The wedgie

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The 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 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.

Truncation of wedgies

A useful operation to perform on any multivector, including wedgies, is truncation of the wedgie to a lower prime limit. This in effect sets all the basis vectors of a p-limit wedgie which are greater than q, the prime limit being truncated to, to zero. An algorithm to produce the truncation is to list the r-subsets of the primes to p in alphabetical order, and add the corresponding coefficient to the list of the q-limit truncation if and only if the maximum prime in the r-subet is less than or equal to q. Truncating a wedgie can lead to a non-wedgie if the GCD of the coefficients is greater than one; this means that in the lower limit, contortion has appeared.

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 dual 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, 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 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 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) hypersurface, known as the 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 algebraic variety in nine-dimensional projective space P⁹. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have w6w1-w5w2+w4w3 = w9w1-w8w2+w7w3 = w10w1-w8w4+w7w5 = w10w2-w9w4+w7w6 = w10w3-w9w5+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 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)||. We now have

[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]

From this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2Eq3q5, |e - q3c + q7a| ≤ 2Eq3q7 and |f - q5c + q7d| ≤ 2Eq5q7. 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/(4q5q7), 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 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/(4q5q7), B ≤ 20/(4q5q7) = 0.767. This badness figure is easily met. 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.

Reconstituting wedgies in general

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.

More generally, we can reconstitute 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-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-Y)+Y||=||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 could 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. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call recoverable, can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as TE complexity which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is generator complexity.

In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes E ≤ 1/(2√5 q5q7q11), then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.