Generator complexity: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 295288596 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 295289262 - 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 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-25 14:01:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>295289262</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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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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.</pre></div> | 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 for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Generator complexity</title></head><body>Suppose &lt;0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and &lt;0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, &lt;0 1 -2 -2| is the generator mapping val for seven limit <a class="wiki_link" href="/pajara">pajara</a>, and &lt;0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ &lt;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 /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Generator complexity</title></head><body>Suppose &lt;0 A₃ A₅ A₇ ... Ap| is the generator mapping val for a rank two temperament with P periods to the octave, and &lt;0 B₃ B₅ B₇ ... Bp| is the same val in weighted coordinates. For instance, &lt;0 1 -2 -2| is the generator mapping val for seven limit <a class="wiki_link" href="/pajara">pajara</a>, and &lt;0 1/log2(3) -2/log2(5) -2/log2(7)| ≅ &lt;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 /> | <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.</body></html></pre></div> | 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 for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.</body></html></pre></div> | ||
Revision as of 14:01, 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 14:01:13 UTC.
- The original revision id was 295289262.
- 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. 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 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 for temperaments below a certain complexity and badness bounds, allowing for a more efficient search.</body></html>