Mathematical theory of regular temperaments: Difference between revisions

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| ja = レギュラーテンペラメントとランクrテンペラメント

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~~:''This article focuses on the mathematical tools used to describe a regular temperament. For an introduction to regular temperaments, see [[~~Regular temperament~~]].''~~

A '''regular temperament''' is a homomorphism that maps an abelian group of target/pure intervals to another abelian group of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]), and tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.

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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.

This uses {{w|exterior algebra}} and {{w|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 W = {{multival| 6 -7 -2 -25 -20 15 }} for the wedgie of 7-limit miracle. Then the interior product W ∨ {{monzo| 1 0 0 0 }} is {{val| 0 -6 7 2 }}, with 15/14 we get W ∨ {{monzo| -1 1 1 -1 }} which is {{val| 1 1 3 3 }}, and with 16/15 we get W ∨ {{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.

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== Geometry of regular temperaments ==

Abstract regular temperaments can be identified with ~~[[Wikipedia: Rational point~~|rational ~~points]]~~ on an ~~[[Wikipedia: Algebraic variety~~|algebraic variety]] known as a ~~[[Wikipedia: Grassmannian~~|Grassmannian]]. In particular, if the number of primes in the ''p''-limit is ''n'', and the rank of the temperament is ''r'', then the real Grassmannian '''Gr''' (''r'', ''n'') has points identified with the ''r''-dimensional subspaces of the ''n''-dimensional real vector space '''R'''<sup>''n''</sup>. This has an embedding into a real vector space known as the [[~~Wikipedia: Pl%C3%BCcker embedding|~~Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank ''r'' in the ''p''-limit may be defined as rational points on '''Gr''' (''r'', ''n''), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr''' (''r'', ''n'') can be identified with real symmetric projection matrices with trace ''r''. The rational symmetric projection matrices with trace ''r'' are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr''' (''r'', ''n''). A rational projection matrix of trace ''r'' which is not symmetric is still a [[tuning map]]; minimax and least squares tunings provide examples of this.

Abstract regular temperaments can be identified with {{w|rational point}}s on an {{w|algebraic variety}} known as a {{w|Grassmannian}}. In particular, if the number of primes in the ''p''-limit is ''n'', and the rank of the temperament is ''r'', then the real Grassmannian '''Gr''' (''r'', ''n'') has points identified with the ''r''-dimensional subspaces of the ''n''-dimensional real vector space '''R'''<sup>''n''</sup>. This has an embedding into a real vector space known as the [[Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank ''r'' in the ''p''-limit may be defined as rational points on '''Gr''' (''r'', ''n''), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr''' (''r'', ''n'') can be identified with real symmetric projection matrices with trace ''r''. The rational symmetric projection matrices with trace ''r'' are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr''' (''r'', ''n''). A rational projection matrix of trace ''r'' which is not symmetric is still a [[tuning map]]; minimax and least squares tunings provide examples of this.

Grassmannians have the structure of a smooth, homogenous ~~[[Wikipedia: Metric space~~|metric space]], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr''' (2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space").

Grassmannians have the structure of a smooth, homogenous {{w|metric space}}, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr''' (2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space").

See also [[equivalence continuum]] for a description of the space of rank-''r'' temperaments supported by a given temperament, such as a rank-1 temperament, as an algebraic variety.

@@ Line 5: / Line 5: @@
 | ja = レギュラーテンペラメントとランクrテンペラメント
 }}
-:''This article focuses on the mathematical tools used to describe a regular temperament. For an introduction to regular temperaments, see [[Regular temperament]].''
+{{Expert|Regular temperament}}
 A '''regular temperament''' is a homomorphism that maps an abelian group of target/pure intervals to another abelian group of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]), and tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.
@@ Line 46: / Line 46: @@
 {{Main| Wedgies and multivals }}
-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.
+This uses {{w|exterior algebra}} and {{w|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 W = {{multival| 6 -7 -2 -25 -20 15 }} for the wedgie of 7-limit miracle. Then the interior product W ∨ {{monzo| 1 0 0 0 }} is {{val| 0 -6 7 2 }}, with 15/14 we get W ∨ {{monzo| -1 1 1 -1 }} which is {{val| 1 1 3 3 }}, and with 16/15 we get W ∨ {{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.
@@ Line 91: / Line 91: @@
 == Geometry of regular temperaments ==
-Abstract regular temperaments can be identified with [[Wikipedia: Rational point|rational points]] on an [[Wikipedia: Algebraic variety|algebraic variety]] known as a [[Wikipedia: Grassmannian|Grassmannian]]. In particular, if the number of primes in the ''p''-limit is ''n'', and the rank of the temperament is ''r'', then the real Grassmannian '''Gr''' (''r'', ''n'') has points identified with the ''r''-dimensional subspaces of the ''n''-dimensional real vector space '''R'''<sup>''n''</sup>. This has an embedding into a real vector space known as the [[Wikipedia: Pl%C3%BCcker embedding|Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank ''r'' in the ''p''-limit may be defined as rational points on '''Gr''' (''r'', ''n''), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr''' (''r'', ''n'') can be identified with real symmetric projection matrices with trace ''r''. The rational symmetric projection matrices with trace ''r'' are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr''' (''r'', ''n''). A rational projection matrix of trace ''r'' which is not symmetric is still a [[tuning map]]; minimax and least squares tunings provide examples of this.
+Abstract regular temperaments can be identified with {{w|rational point}}s on an {{w|algebraic variety}} known as a {{w|Grassmannian}}. In particular, if the number of primes in the ''p''-limit is ''n'', and the rank of the temperament is ''r'', then the real Grassmannian '''Gr''' (''r'', ''n'') has points identified with the ''r''-dimensional subspaces of the ''n''-dimensional real vector space '''R'''<sup>''n''</sup>. This has an embedding into a real vector space known as the [[Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank ''r'' in the ''p''-limit may be defined as rational points on '''Gr''' (''r'', ''n''), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr''' (''r'', ''n'') can be identified with real symmetric projection matrices with trace ''r''. The rational symmetric projection matrices with trace ''r'' are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr''' (''r'', ''n''). A rational projection matrix of trace ''r'' which is not symmetric is still a [[tuning map]]; minimax and least squares tunings provide examples of this.
-Grassmannians have the structure of a smooth, homogenous [[Wikipedia: Metric space|metric space]], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr''' (2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space").
+Grassmannians have the structure of a smooth, homogenous {{w|metric space}}, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr''' (2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space").
 See also [[equivalence continuum]] for a description of the space of rank-''r'' temperaments supported by a given temperament, such as a rank-1 temperament, as an algebraic variety.