Cangwu badness
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//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 - na_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 - na_ia_j])<br/>[[math]] --><script type="math/tex">\displaystyle C(x) = det([(1+x)v_i \cdot v_j - na_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>