Cangwu badness: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 242509021 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 288359276 - 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- | : 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> | : The original revision id was <tt>288359276</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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[[math]] | [[math]] | ||
where the | 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. | 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 | 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><!-- ws:end:WikiTextMathRule:0 --><br /> | --><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 /> | <br /> | ||
where the | 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.<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 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.<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 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.<br /> | ||
<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 | 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 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 /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
Revision as of 02:57, 24 December 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 2011-12-24 02:57:27 UTC.
- The original revision id was 288359276.
- 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 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: 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. 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.
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 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.<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 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.<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 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 /> <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.<br /> <br /> 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>