The Wedge Product
If we take a p-limit val a = a2 v2 + a3 v3 + ... + ap vp and another b = b2 v2 + b3 v3 + ... + bp vp, where the coefficients aq and bq are integers and vq is the q-adic valuation; we define the wedge product a /\ b of a and b to be an element of a new abelian group whose basis consists of the elements vq /\ vr for primes q < r <= p, and which again has integer coefficients. The rule for evaluating this product is that the product is distributive and anticommutative; the latter condition means for two vals a and b, a /\ b = - b /\ a, which is true in particular of the q-adic valuations which we use to define the basis. Since these are p-limit vals, we may write them as bra vectors; the wedge product then may be written
<a2 a3 ... ap|/\<b2 b3 ... bp| = <<c23 c25 ... cqp||
where q is the largest prime less than p and crs, for primes numbers r and s, is the coefficient for vr/\vs.
We may call vals 1-vals, a wedge a product of two vals a 2-val, and so forth. We may go on to define multiple wedge products up to m=pi(p) and the m-val, and may consider integers to be 0-vals. We get an abelian group from each of the n-vals by taking sums with integer coefficients of wedge products, which we can restrict to just sums of products of basis p-adic vals in ascending prime order. If we have a product of more than two vals, v1/\v2/\.../\vn, then every time we exchange two vals in the product we also change sign; this, of course, also applies to products of our basis p-adic vals. We therefore can write everything in terms of a canonical basis of n-wedges consisting of p-adic vals in ascending prime order. Assuming this canonical basis, we can write 2-vals using the notation <<...||, 3-vals <<<...||| and so forth.
To give an example, let h12 = 12v2 + 19v3 + 28v5 = <12 19 28| be one val, and h7 = 7v2 + 11v3 + 16v5 = <7 11 16| be another val. Then
<12 19 28|/\<7 11 16| = (12v2 + 19v3 + 28v5) /\ (7v2 + 11v3 + 16v5) =
132 v2/\v3 + 192 v2/\v5 + 133 v3/\v2 + 304 v3/\v5 + 196 v5 /\v2 + 308 v5/\v3 =
132 v2/\v3 + 192 v2/\v5 - 133 v2/\v3 + 304 v3/\v5 -196 v2/\v5 - 308 v3/\v5 =
-v2/\v3 - 4 v2/\v5 - 4 v3/\v5 = <<-1 -4 -4||
If we wedge the above by yet another val, h9 = 9v2 + 14v3 + 21v5, we get
<<-1 -4 -4||/\<9 14 21| = (-v2/\v3 - 4 v2/\v5 - 4 v3/\v5)/\(9v2 + 14v3 + 21v5) =
-21 v2/\v3/\v5 - 56 v2/\v5/\v3 - 36 v3/\v5/\v2 = -21 v2/\v3/\v5 + 56 v2/\v3/\v5 -36 v2/\v3/\v5 = - v2/\v3/\v5 = <<<-1|||
Here v2/\v3/\v5 is the basis for a one-dimensional space, so we can equate this to a number (what a physicist might call a pseudo-scalar) which in this case is -1.
Everything we have said so far about wedge products of vals can be equally well applied to wedge products of intervals, where now the basis vectors are exponents of one for the prime numbers, which we may write e2, e3, e5 and so forth, so that for instance 81/80 = 2^(-4) 3^4 5^(-1) would be written -4e2 + 4e3 - e5 = |-4 4 -1> where now instead of a bra vector we have a ket vector--ie, a monzo. We obtain in this way two distinct sets of wedge products, the 1, 2, 3, etc. wedges of vals and the 1, 2, 3, etc wedges of monzos. These sets, however, are closely related and may in fact be identified, in the same way that a space with one basis vector may be identified with scalars.
We have between vals <h| and monzos |q> a natural relationship, since the inner product <h|q> is an integer. We can extend this to a relationship between n-vals and n-monzos, since the n by n determinant
[<h1|q1> ... <hn|q1>]
[<h1|qn> ... <hn|qn>]
again gives us an integer.
However, we also get an integer by taking the wedge product of an n-wedge and an (m-n)-wedge, where m = pi(p) is the dimension of the space both of vals and of monzos (or the rank of the group, more precisely.) This is because the product is an m-wedge, and as we have seen, an m-wedge can be equated to a number. Hence any particular n-monzo behaves like a certain (m-n)-val, called its complement, or Hodge dual; we can exploit this fact by mapping the wedge products of monzos to complementary wedge products of vals, and identifying them as such. This means in particular that an m-1 wedge product of vals may be identified as a monzo, and an m-1 product of monzos as a val.
Since any wedge product can be taken as either an n-val or an (m-n)-monzo, taking the product of an n-val with a k-val leads to an (n+k)-val, but taking the product of its complement with a k-monzo leads to an (m-n+k)-monzo, which is complementary to and which we may equate with a certain (n-k)-val. In other words, wedging by vals sends the n-val up to an (n+k)-val, but wedging by monzos sends it down to an (n-k)-monzo.
Let us look at the situation in the 5 and 7 limits. In the 5 limit, the product of two vals is given by
<a2 a3 a5|/\<b2 b3 b5| = (a2 v2 + a3 v3 + a5 v5)/\(b2 v2 + b3 v3 + b5 v5) =
(a3 b5 - a5 b3) v3 /\ v5 + (a2 b5 - a5 b2) v2/\v5 + (a2 b3 - a3 b2) v2/\v3 = <<a2b3 - a3b2 , a2b5 - a5b2, a3b5 - a5b3||
This is to be equated with a monzo; how to do the equating? Note that
v3/\v5/\v2 = v2/\v3/\v5, v2/\v5/\v3 = - v2/\v3/\v5, and v2/\v3/\v5 = v2/\v3/\v5; we may therefore equate the above with
|a3 b5 - a5 b3, a5 b2 - a2 b5, a2 b3 - a3 b2>,
which is its complement. The rule for complements in the 5-limit is
<<c1 c2 c3||* = |c3 -c2 c1> and ||c1 c2 c3>>* = <c3 -c2 c1||
where the asterisk denotes taking the complement.
In general, we may represent the n-wedgies in terms of the complementary vals, but we want to express the (m-1)-vals as an interval, whether in monzo, prime power product, or ratio of integer form.
In the case of the 7-limit, a product of two vals or a product of two intervals will alike lead to a C(4, 2) = 6 dimensional expression. The generic product of two vals is
<a2 a3 a5 a7|/\<b2 b3 b5 b7| =
(a2 v2 + a3 v3 + a5 v5 + a7 v7)/\(b2 v2 + b3 v3 + b5 v5 + b7 v7) =
(a2 b3 - a3 b2) v2/\v3 + (a2 b5-a5 b2) v2/\v5 + (a2 b7-b2 a7) v2/\v7 +
(a3 b5 - a5 b3) v3/\v5 + (a3 b7-a7 b3) v3/\v7 + (a5 b7-b5 a7) v5/\v7 =
<<a2 b3 - a3 b2, a2 b5 - a5 b2, a2 b7 - a7 b2, a3 b5 - a5 b3, a3 b7 - a7 b3, a5 b7 - a7 b5||
The ordering of the basis elements (alphabetical ordering of the corresponding interval basis elements) is of some significance; the first three (the v2-free elements, whose coefficients always have an a2 and a b2) are of more significance to the theory of temperaments.
Duality for 7-limit 2-vals or 2-monzos gives us this:
<<c1 c2 c3 c4 c5 c6||* = ||c6 -c5 c4 c3 -c2 c1>>
||c1 c2 c3 c4 c5 c6>>* = <<c6 -c5 c4 c3 -c2 c1||
If we take the dual of the product of two monzos we obtain the following:
(|a2, a3 a5, a7>/\|b2, b3, b5, b7>)* =
<<a5 b7-b5 a7, a7 b3-a3 b7, a3 b5 -a5 b3, a2 b7-b2 a7, a5 b2-a2 b5, a2 b3 - a3 b2||
This bival, and not the bimonzo itself, is what we would use to define the wedgie of the temperament defined by the two monzos.
For instance, the monzo for 126/125 is |1 2 -3 1> and for 81/80 is |-4 4 -1 0>. The wedge product of these two monzos is ||12 -13 4 10 -4 1>>, and ||12 -13 10 4 -1 1>>* = <<1 4 10 4 13 12||. We may obtain the same bival directly as a product of vals; <19 30 44 53|/\<12 19 28 34| = <<1 4 10 4 13 12|| . Whichever way we obtain it, this bival is the wedgie for the septimal meantone linear temperament.