Cangwu badness
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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. If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a //dominates// b if Cb(x)-Ca(x) is a positive function for x>=0, which says that the badness of 'a' is always less than the badness of 'b' for ever choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is //indomitable//. Examples of 5-limit indomitable temperaments are: Father 16/15 |4 -1 -1> Dicot 25/24 |-3 -1 2> Meantone 81/80 |-4 4 -1> Srutal/Diaschismic 2048/2025 |11 -4 -2> Hanson/Kleismic 15625/15552 |-6 -5 6> Helmholtz/Schismic 32805/32768 |-15 8 1> Hemithirds |38 -2 -15> Ennealimmal |1 -27 18> Kwazy |-53 10 16> Monzismic |54 -37 2> Senior |-17 62 -35> Pirate |-90 -15 49> Atomic |161 -84 -12> Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.
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.<br /> <br /> If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a <em>dominates</em> b if Cb(x)-Ca(x) is a positive function for x>=0, which says that the badness of 'a' is always less than the badness of 'b' for ever choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is <em>indomitable</em>. Examples of 5-limit indomitable temperaments are:<br /> <br /> <br /> Father 16/15 |4 -1 -1><br /> Dicot 25/24 |-3 -1 2><br /> Meantone 81/80 |-4 4 -1><br /> Srutal/Diaschismic 2048/2025 |11 -4 -2><br /> Hanson/Kleismic 15625/15552 |-6 -5 6><br /> Helmholtz/Schismic 32805/32768 |-15 8 1><br /> Hemithirds |38 -2 -15><br /> Ennealimmal |1 -27 18><br /> Kwazy |-53 10 16><br /> Monzismic |54 -37 2><br /> Senior |-17 62 -35><br /> Pirate |-90 -15 49><br /> Atomic |161 -84 -12><br /> <br /> Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.</body></html>