Cangwu badness: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 242415221 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 242428747 - 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 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-22 12:02:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>242428747</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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 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.</pre></div> | 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. | |||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 /> | ||
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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 /> | 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 /> | <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></pre></div> | 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&gt;=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&gt;<br /> | |||
Dicot 25/24 |-3 -1 2&gt;<br /> | |||
Meantone 81/80 |-4 4 -1&gt;<br /> | |||
Srutal/Diaschismic 2048/2025 |11 -4 -2&gt;<br /> | |||
Hanson/Kleismic 15625/15552 |-6 -5 6&gt;<br /> | |||
Helmholtz/Schismic 32805/32768 |-15 8 1&gt;<br /> | |||
Hemithirds |38 -2 -15&gt;<br /> | |||
Ennealimmal |1 -27 18&gt;<br /> | |||
Kwazy |-53 10 16&gt;<br /> | |||
Monzismic |54 -37 2&gt;<br /> | |||
Senior |-17 62 -35&gt;<br /> | |||
Pirate |-90 -15 49&gt;<br /> | |||
Atomic |161 -84 -12&gt;<br /> | |||
<br /> | |||
Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.</body></html></pre></div> | |||
Revision as of 12:02, 22 July 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-07-22 12:02:22 UTC.
- The original revision id was 242428747.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
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>