The Wedgie

We have defined a regular temperament in terms of icons. Part of this definition requires that icons be epimorphic mappings, and so we need a way to insure this. Another part says that an icon related by a unimodular matrix transformation gives the same temperament, and so we would like a way of labeling temperaments with something invariant under such transformations. A slick answer to both problems is to label regular temperaments by means of their wedgie.

Suppose w is a multival--that is, a wedge product of vals, using the canonical ordering of the basis elements (alphabetical on the dual monzo basis elements.) If at least one of the coefficients not zero, we take the gcd of all of the coefficients and reduce by dividing by this. (If the wedge product is a zero vector then the vals were not linearly independent and we do not have an icon.) We also change signs if the first nonzero coefficient is negative. We add an exception to this rule in the case of codimension one multivals, meaning products of pi(p)-1 vals; in this case we normalize by making the wedgie, which now is a monzo, represent a rational number greater than one.

By reducing the wedge product, we eliminate the possibility that our mappings are not epimorphisms. For instance, we might take h2 = <2 3 5| and h8 = <8 13 19|, and hope to produce an icon by [h2, h8]. However, h2+h8 = <10 16 24|; hence the sum of the two coordinates is always even, and the mapping is not epimorphic. The cokernel of the mapping is a cyclic group of order two, so we may say this mapping has torsion problems.

Taking wedge products gives us h2/\h8 = <<2 -2 -8||; the complement of this is |-8 2 2>, which represents 225/256 = (15/16)^2; hence the wedgie is <<1 -1 -4||, which we may equate with |-4 1 1>, or 15/16; reducing this a standard form where the comma in question is greater than one gives 16/15. If we now find an icon which gives this for a wedgie, we have eliminated the torsion problem. All such icons will have the same wedgie, so 16/15 defines a 5-limit linear temperament (the fourth-thirds temperament.)

Since we may also obtain wedge products as products of intervals, we may equally well define wedgies in terms of these. Whether the wedgie comes from a product of vals or a product of intervals, it defines the same temperament. Products of intervals, like products of vals, can have a gcd greater than one, indicating a torsion problem. For instance, 2048/2025/\648/625 = |11 -4 -2>/\|3 4 -4> = ||56 -28 24>>, taking the dual gives <24 38 56|, and reducing to the wedgie gives <12 19 28|. Hence they define this val, but the reduction has eliminated a torsion problem, in that the abelian group defined by the quotient of the 5-limit by the subgroup generated by 2048/2025 and 648/625 is Z x C2, ordered pairs (a, b) where a is an integer and b is an integer modulo 2. C2 is the torsion part of this group and is nontrivial, so again we may call this a torsion problem.