Cangwu badness: Difference between revisions
Wikispaces>genewardsmith **Imported revision 242428747 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 242432815 - 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 12: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-22 12:36:13 UTC</tt>.<br> | ||
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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. | 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 | 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 every 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: | ||
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Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal. | Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal. | ||
If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better. | |||
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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 /> | 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 /> | ||
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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&gt;=0, which says that the badness of 'a' is always less than the badness of 'b' for | 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&gt;=0, which says that the badness of 'a' is always less than the badness of 'b' for every 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 /> | ||
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Atomic |161 -84 -12&gt;<br /> | Atomic |161 -84 -12&gt;<br /> | ||
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Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.</body></html></pre></div> | Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.<br /> | ||
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If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.</body></html></pre></div> | |||