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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-{{nowrap|(''r'' − 1)}} temperament {{nowrap|''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'').
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-{{nowrap|(''r'' − 1)}} temperament {{nowrap|''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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== Definition ==
== 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 ''n''-form, and ''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub> be a group of ''n'' vectors. Take the wedge product of these vectors, producing the multivector ''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 {{nowrap|''W'' {{=}} {{multival| 6 -7 -2 -25 -20 15 }}}}, the [[plücker coordinates|coordinates]] for 7-limit miracle.
If our two vectors are the vectors for 2 and 15/14, namely {{monzo| 1 0 0 0 }} and {{monzo| -1 1 1 -1 }};, then taking their wedge product gives the bivector {{multivector| 1 1 -1 0 0 0 }}. The dot product with ''W'' is {{wmp|6 -7 -2 -25 -20 15|1 1 -1 0 0 0}}, which is {{nowrap|6 − 7 − (−2) {{=}} 1}}, so {{nowrap|W(2, {{frac|15|14}}) {{=}} W({{monzo| 1 0 0 0 }}, {{monzo| -1 1 1 1 }}) {{=}} 1}}. The fact that the result is &#177;1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the result is ''N'', then the vectors 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 &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 an ''n''-form and ''m'' is a vector of the same prime limit ''p'', then form a list of ({{nowrap|''n'' − 1}}) tuples of primes less than or equal to ''p'' in lexicographic order. Taking these in order, the ''i''-th element of {{nowrap|''W'' ⨼ ''m''}}. In a slight abuse of notation, we may also write {{nowrap|''W'' ⨼ ''q''}} where ''q'' is a rational number with vector representation ''m''.
The result of this 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}}).


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 [''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|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}}).
For instance, let {{nowrap|''M''<sub>Marvel</sub> {{=}} {{multivector| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, the coordinates for 11-limit Marvel temperament. To find {{nowrap|''M''<sub>Marvel</sub> ⨼ 441/440}}, we form the list {{nowrap|{{!((}}2, 3{{)!}}|[2, 5]|[2, 7]|[2, 11]|[3, 5]|[3, 7]|[3, 11]|[5, 7]|[5, 11]|{{!(}}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 {{multival| 6 -7 -2 15 -25 -20 3 15 59 49 }}, which are the coordinates for 11-limit Miracle. The interior product has added a comma to Marvel to produce Miracle.


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 [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [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 multivector ''M'' of rank {{nowrap|''r'' &lt; ''n''}}, by forming a list of ({{nowrap|''n'' − ''r''}})-tuples of primes in lexicographic 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 &minus; 1}}</sub>) instead of ''W''(''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>{{nowrap|n &minus; 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'' &lt; ''n''}}, by forming a list of ({{nowrap|''n'' &minus; ''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''}}.


== Applications ==
== Applications ==
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 {{nowrap|''r'' &minus; 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival&mdash;that is, if all the coefficients are zero.
One very useful application is testing whether a temperament tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a ''k''-form if and only if the {{nowrap|(''k'' − 1)}}-form obtained by taking the interior product of the temperament with the interval is the zero form&mdash;that is, if all the coefficients are zero.


Another application is the use of the interior product to define the intervals of the [[Abstract_regular_temperament|abstract regular temperament]] given by a wedgie ''W''. In this case, we use {{nowrap|''W'' ∨ ''q''}} to define a multival which represents the tempered interval which ''q'' is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let ''S'' be an element of tuning space defining a tuning for the abstract regular temperament denoted by ''W'', and ''T'' a truncated version of ''S'' where ''S'' is shortened to only the first ''r'' primes, where ''r'' is the rank of ''W''. Form the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''R''}}], where ''R'' is the ''r''-th prime number. Let ''U'' be the transpose of the pseudoinverse of this matrix, and let {{nowrap|''V'' = ''TU''}} (the matrix product), which can be taken to be an ({{nowrap|''r'' &minus; 1}})-multimonzo. Then for any ({{nowrap|''r'' &minus; 1}})-multival {{nowrap|''W'' ∨ ''q''}} in the abstract regular temperament, the dot product {{nowrap|(''W'' ∨ ''q'') ∙ ''V''}} gives the tuning of {{nowrap|''W'' ∨ ''q''}}. It should be noted that ''V'' with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''q''}}] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of {{nowrap|''M''<sub>meantone</sub> ∨ ''q''}}, where "Meantone" is the 7-limit wedgie, with <math>\bival{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> gives the value in cents of the [[Quarter-comma_meantone|quarter-comma meantone]] tuning of the interval denoted by {{nowrap|''M''<sub>meantone</sub> ∨ ''q''}}.
Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by coordinates ''W''. In this case, we use {{nowrap|''W'' ⨼ ''q''}} to define a multilinear form which represents the tempered interval which ''q'' is tempered to.
For this to make sense, we need a way to define the tuning for such multilinear forms, which can be done in a variety of ways. One is as follows: let ''S'' be an element of tuning space defining a tuning for the abstract regular temperament denoted by ''W'', and ''T'' a truncated version of ''S'' where ''S'' is shortened to only the first ''r'' primes, where ''r'' is the rank of ''W''. Form the matrix {{nowrap|[''W'' ⨼ 2}}, {{nowrap|''W'' ⨼ 3}}, ..., {{nowrap|''W'' ⨼ ''R'']}}, where ''R'' is the ''r''-th prime number. Let ''U'' be the transpose of the pseudoinverse of this matrix, and let {{nowrap|''V'' {{=}} ''TU''}} (the matrix product), which can be taken to be an ({{nowrap|''r'' − 1}})-multivector. Then for any ({{nowrap|''r'' − 1}})-multival {{nowrap|''W'' ⨼ ''q''}} in the abstract regular temperament, the dot product {{nowrap|(''W'' ⨼ ''q'') ∙ ''V''}} gives the tuning of {{nowrap|''W'' ⨼ ''q''}}. It should be noted that ''V'' with this property is underdetermined, so that many possible multivectors can be used to the same effect. An alternative approach is to hermite reduce the matrix {{nowrap|[''W'' ⨼ 2}}, {{nowrap|''W'' ⨼ 3}}, ..., {{nowrap|''W'' ⨼ ''q'']}} and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of {{nowrap|''M''<sub>meantone</sub> ⨼ ''q''}}, where "Meantone" are the coordinates for 7-limit meantone, with {{monzo| $1 }} giving the value in cents of the [[quarter-comma meantone]] tuning of the interval denoted by {{nowrap|''M''<sub>meantone</sub> ⨼ ''q''}}.


The interior product can also be used to add a comma to a ''p''-limit temperament of rank ''r'', producing a rank-({{nowrap|''r'' &minus; 1}}) temperament which [[support]]s it. For instance, <math>\tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}</math> is the wedgie for 11-limit [[Marvel_family#Marvel|Marvel temperament]]. Then:
The interior product can also be used to add a comma to a ''p''-limit temperament of rank ''r'', producing a rank-({{nowrap|''r'' − 1}}) temperament which [[support]]s it. For instance, {{multivector|nullity=3| 1 2 -3 -2 1 -4 -5 12 9 -19 }} are the coordinates for 11-limit [[Marvel_family#Marvel|Marvel temperament]]. Then:


: <math>M_\text{Marvel} ∨ \frac{45}{44} = \bitval{4 & -3 & 2 & 5 & -14 & -8 & -6 & 13 & 22 & 7}</math> gives 11-limit negri,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{45}{44} = \bitval{4 & -3 & 2 & 5 & -14 & -8 & -6 & 13 & 22 & 7}</math> gives 11-limit negri,
: <math>M_\text{Marvel} ∨ \frac{64}{63} = \bitval{-2 & 4 & 4 & -10 & 11 & 12 & -9 & -2 & -37 & -42}</math> gives pajarous,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{64}{63} = \bitval{-2 & 4 & 4 & -10 & 11 & 12 & -9 & -2 & -37 & -42}</math> gives pajarous,
: <math>M_\text{Marvel} ∨ \frac{245}{242} = \bitval{11 & -6 & 10 & 7 & -35 & -15 & -27 & 40 & 37 & -15}</math> gives septimin,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{245}{242} = \bitval{11 & -6 & 10 & 7 & -35 & -15 & -27 & 40 & 37 & -15}</math> gives septimin,
: <math>M_\text{Marvel} ∨ \frac{99}{98} = \bitval{-7 & 3 & -8 & -2 & 21 & 7 & 21 & -27 & -15 & 22}</math> gives orwell,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{99}{98} = \bitval{-7 & 3 & -8 & -2 & 21 & 7 & 21 & -27 & -15 & 22}</math> gives orwell,
: <math>M_\text{Marvel} ∨ \frac{100}{99} = \bitval{5 & 1 & 12 & -8 & -10 & 5 & -30 & 25 & -22 & -64}</math> gives magic,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{100}{99} = \bitval{5 & 1 & 12 & -8 & -10 & 5 & -30 & 25 & -22 & -64}</math> gives magic,
: <math>M_\text{Marvel} ∨ \frac{243}{242} = \bitval{6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49}</math> gives miracle,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{243}{242} = \bitval{6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49}</math> gives miracle,
: <math>M_\text{Marvel} ∨ \frac{3136}{3125} = \bitval{-1 & -4 & -10 & 13 & -4 & -13 & 24 & -12 & 44 & 71}</math> gives meanpop,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{3136}{3125} = \bitval{-1 & -4 & -10 & 13 & -4 & -13 & 24 & -12 & 44 & 71}</math> gives meanpop,
: <math>M_\text{Marvel} ∨ \frac{6250}{6237} = \bitval{6 & 5 & 22 & -21 & -6 & 18 & -54 & 37 & -66 & -135}</math> gives catakleismic,
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{6250}{6237} = \bitval{6 & 5 & 22 & -21 & -6 & 18 & -54 & 37 & -66 & -135}</math> gives catakleismic,
: <math>M_\text{Marvel} ∨ \frac{2200}{2187} = \bitval{-1 & 8 & 14 & -23 & 15 & 25 & -33 & 10 & -81 & -113}</math> gives garibaldi, and
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{2200}{2187} = \bitval{-1 & 8 & 14 & -23 & 15 & 25 & -33 & 10 & -81 & -113}</math> gives garibaldi, and
: <math>M_\text{Marvel} ∨ \frac{9801}{9800} = \bitval{-12 & 2 & -20 & 6 & 31 & 2 & 51 & -52 & 7 & 86}</math> gives wizard.
: <math>M_\text{Marvel} \mathbin{\lrcorner} \frac{9801}{9800} = \bitval{-12 & 2 & -20 & 6 & 31 & 2 & 51 & -52 & 7 & 86}</math> gives wizard.


The interior product is also useful in finding the temperament mapping 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 {{nowrap|''r'' &minus; 1}} primes less than or equal to ''p'', and reducing this to the mapping. For instance, for ''M''<sub>Marvel</sub> we take [{{nowrap|''M''<sub>Marvel</sub> ∨ 2 ∨ 3}}, {{nowrap|''M''<sub>Marvel</sub> ∨ 2 ∨ 5}}, ..., {{nowrap|''M''<sub>Marvel</sub> ∨ 7 ∨ 11}}], which gives:
The interior product is also useful in finding the temperament mapping its coordinates. Given a ''p''-limit temperament of rank-''r'', we can find a collection of linear maps belonging to it by taking the interior product with every set of {{nowrap|''r'' − 1}} primes less than or equal to ''p'', and reducing this to the mapping.
For instance, for ''M''<sub>Marvel</sub> we take {{nowrap|[''M''<sub>Marvel</sub> ⨼ 2 ⨼ 3}}, {{nowrap|''M''<sub>Marvel</sub> ⨼ 2 ⨼ 5}}, ..., {{nowrap|''M''<sub>Marvel</sub> ⨼ 7 ⨼ 11]}}, which gives:


<math>\left[\tmonzo{0 & 0 & -1 & -2 & 3},\right.</math> <math>\tmonzo{0 & 1 & 0 & 2 & -1},</math> <math>\tmonzo{0 & 2 & -2 & 0 & 4},</math> <math>\tmonzo{0 & -3 & 1 & -4 & 0},</math> <math>\tmonzo{-1 & 0 & 0 & 5 & -12},</math> <math>\tmonzo{-2 & 0 & -5 & 0 & -9},</math> <math>\tmonzo{3 & 0 & 12 & 9 & 0},</math> <math>\tmonzo{2 & 5 & 0 & 0 & 19},</math> <math>\tmonzo{-1 & -12 & 0 & -19 & 0},</math> <math>\left.\tmonzo{4 & -9 & 19 & 0 & 0}\right].</math>
[{{monzo| 0, 0, -1, -2, 3 }} {{monzo| 0, 1, 0, 2, -1 }} {{monzo| 0, 2, -2, 0, 4 }} {{monzo| 0, -3, 1, -4, 0 }} {{monzo| -1, 0, 0, 5, -12 }} {{monzo| -2, 0, -5, 0, -9 }} {{monzo| 3, 0, 12, 9, 0 }} {{monzo| 2, 5, 0, 0, 19 }} {{monzo| -1, -12, 0, -19, 0 }} {{monzo| 4, -9, 19, 0, 0 }}].


Hermite-reducing this to a normal val list results in <math>\left[\tmonzo{-1 & 0 & 0 & 5 & -12},\right.</math> <math>\tmonzo{0 & 1 & 0 & 2 & -1},</math> <math>\left.\tmonzo{0 & 0 & -1 & -2 & 3}\right]</math>, the normal val list for 11-limit Marvel. In practice, this method nearly always suffices.
Hermite-reducing this results in {{monzo| -1, 0, 0, 5, -12 }} {{monzo| 0, 1, 0, 2, -1 }} {{monzo| 0, 0, -1, -2, 3 }}, the [[normal form]] for 11-limit Marvel. In practice, this method nearly always suffices.


[[Category:Math]]
[[Category:Math]]
[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]
[[Category:Exterior algebra]]
[[Category:Exterior algebra]]

Latest revision as of 15:02, 21 June 2025

This page or section may be difficult to understand to those unfamiliar with the mathematical concepts involved. A more accessible version will be worked on; in the meantime, feel free to ask questions in the Xenharmonic Alliance Discord server or Facebook group.
English Wikipedia has an article on:

[math]\displaystyle{ \def\hs{\hspace{-3px}} \def\lvsp{{}\mkern-5.5mu}{} \def\rvsp{{}\mkern-2.5mu}{} \def\llangle{\left\langle\lvsp\left\langle} \def\lllangle{\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llllangle{\left\langle\lvsp\left\langle\lvsp\left\langle\lvsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\rvsp\right\rangle} \def\rrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrrrangle{\right\rangle\rvsp\right\rangle\rvsp\right\rangle\rvsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} }[/math][math]\displaystyle{ \def\val#1{\left\langle\begin{matrix}#1\end{matrix}\right]} \def\tval#1{\left\langle\begin{matrix}#1\end{matrix}\right\vert} \def\bival#1{\llangle\begin{matrix}#1\end{matrix}\rrbrack} \def\bitval#1{\llangle\begin{matrix}#1\end{matrix}\rrvert} \def\trival#1{\lllangle\begin{matrix}#1\end{matrix}\rrrbrack} \def\tritval#1{\lllangle\begin{matrix}#1\end{matrix}\rrrvert} \def\quadval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrbrack} \def\quadtval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrvert} \def\monzo#1{\left[\begin{matrix}#1\end{matrix}\right\rangle} \def\tmonzo#1{\left\vert\begin{matrix}#1\end{matrix}\right\rangle} \def\bimonzo#1{\llbrack\begin{matrix}#1\end{matrix}\rrangle} \def\bitmonzo#1{\llvert\begin{matrix}#1\end{matrix}\rrangle} \def\trimonzo#1{\lllbrack\begin{matrix}#1\end{matrix}\rrrangle} \def\tritmonzo#1{\lllvert\begin{matrix}#1\end{matrix}\rrrangle} \def\quadmonzo#1{\llllbrack\begin{matrix}#1\end{matrix}\rrrrangle} \def\quadtmonzo#1{\llllvert\begin{matrix}#1\end{matrix}\rrrrangle} \def\rbra#1{\left\{\begin{matrix}#1\end{matrix}\right]} \def\rket#1{\left[\begin{matrix}#1\end{matrix}\right\}} \def\vmp#1#2{\left\langle\begin{matrix}#1\end{matrix}\,\vert\,\begin{matrix}#2\end{matrix}\right\rangle} \def\wmp#1#2{\llangle\begin{matrix}#1\end{matrix}\,\vert\vert\,\begin{matrix}#2\end{matrix}\rrangle} }[/math] Given a rank-r temperament W and a comma m not 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 codimension is one dimension higher than that of W).

Definition

Let W be a n-form, and m1, m2, ..., mn be a group of n vectors. Take the wedge product of these vectors, producing the multivector 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 coordinates for 7-limit miracle. If our two vectors are the vectors for 2 and 15/14, namely [1 0 0 0⟩ and [-1 1 1 -1⟩;, then taking their wedge product gives the bivector [[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 result is N, then the vectors it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.

If W is an n-form and m is a vector of the same prime limit p, then form a list of (n − 1) tuples of primes less than or equal to p in lexicographic order. Taking these in order, the i-th element of W ⨼ m. In a slight abuse of notation, we may also write W ⨼ q where q is a rational number with vector representation m. The result of this will be W(s1, s2, ..., sn − 1, q), where [s1, s2, ..., sn − 1] is the i-th tuple on the list of (n − 1)-tuples of primes. This will result in W ⨼ m, a multival of rank (n − 1).

For instance, let MMarvel = [[1 2 -3 -2 1 -4 -5 12 9 -19⟩⟩, the coordinates for 11-limit Marvel temperament. To find MMarvel ⨼ 441/440, we form the list [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [7, 11]]. The first element of MMarvel ⨼ 441/440 will be MMarvel(2, 3, 441/440), the second element MMarvel(2, 5, 441/440) and so on down to the last element, MMarvel(7, 11, 441/440). This gives us ⟨⟨ 6 -7 -2 15 -25 -20 3 15 59 49 ]], which are the coordinates 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 m ⨼ W from the front by using W(q, s1, s2, ..., sn − 1) instead of W(s1, s2, ..., sn − 1, q), but this can only lead to a difference in sign. We can also define the interior product of W with a multivector M of rank r < n, by forming a list of (n − r)-tuples of primes in lexicographic order, wedging these together with M, and taking the dot product with W to get a coefficient of W ⨼ M.

Applications

One very useful application is testing whether a temperament tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a k-form if and only if the (k − 1)-form obtained by taking the interior product of the temperament with the interval is the zero form—that is, if all the coefficients are zero.

Another application is the use of the interior product to define the intervals of the abstract regular temperament given by coordinates W. In this case, we use W ⨼ q to define a multilinear form which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multilinear forms, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r primes, where r is the rank of W. Form the matrix [W ⨼ 2, W ⨼ 3, ..., W ⨼ R], where R is the r-th prime number. Let U be the transpose of the pseudoinverse of this matrix, and let V = TU (the matrix product), which can be taken to be an (r − 1)-multivector. Then for any (r − 1)-multival W ⨼ q in the abstract regular temperament, the dot product (W ⨼ q) ∙ V gives the tuning of W ⨼ q. It should be noted that V with this property is underdetermined, so that many possible multivectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [W ⨼ 2, W ⨼ 3, ..., W ⨼ q] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of Mmeantone ⨼ q, where "Meantone" are the coordinates for 7-limit meantone, with [$1⟩ giving the value in cents of the quarter-comma meantone tuning of the interval denoted by Mmeantone ⨼ q.

The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank-(r − 1) temperament which supports it. For instance, [[[1 2 -3 -2 1 -4 -5 12 9 -19⟩⟩⟩ are the coordinates for 11-limit Marvel temperament. Then:

[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{45}{44} = \bitval{4 & -3 & 2 & 5 & -14 & -8 & -6 & 13 & 22 & 7} }[/math] gives 11-limit negri,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{64}{63} = \bitval{-2 & 4 & 4 & -10 & 11 & 12 & -9 & -2 & -37 & -42} }[/math] gives pajarous,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{245}{242} = \bitval{11 & -6 & 10 & 7 & -35 & -15 & -27 & 40 & 37 & -15} }[/math] gives septimin,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{99}{98} = \bitval{-7 & 3 & -8 & -2 & 21 & 7 & 21 & -27 & -15 & 22} }[/math] gives orwell,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{100}{99} = \bitval{5 & 1 & 12 & -8 & -10 & 5 & -30 & 25 & -22 & -64} }[/math] gives magic,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{243}{242} = \bitval{6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49} }[/math] gives miracle,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{3136}{3125} = \bitval{-1 & -4 & -10 & 13 & -4 & -13 & 24 & -12 & 44 & 71} }[/math] gives meanpop,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{6250}{6237} = \bitval{6 & 5 & 22 & -21 & -6 & 18 & -54 & 37 & -66 & -135} }[/math] gives catakleismic,
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{2200}{2187} = \bitval{-1 & 8 & 14 & -23 & 15 & 25 & -33 & 10 & -81 & -113} }[/math] gives garibaldi, and
[math]\displaystyle{ M_\text{Marvel} \mathbin{\lrcorner} \frac{9801}{9800} = \bitval{-12 & 2 & -20 & 6 & 31 & 2 & 51 & -52 & 7 & 86} }[/math] gives wizard.

The interior product is also useful in finding the temperament mapping its coordinates. Given a p-limit temperament of rank-r, we can find a collection of linear maps 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 mapping. For instance, for MMarvel we take [MMarvel ⨼ 2 ⨼ 3, MMarvel ⨼ 2 ⨼ 5, ..., MMarvel ⨼ 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 results in [-1, 0, 0, 5, -12⟩ [0, 1, 0, 2, -1⟩ [0, 0, -1, -2, 3⟩, the normal form for 11-limit Marvel. In practice, this method nearly always suffices.

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