Interior product - Revision history

Sintel: use standard terminology (page is still a confusing mess unfortunately)

2025-06-21T15:02:59Z

use standard terminology (page is still a confusing mess unfortunately)

ArrowHead294 at 00:00, 4 June 2025

2025-06-04T00:00:12Z

← Older revision		Revision as of 00:00, 4 June 2025
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	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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—that is, if all the coefficients are zero.		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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—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 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'' − 1}})-multimonzo. 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 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 {{monzo\| $1 }} giving the value in cents of the [[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 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'' − 1}})-multimonzo. 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 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 {{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'' − 1}}) temperament which [[support]]s it. For instance, {{multimonzo\|nullity=3\| 1 2 -3 -2 1 -4 -5 12 9 -19 }} 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, {{multimonzo\|nullity=3\| 1 2 -3 -2 1 -4 -5 12 9 -19 }} is the wedgie for 11-limit [[Marvel_family#Marvel\|Marvel temperament]]. Then:

Sintel: add wikipedia link

2025-04-19T12:55:33Z

add wikipedia link

← Older revision		Revision as of 12:55, 19 April 2025
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	{{inacc}}		{{inacc}}
			{{wikipedia\|Interior product}}
	{{texmap}}		{{texmap}}
	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'').

Sintel: Inaccessible

2025-04-18T11:14:03Z

Inaccessible

← Older revision		Revision as of 11:14, 18 April 2025
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			{{inacc}}
	{{texmap}}		{{texmap}}
	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'').

ArrowHead294 at 14:32, 26 February 2025

2025-02-26T14:32:03Z

ArrowHead294: /* Definition */

2025-02-26T14:23:47Z

Definition

← Older revision		Revision as of 14:23, 26 February 2025
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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.		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>).

ArrowHead294 at 14:23, 26 February 2025

2025-02-26T14:23:12Z

← Older revision		Revision as of 14:23, 26 February 2025
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	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 {{nowrap\|''W'' {{=}} {{multival\| 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 {{monzo\| 1 0 0 0 }} and {{monzo\| -1 1 1 -1 }};, then wedging them together gives the bimonzo {{multimonzo\| 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 ±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 {{nowrap\|''W'' {{=}} {{multival\| 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 {{monzo\| 1 0 0 0 }} and {{monzo\| -1 1 1 -1 }};, then wedging them together gives the bimonzo {{multimonzo\| 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 ±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 {{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'' − 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 {{nowrap\|''M''<sub>Marvel</sub> {{=}} {{multimonzo\| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, 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 {{multival\| 6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49 }}, which is the wedgie for 11-limit Miracle. The interior product has added a comma to Marvel to produce Miracle.		For instance, let {{nowrap\|''M''<sub>Marvel</sub> {{=}} {{multimonzo\| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, the wedgie 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 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''}}.

ArrowHead294: These inline maths look really clunky, changing them to text

2025-02-26T14:20:16Z

These inline maths look really clunky, changing them to text

Show changes

ArrowHead294: /* Applications */

2024-11-14T16:38:08Z

Applications

← Older revision		Revision as of 16:38, 14 November 2024
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	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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—that is, if all the coefficients are zero.		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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—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 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'' − 1}})-multimonzo. 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 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>\tmonzo{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> giving the value in cents of the [[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 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'' − 1}})-multimonzo. 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 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>\tmonzo{1200 + 300 \log_{2}(5) & -1200 & 0 & 0}</math> 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'' − 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, <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:

ArrowHead294: /* Definition */

2024-11-14T16:37:41Z

Definition

← Older revision		Revision as of 16:37, 14 November 2024
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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.		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 − 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.

← Older revision		Revision as of 00:00, 4 June 2025
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	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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—that is, if all the coefficients are zero.		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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—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 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'' − 1}})-multimonzo. 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 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 {{monzo\| $1 }} giving the value in cents of the [[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 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'' − 1}})-multimonzo. 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 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 {{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'' − 1}}) temperament which [[support]]s it. For instance, {{multimonzo\|nullity=3\| 1 2 -3 -2 1 -4 -5 12 9 -19 }} 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, {{multimonzo\|nullity=3\| 1 2 -3 -2 1 -4 -5 12 9 -19 }} is the wedgie for 11-limit [[Marvel_family#Marvel\|Marvel temperament]]. Then:

← Older revision		Revision as of 14:23, 26 February 2025
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	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 {{nowrap\|''W'' {{=}} {{multival\| 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 {{monzo\| 1 0 0 0 }} and {{monzo\| -1 1 1 -1 }};, then wedging them together gives the bimonzo {{multimonzo\| 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 ±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 {{nowrap\|''W'' {{=}} {{multival\| 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 {{monzo\| 1 0 0 0 }} and {{monzo\| -1 1 1 -1 }};, then wedging them together gives the bimonzo {{multimonzo\| 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 ±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 {{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'' − 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 {{nowrap\|''M''<sub>Marvel</sub> {{=}} {{multimonzo\| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, 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 {{multival\| 6 & -7 & -2 & 15 & -25 & -20 & 3 & 15 & 59 & 49 }}, which is the wedgie for 11-limit Miracle. The interior product has added a comma to Marvel to produce Miracle.		For instance, let {{nowrap\|''M''<sub>Marvel</sub> {{=}} {{multimonzo\| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, the wedgie 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 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''}}.

← Older revision		Revision as of 16:38, 14 November 2024
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	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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—that is, if all the coefficients are zero.		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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—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 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'' − 1}})-multimonzo. 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 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>\tmonzo{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> giving the value in cents of the [[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 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'' − 1}})-multimonzo. 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 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>\tmonzo{1200 + 300 \log_{2}(5) & -1200 & 0 & 0}</math> 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'' − 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, <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:

← Older revision		Revision as of 14:32, 26 February 2025
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	{{texmap}}		{{texmap}}
	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'').

	__TOC__		__TOC__
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	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 {{nowrap\|''W'' {{=}} {{multival\| 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 {{monzo\| 1 0 0 0 }} and {{monzo\| -1 1 1 -1 }};, then wedging them together gives the bimonzo {{multimonzo\| 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 ±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 {{nowrap\|''W'' {{=}} {{multival\| 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 {{monzo\| 1 0 0 0 }} and {{monzo\| -1 1 1 -1 }};, then wedging them together gives the bimonzo {{multimonzo\| 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 ±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 {{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'' − 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 {{nowrap\|''M''<sub>Marvel</sub> {{=}} {{multimonzo\| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, the wedgie 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 is the wedgie for 11-limit Miracle. The interior product has added a comma to Marvel to produce Miracle.		For instance, let {{nowrap\|''M''<sub>Marvel</sub> {{=}} {{multimonzo\| 1 2 -3 -2 1 -4 -5 12 9 -19 }}}}, the wedgie 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 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''}}.

	== 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'' ~~−~~ 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—that is, if all the coefficients are zero.		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'' − 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival—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 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'' ~~−~~ 1}})-multimonzo. 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 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 {{monzo\| $1 }} giving the value in cents of the [[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 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'' − 1}})-multimonzo. 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 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 {{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'' ~~−~~ 1}}) temperament which [[support]]s it. For instance, {{multimonzo\|nullity=3\| 1 2 -3 -2 1 -4 -5 12 9 -19 }} 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, {{multimonzo\|nullity=3\| 1 2 -3 -2 1 -4 -5 12 9 -19 }} is the wedgie 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} ∨ \frac{45}{44} = \bitval{4 & -3 & 2 & 5 & -14 & -8 & -6 & 13 & 22 & 7}</math> gives 11-limit negri,
Line 35:		Line 35:
	: <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} ∨ \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'' ~~−~~ 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 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'' − 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:

	{{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 }}.		[{{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 {{monzo\| -1, 0, 0, 5, -12 }} {{monzo\| 0, 1, 0, 2, -1 }} {{monzo\| 0, 0, -1, -2, 3 }}, the normal val list for 11-limit Marvel. In practice, this method nearly always suffices.		Hermite-reducing this to a normal val list results in {{monzo\| -1, 0, 0, 5, -12 }} {{monzo\| 0, 1, 0, 2, -1 }} {{monzo\| 0, 0, -1, -2, 3 }}, the normal val list for 11-limit Marvel. In practice, this method nearly always suffices.