Generator complexity
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Original Wikitext content:
Suppose A = <0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = <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. For any vector V, let max(V) - min(V) = span(V). The //generator complexity// of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit. Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C KE(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 KE(I) is the [[Kees height|Kees expressibility]] of I. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)^2 * 45/64), and P = 2, so that G(5/4) = 2*2. This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/KE(I) over non-octave intervals, where KE(I)>0. A related definition can be extended to higher ranks: since the [[Tenney-Euclidean metrics#The OETES|OETES]] in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/KE(I). Generator complexity has the nice property that for any MOS of size N, floor(N/(C KE(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 A = <0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and B = <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. For any vector V, let max(V) - min(V) = span(V). The <em>generator complexity</em> of the temperament is P span(B). In the case of pajara, which has two periods to the octave, this would be 2*(0.631 - (-0.861)) = 2.984. This can also be described in terms of the wedgie W of the temperament, as span(2∨W), which is the span of 0 followed by the first n-1 elements of W, where n is the number of primes in the p-limit. <br /> <br /> Generator complexity satisfies the inequality, for any p-limit interval I, G(I) ≤ C KE(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 KE(I) is the <a class="wiki_link" href="/Kees%20height">Kees expressibility</a> of I. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and KE(5/4) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4 (5/4 = (4/3)^2 * 45/64), and P = 2, so that G(5/4) = 2*2.<br /> <br /> This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/KE(I) over non-octave intervals, where KE(I)>0. A related definition can be extended to higher ranks: since the <a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES">OETES</a> in the case of a rank two temperament is proportional (albeit with a different proportionality factor for each temperament) to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/KE(I).<br /> <br /> Generator complexity has the nice property that for any MOS of size N, floor(N/(C KE(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>