Tenney–Euclidean temperament measures: Difference between revisions
Wikispaces>genewardsmith **Imported revision 471861376 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 472088034 - 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>2013-11- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-25 11:42:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>472088034</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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TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents. | TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents. | ||
From the ratio (||J∧M||/||M||)^2 we obtain C(n, r+1)/(n C(n, r)) = (n-r)/(n (r+1)). If we take the ratio of this for rank one with this for rank r, the "n" cancels, and we get (n-1)/2 * (r+1)/(n-r) = (r+1)(n-1)/(2(n-r)). It follows that | From the ratio (||J∧M||/||M||)^2 we obtain C(n, r+1)/(n C(n, r)) = (n-r)/(n (r+1)). If we take the ratio of this for rank one with this for rank r, the "n" cancels, and we get (n-1)/2 * (r+1)/(n-r) = (r+1)(n-1)/(2(n-r)). It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank r temperament then | ||
[[math]] | [[math]] | ||
\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi | \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi | ||
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[[math]] | [[math]] | ||
\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi | \displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi | ||
[[math]]</pre></div> | [[math]] | ||
Graham's TE error and val normalized TE error both have the advantage that higher rank temperament error corresponds directly to rank one error: if ξ is the error function and T the TE tuning map, then ξ(T) = ξ(M). On the other hand, TE error as defined here and val normalized error both have the advantage that for tunings T, which are rank one and hence where the two are identical, ξ(T) = ||T∧J||/||T|| is sin θ, where θ is the angle between the J (a unit vector in tuning space under the normalization here) and u = T/||T||, the direction vector for T.</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>Tenney-Euclidean temperament measures</title></head><body><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#TE Complexity">TE Complexity</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#TE simple badness">TE simple badness</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#TE error">TE error</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <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>Tenney-Euclidean temperament measures</title></head><body><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#TE Complexity">TE Complexity</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#TE simple badness">TE simple badness</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#TE error">TE error</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | ||
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TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents. <br /> | TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents. <br /> | ||
<br /> | <br /> | ||
From the ratio (||J∧M||/||M||)^2 we obtain C(n, r+1)/(n C(n, r)) = (n-r)/(n (r+1)). If we take the ratio of this for rank one with this for rank r, the &quot;n&quot; cancels, and we get (n-1)/2 * (r+1)/(n-r) = (r+1)(n-1)/(2(n-r)). It follows that | From the ratio (||J∧M||/||M||)^2 we obtain C(n, r+1)/(n C(n, r)) = (n-r)/(n (r+1)). If we take the ratio of this for rank one with this for rank r, the &quot;n&quot; cancels, and we get (n-1)/2 * (r+1)/(n-r) = (r+1)(n-1)/(2(n-r)). It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank r temperament then<br /> | ||
<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi&lt;br/&gt;[[math]] | \displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi</script><!-- ws:end:WikiTextMathRule:3 --></body></html></pre></div> | --><script type="math/tex">\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
Graham's TE error and val normalized TE error both have the advantage that higher rank temperament error corresponds directly to rank one error: if ξ is the error function and T the TE tuning map, then ξ(T) = ξ(M). On the other hand, TE error as defined here and val normalized error both have the advantage that for tunings T, which are rank one and hence where the two are identical, ξ(T) = ||T∧J||/||T|| is sin θ, where θ is the angle between the J (a unit vector in tuning space under the normalization here) and u = T/||T||, the direction vector for T.</body></html></pre></div> | |||