exist in the MOS. Generator complexity is also useful in making complete searches using [[the wedgie]] for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.
Original HTML content:
<html><head><title>Generator complexity</title></head><body>Suppose <0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and <0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, <0 1 -2 -2| is the generator mapping val for seven limit <a class="wiki_link" href="/pajara">pajara</a>, and <0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ <0 0.631 -0.831 -0.712| is the val in weighted coordinates. Then the <em>generator complexity</em> of the temperament is P*(max(0 B₃ B₅ B₇ ... Bp) - min(0 B₃ B₅ B₇ ... Bp)). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984.<br /> <br /> Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C OH(I), where C is the generator complexity of the temperament, G(I) is the number of generator steps, times P, required to reach the tempered version of I, and OH(I) is the <em>odd height</em> of I, that is the <a class="wiki_link" href="/Tenney%20height">Tenney height</a> of K where K has the factorization of I without any factors of 2. Hence for any MOS of size N, floor(N/(C OH(I))) intervals with pitch class corresponding to I are guaranteed to exist in the MOS. Generator complexity is also useful in making complete searches using <a class="wiki_link" href="/the%20wedgie">the wedgie</a> for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.</body></html>