Regular temperament: Difference between revisions
Wikispaces>genewardsmith **Imported revision 203408226 - 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>2011-06-18 17:18:55 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>237477747</tt>.<br> | ||
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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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For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6. | For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6. | ||
=The Geometry of Regular Temperaments= | |||
Abstract regular temperaments can be identified with [[http://en.wikipedia.org/wiki/Rational_point|rational points]] on an [[http://en.wikipedia.org/wiki/Algebraic_variety|algebraic variety]] known as a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]]. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian **Gr**(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space **R**^n. This has an embedding into a real vector space known as the [[http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding|Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on **Gr**(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. | Abstract regular temperaments can be identified with [[http://en.wikipedia.org/wiki/Rational_point|rational points]] on an [[http://en.wikipedia.org/wiki/Algebraic_variety|algebraic variety]] known as a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]]. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian **Gr**(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space **R**^n. This has an embedding into a real vector space known as the [[http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding|Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on **Gr**(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. | ||
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Grassmannians have the structure of a smooth, homogenous [[http://en.wikipedia.org/wiki/Metric_space|metric space]], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian **Gr**(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below. | Grassmannians have the structure of a smooth, homogenous [[http://en.wikipedia.org/wiki/Metric_space|metric space]], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian **Gr**(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below. | ||
[[image:dualzoom.gif]]</pre></div> | [[image:dualzoom.gif]] | ||
=Translation between methods of specifying temperaments= | |||
The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while in some ways not the best system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in full generality. | |||
==Wedgies== | |||
To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arthimetic, not floating point numbers. | |||
==Frobenius projection maps== | |||
To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to | |||
the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.</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>Regular temperament</title></head><body>An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<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>Regular temperament</title></head><body>An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<br /> | ||
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For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.<br /> | For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->The Geometry of Regular Temperaments</h1> | ||
<br /> | <br /> | ||
Abstract regular temperaments can be identified with <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_point" rel="nofollow">rational points</a> on an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">algebraic variety</a> known as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmannian</a>. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian <strong>Gr</strong>(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space <strong>R</strong>^n. This has an embedding into a real vector space known as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow">Plücker embedding</a>, which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on <strong>Gr</strong>(r, n), though we should note that most of these do not correspond to anything worth much as a temperament.<br /> | Abstract regular temperaments can be identified with <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_point" rel="nofollow">rational points</a> on an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">algebraic variety</a> known as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmannian</a>. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian <strong>Gr</strong>(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space <strong>R</strong>^n. This has an embedding into a real vector space known as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow">Plücker embedding</a>, which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on <strong>Gr</strong>(r, n), though we should note that most of these do not correspond to anything worth much as a temperament.<br /> | ||
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Grassmannians have the structure of a smooth, homogenous <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow">metric space</a>, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian <strong>Gr</strong>(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below.<br /> | Grassmannians have the structure of a smooth, homogenous <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow">metric space</a>, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian <strong>Gr</strong>(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:32:&lt;img src=&quot;/file/view/dualzoom.gif/203369636/dualzoom.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/dualzoom.gif/203369636/dualzoom.gif" alt="dualzoom.gif" title="dualzoom.gif" /><!-- ws:end:WikiTextLocalImageRule:32 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Translation between methods of specifying temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Translation between methods of specifying temperaments</h1> | |||
The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while in some ways not the best system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in full generality.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Translation between methods of specifying temperaments-Wedgies"></a><!-- ws:end:WikiTextHeadingRule:4 -->Wedgies</h2> | |||
To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements represent primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arthimetic, not floating point numbers.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Translation between methods of specifying temperaments-Frobenius projection maps"></a><!-- ws:end:WikiTextHeadingRule:6 -->Frobenius projection maps</h2> | |||
To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to <br /> | |||
the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.</body></html></pre></div> | |||