Cangwu badness: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 242414813 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 242415221 - 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>2011-07-22 10: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-22 10:13:29 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>242415221</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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[[math]] | [[math]] | ||
\displaystyle C(x) = det([(1+x)v_i \cdot v_j - | \displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j]) | ||
[[math]] | [[math]] | ||
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<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\displaystyle C(x) = det([(1+x)v_i \cdot v_j - | \displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\displaystyle C(x) = det([(1+x)v_i \cdot v_j - | --><script type="math/tex">\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
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where the v_i are r independent weighted vals of dimension n defining a rank r regular temperament, and the a_i are the average of the values of v_i; that is, the sum of the values of v_i divided by n.<br /> | where the v_i are r independent weighted vals of dimension n defining a rank r regular temperament, and the a_i are the average of the values of v_i; that is, the sum of the values of v_i divided by n.<br /> | ||
<br /> | <br /> | ||
From this definition, it follows that C(0) is proportional to the square of <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">simple badness</a>, aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE Complexity">TE complexity</a>. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.</body></html></pre></div> | From this definition, it follows that C(0) is proportional to the square of <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">simple badness</a>, aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE Complexity">TE complexity</a>. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.</body></html></pre></div> | ||
Revision as of 10:13, 22 July 2011
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 2011-07-22 10:13:29 UTC.
- The original revision id was 242415221.
- 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:
//Cangwu badness// is a polynomial function badness measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham Breed]] thought it up. It is defined in terms of a matrix determinant as [[math]] \displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j]) [[math]] where the v_i are r independent weighted vals of dimension n defining a rank r regular temperament, and the a_i are the average of the values of v_i; that is, the sum of the values of v_i divided by n. From this definition, it follows that C(0) is proportional to the square of [[Tenney-Euclidean temperament measures#TE simple badness|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to [[Tenney-Euclidean temperament measures#TE Complexity|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.
Original HTML content:
<html><head><title>Cangwu badness</title></head><body><em>Cangwu badness</em> is a polynomial function badness measure; the name stems from Cangwu Green Park, Lianyungang, China, where <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a> thought it up. It is defined in terms of a matrix determinant as<br /> <br /> <!-- ws:start:WikiTextMathRule:0: [[math]]<br/> \displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])<br/>[[math]] --><script type="math/tex">\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])</script><!-- ws:end:WikiTextMathRule:0 --><br /> <br /> where the v_i are r independent weighted vals of dimension n defining a rank r regular temperament, and the a_i are the average of the values of v_i; that is, the sum of the values of v_i divided by n.<br /> <br /> From this definition, it follows that C(0) is proportional to the square of <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">simple badness</a>, aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE Complexity">TE complexity</a>. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.</body></html>