Regular temperament: Difference between revisions
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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> | ||
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=The Geometry of Regular Temperaments= | =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. In matrix terms, the real Grassmannian **Gr**(r, n) can be identified with real symmetric projection matrices with trace r. The rational symmetric projection matrices with trace r are precisely the Frobenius projections, so under this identification it is clear they represent rational points on **Gr**(r, n). A rational projection matrix of trace r which is not symmetric is still a tuning map; minimax and least squares tunings provide examples of this. | ||
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. | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->The Geometry of Regular Temperaments</h1> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->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. In matrix terms, the real Grassmannian <strong>Gr</strong>(r, n) can be identified with real symmetric projection matrices with trace r. The rational symmetric projection matrices with trace r are precisely the Frobenius projections, so under this identification it is clear they represent rational points on <strong>Gr</strong>(r, n). A rational projection matrix of trace r which is not symmetric is still a tuning map; minimax and least squares tunings provide examples of this.<br /> | ||
<br /> | <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 /> | 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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