Interior product: Difference between revisions

Given a [[rank]]-''r'' [[regular temperament|temperament]] ''W'' and a [[comma]] ''m'' not [[tempering out|tempered out]] by ''W'', the '''interior product''' of ''W'' and ''m'' is the rank-(''r''-1) temperament ''W''∨''m'' which tempers out ''m'' in addition to all the commas that are tempered out by ''W'' (thus its [[Rank and codimension|codimension]] is one dimension higher than that of ''W'').

The '''interior product''' is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[Wedgies_and_Multivals|n-map]], a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).

For example, suppose W = <<6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0> and |-1 1 1 -1>, then wedging them together gives the bimonzo ||1 1 -1 0 0 0>>. The dot product with W is <<6 -7 -2 -25 -20 15||1 1 -1 0 0 0>>, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0>, |-1 1 1 1>) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.

If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n-1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the ith element of W∨m, which we may also write W∨q where q is the rational number with monzo m, will be W(s1, s2, s3 ... s_(n-1), q), where [s1, s2, ..., s_(n-1)] is the ith tuple on the list of (n-1)-tuples of primes. This will result in W∨m, a multival of rank n-1. For instance, let Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperament. To find Marv∨441/440, we form the list [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [7, 11]]. The first element of Marv∨441/440 will be Marv(2, 3, 441/440), the second element Marv(2, 5, 441/440) and so on down to the last element, Marv(7, 11, 441/440). This gives us <<6 -7 -2 15 -25 -20 3 15 59 49||, which is the wedgie for 11-limit miracle. The interior product has added a comma to marvel to produce miracle.

If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign. We can also define the interior product of W with a multimonzo M of rank r < n, by forming a list of (n-r)-tuples of primes in alphabetical order, wedging these together with M, and taking the dot product with W to get a coefficient of W∨M.