Interior product: Difference between revisions

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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 ''m''1, ''m''2, ..., ''m''n be a group of ''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''(''m''1, ''m''2, ..., ''m''''n'').

Let ''W'' be a multival of rank ''n'', and ''m''1, ''m''2, ..., ''m''''n'' be a group of ''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''(''m''1, ''m''2, ..., ''m''''n'').

For example, suppose <math>W = \bitval{6 & -7 & -2 & -25 & -20 & 15}</math>, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely <math>\tmonzo{1 & 0 & 0 & 0}</math> and <math>\tmonzo{-1 & 1 & 1 & -1}</math>;, then wedging them together gives the bimonzo <math>\bitmonzo{1 & 1 & -1 & 0 & 0 & 0}</math>. The dot product with ''W'' is <math>\wmp{6 & -7 & -2 & -25 & -20 & 15}{1 & 1 & -1 & 0 & 0 & 0}</math>, which is {{nowrap|6 − 7 − (−2) {{=}} 1}}, so <math>W\left(2, \frac{15}{14}\right) = W\left(\tmonzo{1 & 0 & 0 & 0}, \tmonzo{-1 & 1 & 1 & 1}\right) = 1</math>. 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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 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 ''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>n</sub> be a group of ''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''(''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub>).
+Let ''W'' be a multival of rank ''n'', and ''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub> be a group of ''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''(''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub>).
 For example, suppose <math>W = \bitval{6 & -7 & -2 & -25 & -20 & 15}</math>, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely <math>\tmonzo{1 & 0 & 0 & 0}</math> and <math>\tmonzo{-1 & 1 & 1 & -1}</math>;, then wedging them together gives the bimonzo <math>\bitmonzo{1 & 1 & -1 & 0 & 0 & 0}</math>. The dot product with ''W'' is <math>\wmp{6 & -7 & -2 & -25 & -20 & 15}{1 & 1 & -1 & 0 & 0 & 0}</math>, which is {{nowrap|6 &minus; 7 &minus; (&minus;2) {{=}} 1}}, so <math>W\left(2, \frac{15}{14}\right) = W\left(\tmonzo{1 & 0 & 0 & 0}, \tmonzo{-1 & 1 & 1 & 1}\right) = 1</math>. The fact that the result is &#x2213;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.