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'').

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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).

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.

One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if and only if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.

The interior product is also useful in finding the temperament map given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the map. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [<0 0 -1 -2 3|, <0 1 0 2 -1|, <0 2 -2 0 4|, <0 -3 1 -4 0|, <-1 0 0 5 -12|, <-2 0 -5 0 -9|, <3 0 12 9 0|, <2 5 0 0 19|, <-1 -12 0 -19 0|, <4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|], the normal val list for 11-limit marvel. In practice this method nearly always suffices.