The wedgie: Difference between revisions
Wikispaces>genewardsmith **Imported revision 289641923 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 289643227 - 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>2012-01-04 18: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-04 18:35:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>289643227</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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If we take any three integers <<a b c|| such that GCD(a, b, c) = 1 and a ≥ 1 the result is always a wedgie, the wedgie tempering out the [[The dual|dual]] [[monzos|monzo]] |c -b a>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined. | If we take any three integers <<a b c|| such that GCD(a, b, c) = 1 and a ≥ 1 the result is always a wedgie, the wedgie tempering out the [[The dual|dual]] [[monzos|monzo]] |c -b a>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined. | ||
However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that W∧W = 0, where the "0" means the multival of rank 2r obtained by wedging W with W. For lower prime limits this condition or else Wº∧Wº = 0 suffices, but in general we need to check, for every prime q ≤ p, that (W∨q)∧W = 0, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank r multival, W∨q is a rank r-1 multival corresponding to tempering out all the commas of W, as well as q. | However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that W∧W = 0, where the "0" means the multival of rank 2r obtained by wedging W with W. For lower prime limits this condition or else Wº∧Wº = 0 suffices, but in general we need to check, for every prime q ≤ p, that (W∨q)∧W and (Wº∨qº)∨Wº = 0, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank r multival, W∨q is a rank r-1 multival corresponding to tempering out all the commas of W, as well as q. | ||
In the 7-limit case, if we wedge a prospective rank two multival W = <<a b c d e f|| with itself, we obtain W∧W = 2(af-be+cd). The quantity af-be+cd is the [[http://en.wikipedia.org/wiki/Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space **P⁵** in which wedgies live, the wedgie lies on a (four-dimensional) [[http://en.wikipedia.org/wiki/Hypersurface|hypersurfce]], known as the [[Abstract regular temperament#The Geometry of Regular Temperaments|Grassmannian]] **Gr**(2, 4). For an 11-limit rank-two wedgie W = <<w1 w2 w3 w4 w5 w6 w7 w8 w9 w10|| we have W∧W = 2<<<<w1w8-w2w6+w3w5, w1w9-w2w7+w4w5, w1w10-w3w7+w4w6, w2w10-w3w9+w4w8, w5w10-w6w9+w7w8|||| = 0. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that **Gr**(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[http://en.wikipedia.org/wiki/Algebraic_variety|algebraic variety]] in nine-dimensional projective space **P⁹**. Wedgies correspond to rational points on this variety. For rank three temperaments, the condition Wº∧Wº = 0 leads to (Wº∧Wº)º = 2<w6w1-w5w2+w4w3, w1w9-w8w2+w3w7, w1w10-w4w8+w5w7, w2w10-w4w9+w7w6, w10w3-w5w9+w8w6| = 0; again, this leads to a six-dimensional variety, this time **Gr**(3, 5). In the 13-limit, the rank-two condition for W = <<w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15|| is W∧W = 2<<<<w1w10-w2w7+w3w6, w1w11-w8w2+w4w6, w1w12-w2w9+w5w6, w1w13-w3w8+w4w7, w1w14-w3w9+w5w7, w1w15-w4w9+w5w8, w2w13-w3w11+w4w10, w2w14-w3w12+w5w10, w2w15-w4w12+w5w11, w3w15-w4w14+w5w13, w6w13-w7w11+w8w10, w6w14-w7w12+w9w10, w6w15-w8w12+w9w11, w7w15-w8w14+w9w13, w10w15-w11w14+w12w13|||| = 0; here six can be solved for in terms of the other nine, leading to an eight-dimensional variety of 13-limit rank-two temperaments, **Gr**(2, 6). For rank three, we need to invoke the full set of Plücker relations. | In the 7-limit case, if we wedge a prospective rank two multival W = <<a b c d e f|| with itself, we obtain W∧W = 2(af-be+cd). The quantity af-be+cd is the [[http://en.wikipedia.org/wiki/Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space **P⁵** in which wedgies live, the wedgie lies on a (four-dimensional) [[http://en.wikipedia.org/wiki/Hypersurface|hypersurfce]], known as the [[Abstract regular temperament#The Geometry of Regular Temperaments|Grassmannian]] **Gr**(2, 4). For an 11-limit rank-two wedgie W = <<w1 w2 w3 w4 w5 w6 w7 w8 w9 w10|| we have W∧W = 2<<<<w1w8-w2w6+w3w5, w1w9-w2w7+w4w5, w1w10-w3w7+w4w6, w2w10-w3w9+w4w8, w5w10-w6w9+w7w8|||| = 0. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that **Gr**(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[http://en.wikipedia.org/wiki/Algebraic_variety|algebraic variety]] in nine-dimensional projective space **P⁹**. Wedgies correspond to rational points on this variety. For rank three temperaments, the condition Wº∧Wº = 0 leads to (Wº∧Wº)º = 2<w6w1-w5w2+w4w3, w1w9-w8w2+w3w7, w1w10-w4w8+w5w7, w2w10-w4w9+w7w6, w10w3-w5w9+w8w6| = 0; again, this leads to a six-dimensional variety, this time **Gr**(3, 5). In the 13-limit, the rank-two condition for W = <<w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15|| is W∧W = 2<<<<w1w10-w2w7+w3w6, w1w11-w8w2+w4w6, w1w12-w2w9+w5w6, w1w13-w3w8+w4w7, w1w14-w3w9+w5w7, w1w15-w4w9+w5w8, w2w13-w3w11+w4w10, w2w14-w3w12+w5w10, w2w15-w4w12+w5w11, w3w15-w4w14+w5w13, w6w13-w7w11+w8w10, w6w14-w7w12+w9w10, w6w15-w8w12+w9w11, w7w15-w8w14+w9w13, w10w15-w11w14+w12w13|||| = 0; here six can be solved for in terms of the other nine, leading to an eight-dimensional variety of 13-limit rank-two temperaments, **Gr**(2, 6). For rank three, we need to invoke the full set of Plücker relations. | ||
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If we take any three integers &lt;&lt;a b c|| such that GCD(a, b, c) = 1 and a ≥ 1 the result is always a wedgie, the wedgie tempering out the <a class="wiki_link" href="/The%20dual">dual</a> <a class="wiki_link" href="/monzos">monzo</a> |c -b a&gt;. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined. <br /> | If we take any three integers &lt;&lt;a b c|| such that GCD(a, b, c) = 1 and a ≥ 1 the result is always a wedgie, the wedgie tempering out the <a class="wiki_link" href="/The%20dual">dual</a> <a class="wiki_link" href="/monzos">monzo</a> |c -b a&gt;. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined. <br /> | ||
<br /> | <br /> | ||
However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that W∧W = 0, where the &quot;0&quot; means the multival of rank 2r obtained by wedging W with W. For lower prime limits this condition or else Wº∧Wº = 0 suffices, but in general we need to check, for every prime q ≤ p, that (W∨q)∧W = 0, where &quot;∨&quot; denotes the <a class="wiki_link" href="/interior%20product">interior product</a>. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow">Plücker relations</a>. Note that the Plücker relations must be satisfied, since for a rank r multival, W∨q is a rank r-1 multival corresponding to tempering out all the commas of W, as well as q.<br /> | However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that W∧W = 0, where the &quot;0&quot; means the multival of rank 2r obtained by wedging W with W. For lower prime limits this condition or else Wº∧Wº = 0 suffices, but in general we need to check, for every prime q ≤ p, that (W∨q)∧W and (Wº∨qº)∨Wº = 0, where &quot;∨&quot; denotes the <a class="wiki_link" href="/interior%20product">interior product</a>. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow">Plücker relations</a>. Note that the Plücker relations must be satisfied, since for a rank r multival, W∨q is a rank r-1 multival corresponding to tempering out all the commas of W, as well as q.<br /> | ||
<br /> | <br /> | ||
In the 7-limit case, if we wedge a prospective rank two multival W = &lt;&lt;a b c d e f|| with itself, we obtain W∧W = 2(af-be+cd). The quantity af-be+cd is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pfaffian" rel="nofollow">Pfaffian</a> of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space <strong>P⁵</strong> in which wedgies live, the wedgie lies on a (four-dimensional) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hypersurface" rel="nofollow">hypersurfce</a>, known as the <a class="wiki_link" href="/Abstract%20regular%20temperament#The Geometry of Regular Temperaments">Grassmannian</a> <strong>Gr</strong>(2, 4). For an 11-limit rank-two wedgie W = &lt;&lt;w1 w2 w3 w4 w5 w6 w7 w8 w9 w10|| we have W∧W = 2&lt;&lt;&lt;&lt;w1w8-w2w6+w3w5, w1w9-w2w7+w4w5, w1w10-w3w7+w4w6, w2w10-w3w9+w4w8, w5w10-w6w9+w7w8|||| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x0. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that Gr**(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective algebraic variety in nine-dimensional projective space **P⁹. Wedgies correspond to rational points on this variety. For rank three temperaments, the condition Wº∧Wº"></a><!-- ws:end:WikiTextHeadingRule:4 --> 0. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that <strong>Gr</strong>(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">algebraic variety</a> in nine-dimensional projective space <strong>P⁹</strong>. Wedgies correspond to rational points on this variety. For rank three temperaments, the condition Wº∧Wº </h1> | In the 7-limit case, if we wedge a prospective rank two multival W = &lt;&lt;a b c d e f|| with itself, we obtain W∧W = 2(af-be+cd). The quantity af-be+cd is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pfaffian" rel="nofollow">Pfaffian</a> of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space <strong>P⁵</strong> in which wedgies live, the wedgie lies on a (four-dimensional) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hypersurface" rel="nofollow">hypersurfce</a>, known as the <a class="wiki_link" href="/Abstract%20regular%20temperament#The Geometry of Regular Temperaments">Grassmannian</a> <strong>Gr</strong>(2, 4). For an 11-limit rank-two wedgie W = &lt;&lt;w1 w2 w3 w4 w5 w6 w7 w8 w9 w10|| we have W∧W = 2&lt;&lt;&lt;&lt;w1w8-w2w6+w3w5, w1w9-w2w7+w4w5, w1w10-w3w7+w4w6, w2w10-w3w9+w4w8, w5w10-w6w9+w7w8|||| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x0. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that Gr**(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective algebraic variety in nine-dimensional projective space **P⁹. Wedgies correspond to rational points on this variety. For rank three temperaments, the condition Wº∧Wº"></a><!-- ws:end:WikiTextHeadingRule:4 --> 0. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that <strong>Gr</strong>(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">algebraic variety</a> in nine-dimensional projective space <strong>P⁹</strong>. Wedgies correspond to rational points on this variety. For rank three temperaments, the condition Wº∧Wº </h1> | ||
0 leads to (Wº∧Wº)º = 2&lt;w6w1-w5w2+w4w3, w1w9-w8w2+w3w7, w1w10-w4w8+w5w7, w2w10-w4w9+w7w6, w10w3-w5w9+w8w6| = 0; again, this leads to a six-dimensional variety, this time <strong>Gr</strong>(3, 5). In the 13-limit, the rank-two condition for W = &lt;&lt;w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15|| is W∧W = 2&lt;&lt;&lt;&lt;w1w10-w2w7+w3w6, w1w11-w8w2+w4w6, w1w12-w2w9+w5w6, w1w13-w3w8+w4w7, w1w14-w3w9+w5w7, w1w15-w4w9+w5w8, w2w13-w3w11+w4w10, w2w14-w3w12+w5w10, w2w15-w4w12+w5w11, w3w15-w4w14+w5w13, w6w13-w7w11+w8w10, w6w14-w7w12+w9w10, w6w15-w8w12+w9w11, w7w15-w8w14+w9w13, w10w15-w11w14+w12w13|||| = 0; here six can be solved for in terms of the other nine, leading to an eight-dimensional variety of 13-limit rank-two temperaments, <strong>Gr</strong>(2, 6). For rank three, we need to invoke the full set of Plücker relations.</body></html></pre></div> | 0 leads to (Wº∧Wº)º = 2&lt;w6w1-w5w2+w4w3, w1w9-w8w2+w3w7, w1w10-w4w8+w5w7, w2w10-w4w9+w7w6, w10w3-w5w9+w8w6| = 0; again, this leads to a six-dimensional variety, this time <strong>Gr</strong>(3, 5). In the 13-limit, the rank-two condition for W = &lt;&lt;w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15|| is W∧W = 2&lt;&lt;&lt;&lt;w1w10-w2w7+w3w6, w1w11-w8w2+w4w6, w1w12-w2w9+w5w6, w1w13-w3w8+w4w7, w1w14-w3w9+w5w7, w1w15-w4w9+w5w8, w2w13-w3w11+w4w10, w2w14-w3w12+w5w10, w2w15-w4w12+w5w11, w3w15-w4w14+w5w13, w6w13-w7w11+w8w10, w6w14-w7w12+w9w10, w6w15-w8w12+w9w11, w7w15-w8w14+w9w13, w10w15-w11w14+w12w13|||| = 0; here six can be solved for in terms of the other nine, leading to an eight-dimensional variety of 13-limit rank-two temperaments, <strong>Gr</strong>(2, 6). For rank three, we need to invoke the full set of Plücker relations.</body></html></pre></div> | ||