Wedgies and Multivals
An alternating multilinear map which is a multilinear function taking a certain number n of monzos as arguments and returning an integer as a value we may call an n-map. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
The simplest kind of n-map is the 1-map, or val. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a group homomorphism and a module homomorphism. Vals are linear: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).
One use for such things is as "machines" for measuring complexity. If we consider the 1-map which is the val for 11-limit 31et, we find we have <31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 72 in this system. Now consider a 2-map "meantone(u,v)" which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, in which case we have meantone(2,3)=1, meantone(2,5)=4, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.
Given an n-map f and an m-map g we may define a new (n+m)-map, the wedge product of f and g, written f∧g, as follows:
where the sum is taken over S(n,m), the set of all permutations of the first n+m integers which are an (n,m) shuffles, and sgn(t) is the parity of the permutation t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.
If f and g are both vals (1-maps) then this becomes especially easy: f∧g(u,v) = f(u)g(v) - f(v)g(u). Let's consider a specific example. Suppose E19 = <19 30 44 53| is the equal temperament val for septimal 19et, and E31 = <31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have
We may continue in this way to consider (2,5), (2,7),, (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a bival. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to rank two temperaments such as meantone, trivals to rank three temperaments, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone.
This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the GCD of all of the coordinates is 1. An n-map with these properties we may call reduced, and reduced n-vals can be used to give unique names to regular temperaments.
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 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.