Interior product: Difference between revisions

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=Definition=

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).

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.

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 =Definition=
-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).
+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 = &lt;&lt;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&gt; and |-1 1 1 -1&gt;, then wedging them together gives the bimonzo ||1 1 -1 0 0 0&gt;&gt;. The dot product with W is &lt;&lt;6 -7 -2 -25 -20 15||1 1 -1 0 0 0&gt;&gt;, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0&gt;, |-1 1 1 1&gt;) = 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.