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; 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.

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 that W∧W = 2<<<<w1w8-w2w6+w3w5, w1w9-w2w7+w4w5, w1w10-w3w7+w4w6, w2w10-w3w9+w4w8, w5w10-w6w9+w7w8|||| is zero. 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.

=Constrained wedgies=
Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank two wedgie W = <<a b c d e f||.

By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = <<a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that

[[math]]
\displaystyle (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\
+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^2
[[math]]

For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2E√q3q5, |e - q3c + q7a| ≤ 2E√q3q7 and |f - q5c + q7d| ≤ 2E√q5q7. This has an interesting interpretation: since <1 q3 q5 q7|∧<0 a b c| = <<a  b  c  q3b-q5a  q3c-q7a  q5c-q7b||, if E ≤ 1/(4√q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples <<a b d ...|| (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound. 

If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean+metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE.

Original HTML content:

<html><head><title>The wedgie</title></head><body><br />
<!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:1 -->Basics</h1>
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 />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Conditions on being a wedgie"></a><!-- ws:end:WikiTextHeadingRule:3 -->Conditions on being a wedgie</h1>
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 />
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 />
<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 that W∧W = 2&lt;&lt;&lt;&lt;w1w8-w2w6+w3w5, w1w9-w2w7+w4w5, w1w10-w3w7+w4w6, w2w10-w3w9+w4w8, w5w10-w6w9+w7w8|||| is zero. 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 = 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="Constrained wedgies"></a><!-- ws:end:WikiTextHeadingRule:5 -->Constrained wedgies</h1>
Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">relative error</a> E, aka simple badness, constrains a 7-limit rank two wedgie W = &lt;&lt;a b c d e f||.<br />
<br />
By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &lt;&lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. From this we may conclude that<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
\displaystyle (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\&lt;br /&gt;
+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^2&lt;br/&gt;[[math]]
 --><script type="math/tex">\displaystyle (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\
+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^2</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
For this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2E√q3q5, |e - q3c + q7a| ≤ 2E√q3q7 and |f - q5c + q7d| ≤ 2E√q5q7. This has an interesting interpretation: since &lt;1 q3 q5 q7|∧&lt;0 a b c| = &lt;&lt;a  b  c  q3b-q5a  q3c-q7a  q5c-q7b||, if E ≤ 1/(4√q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with &lt;1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming &quot;reasonable&quot; requires this bound to be met, searching through triples &lt;&lt;a b d ...|| (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with &lt;1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound. <br />
<br />
If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean+metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE.</body></html>