Wedgie/Archived version: Difference between revisions

A wedgie is written as a list of entries that give the values of the wedgie on the [[basis]] elements of the [[JI subgroup]] that the temperament is on. By the alternating property [i.e. W('''u''', '''v''') = −W('''v''', '''u''')] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. The simplest example is rank-2 wedgies: Let a and b be (non-[[contorted]]) vals on a [[JI subgroup]] ''q''₁.[…].''q''_''n'' (where the ''q''_''i'' need not be prime). Then the entries of the wedgie W corresponding to the rank-2 temperament a&b of the JI subgroup ''q''₁.[…].''q''_''n'' are (ignoring sign and normalization):

where bolded variables and numbers represent the ordinary numbers written in [[monzo]] form.

For the ''p''_''n''-prime limit, the entries of W are conventionally listed in the order  

For example, a 5-limit wedgie is of the form

and a 7-limit wedgie is of the form

More generally, if one takes ''r'' independent [[vals]] V₁, …, V_''r'' in a rank-''n'' [[JI subgroup]] ''q''₁.[…].''q''_''n'', then the wedgie for the rank-''r'' temperament V₁&…&V_''r'' is defined by:

The result is the wedgie of the rank-''r'' temperament V₁&…&V_''r'', whose entries are (ignoring steps 2 and 3):

== How the period and generator falls out of a rank-2 wedgie ==

These reduced n-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [[Wikipedia: Projective space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called ''contorted''. Wedgies do not name or signify contorted temperaments.

In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes. Expressions where there are [math] dx_i [/math] can be called 1 forms,  [math] dx_i \wedge dx_j [/math] 2 forms etc.

and one can check that  [math] \alpha=b_1\wedge b_2 \wedge b_3[/math]. Note by the way that n-1 forms are always decomposable (here n=4 and we computed the decomposition of 3 form).

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, [[Wedgies_and_Multivals|contortion]] has appeared.

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.  

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 [[Wikipedia: 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) [[Wikipedia: Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#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 [[Wikipedia: Algebraic variety|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).

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||.

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/(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.

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 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.

*[[Intro to exterior algebra for RTT]]: for detailed background about and further explanations for how to work with multivectors and multicovectors such as wedgies

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