Regular temperament: Difference between revisions
Wikispaces>genewardsmith **Imported revision 203405762 - Original comment: ** |
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>2011-02-19 22: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-19 22:52:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>203408226</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html"> | <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;white-space: pre-wrap ! important" class="old-revision-html">An //abstract regular temperament// is a [[regular temperament]] 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 | ||
An //abstract regular temperament// is a [[regular temperament]] 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 | |||
* **The [[Wedgies and Multivals|wedgie]]** | * **The [[Wedgies and Multivals|wedgie]]** | ||
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[[image:dualzoom.gif]]</pre></div> | [[image:dualzoom.gif]]</pre></div> | ||
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<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 | <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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<ul><li><strong>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></strong></li></ul>This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos. <br /> | <ul><li><strong>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></strong></li></ul>This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos. <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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->The Geometry of Regular Temperaments</h3> | ||
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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 /> | ||
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