Regular temperament: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-03 11:25:30 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-03 11:30:32 UTC</tt>.

: The original revision id was <tt>~~198399190~~</tt>.

: The original revision id was <tt>198401242</tt>.

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===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.

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.

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<h3 id="toc0"><a name="x--The Geometry of Regular Temperaments"></a>The Geometry of Regular Temperaments</h3>

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 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 <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 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 <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 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 <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 Gr(r, n), though we should note that most of these do not correspond to anything worth much as a temperament.

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 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.

<img src="/file/view/dualzoomer.gif/198395098/dualzoomer.gif" alt="dualzoomer.gif" title="dualzoomer.gif" /></body></html></pre></div>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-02-03 11:25:30 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-03 11:30:32 UTC</tt>.<br>
-: The original revision id was <tt>198399190</tt>.<br>
+: The original revision id was <tt>198401242</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>
@@ Line 34: / Line 34: @@
 ===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.
 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.
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 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--The Geometry of Regular Temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;The Geometry of Regular Temperaments&lt;/h3&gt;
 &lt;br /&gt;
-Abstract regular temperaments can be identified with &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_point" rel="nofollow"&gt;rational points&lt;/a&gt; on an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow"&gt;algebraic variety&lt;/a&gt; known as a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow"&gt;Grassmannian&lt;/a&gt;. 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 &lt;strong&gt;Gr&lt;/strong&gt;(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space &lt;strong&gt;R&lt;/strong&gt;^n. This has an embedding into a real vector space known as the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow"&gt;Plücker embedding&lt;/a&gt;, 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 &lt;strong&gt;Gr&lt;/strong&gt;(r, n), though we should note that most of these do not correspond to anything worth much as a temperament.&lt;br /&gt;
+Abstract regular temperaments can be identified with &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_point" rel="nofollow"&gt;rational points&lt;/a&gt; on an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow"&gt;algebraic variety&lt;/a&gt; known as a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow"&gt;Grassmannian&lt;/a&gt;. 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 &lt;strong&gt;Gr&lt;/strong&gt;(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space &lt;strong&gt;R&lt;/strong&gt;^n. This has an embedding into a real vector space known as the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow"&gt;Plücker embedding&lt;/a&gt;, 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 &lt;strong&gt;Gr&lt;/strong&gt;(r, n), though we should note that most of these do not correspond to anything worth much as a temperament.&lt;br /&gt;
 &lt;br /&gt;
 Grassmannians have the structure of a smooth, homogenous &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow"&gt;metric space&lt;/a&gt;, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian &lt;strong&gt;Gr&lt;/strong&gt;(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.&lt;br /&gt;
 &lt;br /&gt;
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