The wedgie
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=Basics= 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. =Conditions on being a wedgie= 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; 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. If the rank r is greater than n/2, we are also confronted by the dual condition: Wº∧Wº = 0, where Wº is the dual multimonzo to the multival W. 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).
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<html><head><title>The wedgie</title></head><body><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basics</h1> The <a class="wiki_link" href="/Wedgies%20and%20Multivals">//wedgie//</a> 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 /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Conditions on being a wedgie"></a><!-- ws:end:WikiTextHeadingRule:2 -->Conditions on being a wedgie</h1> 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 <a class="wiki_link" href="/The%20dual">dual</a> <a class="wiki_link" href="/monzos">monzo</a> |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. <br /> <br /> However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; 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. If the rank r is greater than n/2, we are also confronted by the dual condition: Wº∧Wº = 0, where Wº is the dual multimonzo to the multival W.<br /> <br /> 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 <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).</body></html>