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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | #REDIRECT [[Wedgie/Archived version]] |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-04 19:38:31 UTC</tt>.<br>
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| : The original revision id was <tt>289653241</tt>.<br>
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| : The revision comment was: <tt></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>
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| <h4>Original Wikitext content:</h4>
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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;white-space: pre-wrap ! important" class="old-revision-html">
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| =Basics=
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| The //[[Wedgies and Multivals|wedgie]]// is a way of defining and working with an [[abstract regular temperament]]. If one takes r independent [[vals]] in a p-limit group of n primes, then the wedgie is defined by taking the [[Wedgies and Multivals|wedge product]] of the vals, and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, it is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie. Wedgies are in a one-to-one relationship with abstract regular temperaments; that is, regular temperaments where no tuning has been decided on.
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| =Conditions on being a wedgie=
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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.
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| 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 and every basis val v sending q to 1 and everything else to 0, that (W∨q)∧W and (W∧v)º∧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.
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| 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 <<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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| </pre></div>
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| <h4>Original HTML content:</h4>
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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>The wedgie</title></head><body><br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basics</h1>
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| The <em><a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></em> is a way of defining and working with an <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a>. If one takes r independent <a class="wiki_link" href="/vals">vals</a> in a p-limit group of n primes, then the wedgie is defined by taking the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedge product</a> of the vals, and dividing out the greatest common divisior of the coefficients, to produce an r-multival. If the first non-zero coefficient of this multival is negative, it is then scalar multiplied by -1, changing the sign of the first non-zero coefficient to be positive. The result is the wedgie. Wedgies are in a one-to-one relationship with abstract regular temperaments; that is, regular temperaments where no tuning has been decided on.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Conditions on being a wedgie"></a><!-- ws:end:WikiTextHeadingRule:2 -->Conditions on being a wedgie</h1>
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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 />
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| <br />
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| 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 and every basis val v sending q to 1 and everything else to 0, that (W∨q)∧W and (W∧v)º∧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 />
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| <br />
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| 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 &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>
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| 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>
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