Interior product: Difference between revisions

Line 9:

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>\~~wmproduct~~{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.

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.

If ''W'' is a multival of rank ''n'' and ''m'' is a monzo of the same prime limit p, then form a list of ({{nowrap|n − 1}}) tuples of primes less than or equal to ''p'' in alphabetical order. Taking these in order, the ''i''-th element of {{nowrap|''W'' ∨ ''m''}}, which we may also write {{nowrap|''W'' ∨ ''q''}} where ''q'' is the rational number with monzo ''m'', will be W(''s''1, ''s''2, ..., ''s''{{nowrap|n − 1}}, ''q''), where {{nowrap|{{!(}}''s''1, ''s''2, ..., ''s''n − 1{{)!}}}} is the ''i''-th tuple on the list of ({{nowrap|''n'' − 1}})-tuples of primes. This will result in {{nowrap|''W'' ∨ ''m''}}, a multival of rank ({{nowrap|n − 1}}).

@@ Line 9: / Line 9: @@
 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>\wmproduct{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.
+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.
 If ''W'' is a multival of rank ''n'' and ''m'' is a monzo of the same prime limit p, then form a list of ({{nowrap|n &minus; 1}}) tuples of primes less than or equal to ''p'' in alphabetical order. Taking these in order, the ''i''-th element of {{nowrap|''W'' ∨ ''m''}}, which we may also write {{nowrap|''W'' ∨ ''q''}} where ''q'' is the rational number with monzo ''m'', will be W(''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n &minus; 1}}</sub>, ''q''), where {{nowrap|{{!(}}''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>n &minus; 1</sub>{{)!}}}} is the ''i''-th tuple on the list of ({{nowrap|''n'' &minus; 1}})-tuples of primes. This will result in {{nowrap|''W'' ∨ ''m''}}, a multival of rank ({{nowrap|n &minus; 1}}).