Cangwu badness: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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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 the square of [[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 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//. An alternative procedure is to divide the cangwu badness polynomial by the ~~square~~ of ~~TE complexity~~, getting the square of TE absolute error ~~when the polynomial is zero~~. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

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//. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

Examples of 5-limit indomitable temperaments are:

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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 the square of <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.

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. An alternative procedure is to divide the cangwu badness polynomial by the ~~square~~ of ~~TE complexity~~, getting the square of TE absolute error ~~when the polynomial is zero~~. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

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. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

Examples of 5-limit indomitable temperaments are:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-06-11 02:40:08 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-11 02:45:36 UTC</tt>.<br>
-: The original revision id was <tt>344324010</tt>.<br>
+: The original revision id was <tt>344324720</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>
@@ Line 16: / Line 16: @@
 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 the square of [[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 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//. An alternative procedure is to divide the cangwu badness polynomial by the square of TE complexity, getting the square of TE absolute error when the polynomial is zero. When one temperament dominates another using this polynomial, it is lower in both error and complexity.
+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//. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.
 Examples of 5-limit indomitable temperaments are:
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 From this definition, it follows that C(0) is proportional to the square of &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&gt;simple badness&lt;/a&gt;, aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of &lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE Complexity"&gt;TE complexity&lt;/a&gt;. 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.&lt;br /&gt;
 &lt;br /&gt;
-If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a &lt;em&gt;dominates&lt;/em&gt; 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 &lt;em&gt;indomitable&lt;/em&gt;. An alternative procedure is to divide the cangwu badness polynomial by the square of TE complexity, getting the square of TE absolute error when the polynomial is zero. When one temperament dominates another using this polynomial, it is lower in both error and complexity.&lt;br /&gt;
+If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a &lt;em&gt;dominates&lt;/em&gt; 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 &lt;em&gt;indomitable&lt;/em&gt;. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.&lt;br /&gt;
 &lt;br /&gt;
 Examples of 5-limit indomitable temperaments are:&lt;br /&gt;