The wedgie
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-01-08 18:22:21 UTC.
- The original revision id was 290423187.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
=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 prime limits 7 and 11 this condition suffices for rank two, and Wº∧Wº = 0 suffices for 11-limit rank three, 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).
=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√q3√q5, |e - q3c + q7a| ≤ 2E√q3√q7 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√q5√q7), 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 c ...|| (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. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4√q5√q7), B ≤ 20/(4√q5√q7) = 240.250. This is an absurdly high badness figure; while simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.
Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = <1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.
In general, we can reconstruct W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||W|| = ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K|| = ||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded.
Original HTML content:
<html><head><title>The wedgie</title></head><body><br />
<!-- ws:start:WikiTextHeadingRule:1:<h1> --><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:<h1> --><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 <<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; 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 prime limits 7 and 11 this condition suffices for rank two, and Wº∧Wº = 0 suffices for 11-limit rank three, 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 <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 = <<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 = <<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 <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<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). <br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h1> --><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 = <<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 = <<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]]<br/>
\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 \\<br />
+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^2<br/>[[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√q3√q5, |e - q3c + q7a| ≤ 2E√q3√q7 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√q5√q7), 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 c ...|| (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. <br />
<br />
If C = ||W|| is the TE complexity, then the formula for the <a class="wiki_link" href="/Tenney-Euclidean%20metrics#Logflat TE badness">logflat badness</a> B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4√q5√q7), B ≤ 20/(4√q5√q7) = 240.250. This is an absurdly high badness figure; while simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.<br />
<br />
Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)√lb(q)√lb(p)) then wedging K = <1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.<br />
<br />
In general, we can reconstruct W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||W|| <!-- ws:start:WikiTextHeadingRule:7:<h1> --><h1 id="toc3"><a name="x||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K"></a><!-- ws:end:WikiTextHeadingRule:7 --> ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we can be conducting a search. Moreover, we have from Y∧K </h1>
((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K|| = ||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded.</body></html>