Tenney–Euclidean temperament measures: Difference between revisions
Wikispaces>genewardsmith **Imported revision 472327808 - 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>2013-11-26 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-26 11:43:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>472438896</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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\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi | \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi | ||
[[math]] | [[math]] | ||
is an adjusted error | is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's [[http://x31eq.com/temper/|temperament finder]], which we will call "Graham error"; if G denotes Graham error for a temperament, then | ||
[[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]] | [[math]] | ||
Graham | Graham error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the Graham normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. Graham error may also be found using the TE tuning map; if T is the tuning map, then | ||
[[math]] | |||
\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}} | |||
[[math]] | |||
where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = <1200 1200 ... 1200|.</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: | <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:13:&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:13 --><!-- ws:start:WikiTextTocRule:14: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#TE Complexity">TE Complexity</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#TE simple badness">TE simple badness</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#TE error">TE error</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:5 -->Introduction</h1> | ||
Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">multival</a> or multimonzo which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge product</a> of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_mean_square" rel="nofollow">root mean square</a>) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||.<br /> | Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">multival</a> or multimonzo which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge product</a> of weighted vals or monzos, we may define a norm by means of the usual Euclidean norm. We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_mean_square" rel="nofollow">root mean square</a>) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc1"><a name="TE Complexity"></a><!-- ws:end:WikiTextHeadingRule:7 -->TE Complexity</h1> | ||
Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the <em>complexity</em> of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been <a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow">extensively studied</a> by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>, and we may call it Tenney-Euclidean complexity since it can be defined in terms of the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean norm</a>.<br /> | Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> M, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||M|| is a measure of the <em>complexity</em> of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. This complexity and related measures have been <a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow">extensively studied</a> by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>, and we may call it Tenney-Euclidean complexity since it can be defined in terms of the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean norm</a>.<br /> | ||
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If V is a matrix whose rows are the weighted vals vi, we may write det([vi.vj]) as det(VV*), where V* denotes the transpose. If W is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Diagonal_matrix" rel="nofollow">diagonal matrix</a> with 1, 1/log2(3), ..., 1/log2(p) along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV*) = det(AW^2A*). This may be related to the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">TE tuning projection matrix</a> P, which is V*(VV*)^(-1)V, and the corresponding matix for unweighted monzos <strong>P</strong> = A*(AW^2A*)^(-1)A.<br /> | If V is a matrix whose rows are the weighted vals vi, we may write det([vi.vj]) as det(VV*), where V* denotes the transpose. If W is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Diagonal_matrix" rel="nofollow">diagonal matrix</a> with 1, 1/log2(3), ..., 1/log2(p) along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV*) = det(AW^2A*). This may be related to the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">TE tuning projection matrix</a> P, which is V*(VV*)^(-1)V, and the corresponding matix for unweighted monzos <strong>P</strong> = A*(AW^2A*)^(-1)A.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc2"><a name="TE simple badness"></a><!-- ws:end:WikiTextHeadingRule:9 -->TE simple badness</h1> | ||
If J = &lt;1 1 ... 1| is the JI point in weighted coordinates, then the simple badness of M, which we may also call the relative error of M, is defined as ||J∧M||. This may considered to be a sort of badness which heavily favors complex temperaments, or it may be considered error relativized to the complexity of the temperament: relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that ai = J.vi/n is the mean value of the entries of vi. Then note that J∧(v1-a1J)∧(v2-a2J)∧...∧(vr-arJ) = J∧v1∧v2∧...∧vr, since wedge products with more than one term are zero. The Gram matrix of the vectors J and v1-aiJ will have n as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:<br /> | If J = &lt;1 1 ... 1| is the JI point in weighted coordinates, then the simple badness of M, which we may also call the relative error of M, is defined as ||J∧M||. This may considered to be a sort of badness which heavily favors complex temperaments, or it may be considered error relativized to the complexity of the temperament: relative error is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. Once again, if we have a list of vectors we may use a Gramian to compute relative error. First we note that ai = J.vi/n is the mean value of the entries of vi. Then note that J∧(v1-a1J)∧(v2-a2J)∧...∧(vr-arJ) = J∧v1∧v2∧...∧vr, since wedge products with more than one term are zero. The Gram matrix of the vectors J and v1-aiJ will have n as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:<br /> | ||
<br /> | <br /> | ||
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--><script type="math/tex">\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex">\displaystyle ||J \wedge M|| = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc3"><a name="TE error"></a><!-- ws:end:WikiTextHeadingRule:11 -->TE error</h1> | ||
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 /> | ||
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\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi&lt;br/&gt;[[math]] | \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
is an adjusted error | is an adjusted error which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. ψ can be related to TE error as it appears on Graham Breed's <a class="wiki_link_ext" href="http://x31eq.com/temper/" rel="nofollow">temperament finder</a>, which we will call &quot;Graham error&quot;; if G denotes Graham error for a temperament, then<br /> | ||
<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[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 --><br /> | --><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 | Graham error and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the Graham normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. Graham error may also be found using the TE tuning map; if T is the tuning map, then<br /> | ||
<!-- ws:start:WikiTextMathRule:4: | |||
[[math]]&lt;br/&gt; | |||
\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\displaystyle G = \sqrt{\frac{(T -J) \cdot (T - J)}{n}}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | |||
where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = &lt;1200 1200 ... 1200|.</body></html></pre></div> | |||