Generator complexity: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 295327618 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 295329596 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-25 15: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-25 15:39:17 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>295329596</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and OH(5/4) = Tenney(5) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4, and P = 2, so that G(5/4) = 2*2. | 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. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and OH(5/4) = Tenney(5) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4, 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)/OH(I) over non-octave intervals, where OH(I)>0. A related definition can be extended to higher limits: since the [[Tenney-Euclidean metrics#The OETES|OETES]] in the case of a rank two temperament is proportional to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/OH(I). | This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/OH(I) over non-octave intervals, where OH(I)>0. A related definition can be extended to higher limits: 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)/OH(I). | ||
Generator complexity has the nice property that 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.</pre></div> | Generator complexity has the nice property that 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.</pre></div> | ||
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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. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and OH(5/4) = Tenney(5) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4, and P = 2, so that G(5/4) = 2*2.<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. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and OH(5/4) = Tenney(5) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4, and P = 2, so that G(5/4) = 2*2.<br /> | ||
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This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/OH(I) over non-octave intervals, where OH(I)&gt;0. A related definition can be extended to higher limits: since the <a class="wiki_link" href="/Tenney-Euclidean%20metrics#The OETES">OETES</a> in the case of a rank two temperament is proportional to G(I), we can define a complexity measure for any rank of temperament by C = sup OETES(I)/OH(I).<br /> | This inequality can be used to give an alternative definition of generator complexity: C = sup G(I)/OH(I) over non-octave intervals, where OH(I)&gt;0. A related definition can be extended to higher limits: 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)/OH(I).<br /> | ||
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Generator complexity has the nice property that 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></pre></div> | Generator complexity has the nice property that 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></pre></div> | ||
Revision as of 15:39, 25 January 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-01-25 15:39:17 UTC.
- The original revision id was 295329596.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
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. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and OH(5/4) = Tenney(5) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4, 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)/OH(I) over non-octave intervals, where OH(I)>0. A related definition can be extended to higher limits: 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)/OH(I). Generator complexity has the nice property that 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. So for instance, in meantone G(5/4) = 4, since it requires four generator steps to get to 5/4, and OH(5/4) = Tenney(5) = log2(5). In pajara, G(5/4) = 4 also, since two generator steps are required to get to 5/4, 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)/OH(I) over non-octave intervals, where OH(I)>0. A related definition can be extended to higher limits: 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)/OH(I).<br /> <br /> Generator complexity has the nice property that 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>