Interior product: Difference between revisions
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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>\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. | ||
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''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n − 1}}</sub>, ''q''), where | 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''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n − 1}}</sub>, ''q''), where {{nowrap|{{!(}}''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>n − 1</sub>{{)!}}}} 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}}). | ||
For instance, let <math>M_\text{Marvel} = \tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}</math>, the wedgie for 11-limit Marvel temperament. To find {{nowrap|''M''<sub>Marvel</sub> ∨ 441/440}}, we form the list [ | For instance, let <math>M_\text{Marvel} = \tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}</math>, the wedgie for 11-limit Marvel temperament. To find {{nowrap|''M''<sub>Marvel</sub> ∨ 441/440}}, we form the list [{{nowrap|{{!(}}2, 3{{)!}}}}, {{nowrap|{{!(}}2, 5{{)!}}}}, {{nowrap|{{!(}}2, 7{{)!}}}}, {{nowrap|{{!(}}2, 11{{)!}}}}, {{nowrap|{{!(}}3, 5{{)!}}}}, {{nowrap|{{!(}}3, 7{{)!}}}}, {{nowrap|{{!(}}3, 11{{)!}}}}, {{nowrap|{{!(}}5, 7{{)!}}}}, {{nowrap|{{!(}}5, 11{{)!}}}}, {{nowrap|{{!(}}7, 11{{)!}}}}]. The first element of {{nowrap|''M''<sub>Marvel</sub> ∨ 441/440}} will be {{nowrap|''M''<sub>Marvel</sub>(2, 3, 441/440)}}, the second element {{nowrap|''M''<sub>Marvel</sub>(2, 5, 441/440)}} and so on down to the last element, {{nowrap|''M''<sub>Marvel</sub>(7, 11, 441/440)}}. This gives us <math>\bitval{6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49}</math>, 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 {{nowrap|''m'' ∨ ''W''}} from the front by using ''W''(''q'', ''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n − 1}}</sub>) instead of ''W''(''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n − 1}}</sub>, ''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 {{nowrap|''r'' < ''n''}}, by forming a list of ({{nowrap|''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 {{nowrap|''W'' ∨ ''M''}}. | If we like, we can take the wedge product {{nowrap|''m'' ∨ ''W''}} from the front by using ''W''(''q'', ''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n − 1}}</sub>) instead of ''W''(''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n − 1}}</sub>, ''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 {{nowrap|''r'' < ''n''}}, by forming a list of ({{nowrap|''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 {{nowrap|''W'' ∨ ''M''}}. | ||