Cangwu badness: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-~~23 03~~:07:59 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-24 02:57:27 UTC</tt>.

: The original revision id was <tt>~~242509021~~</tt>.

: The original revision id was <tt>288359276</tt>.

: The revision comment was: <tt></tt>

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[[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.

where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ 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 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//. Examples of 5-limit indomitable temperaments are:

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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--><script type="math/tex">\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])</script>

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.

where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.

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 ~~&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 indomitable. Examples of 5-limit indomitable temperaments are:

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:

@@ 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>2011-07-23 03:07:59 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-24 02:57:27 UTC</tt>.<br>
-: The original revision id was <tt>242509021</tt>.<br>
+: The original revision id was <tt>288359276</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 12: / Line 12: @@
 [[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.
+where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ 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 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 &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 //indomitable//. Examples of 5-limit indomitable temperaments are:
+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:
@@ Line 45: / Line 45: @@
   --&gt;&lt;script type="math/tex"&gt;\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-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.&lt;br /&gt;
+where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.&lt;br /&gt;
 &lt;br /&gt;
 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 &amp;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 &lt;em&gt;indomitable&lt;/em&gt;. Examples of 5-limit indomitable temperaments are:&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;. Examples of 5-limit indomitable temperaments are:&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;