Mathematical theory of regular temperaments: Difference between revisions

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This uses [~~http~~:~~//en.wikipedia.org/wiki/Exterior_algebra~~ multilinear algebra] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[interior product]] of a [[~~Wedgies and Multivals|~~wedgie]] for a p-limit temperament with the p-limit monzos.

This uses [[Wikipedia: Exterior algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[interior product]] of a [[wedgie]] for a ''p''-limit temperament with the ''p''-limit monzos.

For example, using "∨" to represent the interior product, we have mir = ~~<<~~6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0~~>~~ is ~~<~~0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1~~>~~ which is ~~<~~1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0~~>~~ which is also ~~<~~1 1 3 3|~~; <~~1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

For example, using "∨" to represent the interior product, we have mir = {{multival| 6 -7 -2 -25 -20 15 }} for the wedgie of 7-limit miracle. Then the interior product mir ∨ {{monzo| 1 0 0 0 }} is {{val| 0 -6 7 2 }}, with 15/14 we get mir ∨ {{monzo| -1 1 1 -1 }} which is {{val| 1 1 3 3 }}, and with 16/15 we get mir ∨ {{monzo| 4 -1 -1 0 }} which is also {{val| 1 1 3 3 }}; {{val| 1 1 3 3 }} tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.

As explained on the [[Interior product#Applications|interior product]] page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (~~W∨q~~) ~~• V~~.

As explained on the [[Interior product #Applications|interior product]] page, if W is the ''r''-wedgie defining the rank-''r'' temperament, then the tuning of map for the temperament can be defined via an (''r'' - 1)-multimonzo V which has the property that for every JI interval ''q'', the tempered value of ''q'' is given by the dot product (W∨''q'')·V.

=== Normal val list ===

Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [~~<~~1 1 3 3|~~, <~~0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }}, {{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

=== Frobenius projection map ===

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A relatively concrete approach, but one which is not canonically defined, is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.

For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank ~~two~~ 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.

For example, for [[Gamelismic clan #Miracle|miracle temperament]] [2, 15/14] defines a rank-2 7-limit subgroup whose [[Normal lists #Normal interval list|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.

=== Normal comma lists ===

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If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.

For example, if we feed [~~<~~22 35 51 62|~~, <~~31 49 72 87|~~, <~~84 133 195 236|] into a reduced row echelon form routine, we obtain [~~<~~1 0 3 1|~~, <~~0 1 -3/7 8/7|~~, <~~0 0 0 0|]. Stripping off the zero val in the final row, we get E = [~~<~~1 0 3 1|~~, <~~0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1~~>~~, and |-1 -1 0 1~~>~~E* = [0 1/7]. Multiply by |1 0 0 0~~>~~, the val for 2, and the result is |1 0 0 0~~>~~E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.

For example, if we feed [{{val| 22 35 51 62 }}, {{val| 31 49 72 87 }}, {{val| 84 133 195 236 }}] into a reduced row echelon form routine, we obtain [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}, {{val| 0 0 0 0 }}]. Stripping off the zero val in the final row, we get E = [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}]. The monzo for 7/6 is {{monzo| -1 -1 0 1 }}, and {{monzo| -1 -1 0 1 }}E* = [0 1/7]. Multiply by {{monzo| 1 0 0 0 }}, the val for 2, and the result is {{monzo| 1 0 0 0 }}E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.

== Translation between methods of specifying temperaments ==

@@ Line 28: / Line 28: @@
 {{main| Wedgies and Multivals }}
-This uses [http://en.wikipedia.org/wiki/Exterior_algebra multilinear algebra] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos.
+This uses [[Wikipedia: Exterior algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[interior product]] of a [[wedgie]] for a ''p''-limit temperament with the ''p''-limit monzos.
-For example, using "∨" to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; &lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.
+For example, using "∨" to represent the interior product, we have mir = {{multival| 6 -7 -2 -25 -20 15 }} for the wedgie of 7-limit miracle. Then the interior product mir ∨ {{monzo| 1 0 0 0 }} is {{val| 0 -6 7 2 }}, with 15/14 we get mir ∨ {{monzo| -1 1 1 -1 }} which is {{val| 1 1 3 3 }}, and with 16/15 we get mir ∨ {{monzo| 4 -1 -1 0 }} which is also {{val| 1 1 3 3 }}; {{val| 1 1 3 3 }} tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.
-As explained on the [[Interior product#Applications|interior product]] page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.
+As explained on the [[Interior product #Applications|interior product]] page, if W is the ''r''-wedgie defining the rank-''r'' temperament, then the tuning of map for the temperament can be defined via an (''r'' - 1)-multimonzo V which has the property that for every JI interval ''q'', the tempered value of ''q'' is given by the dot product (W∨''q'')·V.
 === Normal val list ===
 {{main| Normal lists #Normal val list }}
-Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&lt;1 1 3 3|, &lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].
+Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [{{val| 1 1 3 3 }}, {{val| 0 6 -7 -2 }}] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].
 === Frobenius projection map ===
@@ Line 50: / Line 50: @@
 A relatively concrete approach, but one which is not canonically defined, is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.
-For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.
+For example, for [[Gamelismic clan #Miracle|miracle temperament]] [2, 15/14] defines a rank-2 7-limit subgroup whose [[Normal lists #Normal interval list|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.
 === Normal comma lists ===
@@ Line 64: / Line 64: @@
 If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.
-For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.
+For example, if we feed [{{val| 22 35 51 62 }}, {{val| 31 49 72 87 }}, {{val| 84 133 195 236 }}] into a reduced row echelon form routine, we obtain [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}, {{val| 0 0 0 0 }}]. Stripping off the zero val in the final row, we get E = [{{val| 1 0 3 1 }}, {{val| 0 1 -3/7 8/7 }}]. The monzo for 7/6 is {{monzo| -1 -1 0 1 }}, and {{monzo| -1 -1 0 1 }}E* = [0 1/7]. Multiply by {{monzo| 1 0 0 0 }}, the val for 2, and the result is {{monzo| 1 0 0 0 }}E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.
 == Translation between methods of specifying temperaments ==