Mathematical theory of regular temperaments: Difference between revisions

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== Characterizing a regular temperament ==

=== Normal ~~map lists~~ ===

=== Normal mapping matrices ===

Since an abstract temperament corresponds to some linear map, we can represent it as a matrix. We can [[Mathematical theory of saturation|saturate]] it and reduce it to the [[Hermite normal form]], which gives a unique representation. Applying this map to the vector representation of a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal form for 7-limit miracle is

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and applying this to the vector for either 16/15 or 15/14 leads to [0 1].

=== Normal comma ~~lists~~ ===

=== Normal comma bases ===

A temperament may also be defined by a list of commas. By putting these into a normal form, the representation is also unique. Using commas has the advantage of showing family relationships more clearly.

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=== Just intonation subgroups and transversals ===

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.

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Maple code for the parts of these which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]].

~~=== Wedgies ===~~

To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank ''r'' in ''n'' dimensions (where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes in the ''p''-limit) take a wedge product of basis vectors involving {{nowrap|''r'' − 1}} basis elements (i.e., the wedge product of {{nowrap|''r'' − 1}} elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an ''r''-fold wedge product with sign ±1. Take the corresponding element of the wedgie times the ±1 sign (which is computed from the parity of the permutation of the ''r'' elements.) This gives a val; do this for every combination of {{nowrap|''r'' − 1}} basis elements to obtain ''n'' choose {{nowrap|''r'' − 1}} vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.

~~An alternative explanation of this process is provided here: [[Intro to exterior algebra for RTT#Converting varianced multivectors to matrices]]~~

=== Frobenius projection matrices ===

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== Geometry of regular temperaments ==

~~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 {{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.

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 == Characterizing a regular temperament ==
-=== Normal map lists ===
+=== Normal mapping matrices ===
-{{Main| Normal forms #Normal forms for mappings }}
+{{Main| Temperament mapping matrix }}
+{{See also| Normal forms #Normal forms for mappings }}
 Since an abstract temperament corresponds to some linear map, we can represent it as a matrix. We can [[Mathematical theory of saturation|saturate]] it and reduce it to the [[Hermite normal form]], which gives a unique representation. Applying this map to the vector representation of a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal form for 7-limit miracle is
@@ Line 43: / Line 44: @@
 and applying this to the vector for either 16/15 or 15/14 leads to [0&nbsp;1].
-=== Normal comma lists ===
+=== Normal comma bases ===
-{{Main| Normal forms #Normal forms for commas }}
+{{Main| Comma basis }}
+{{See also| Normal forms #Normal forms for commas }}
 A temperament may also be defined by a list of commas. By putting these into a normal form, the representation is also unique. Using commas has the advantage of showing family relationships more clearly.
@@ Line 60: / Line 62: @@
 === Just intonation subgroups and transversals ===
 {{Main| Just intonation subgroups | Transversal }}
+{{See also| Gencom }}
 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.
@@ Line 69: / Line 72: @@
 Maple code for the parts of these which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]].
-=== Wedgies ===
-To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank ''r'' in ''n'' dimensions (where {{nowrap|''n'' {{=}} π(''p'')}} is the number of primes in the ''p''-limit) take a wedge product of basis vectors involving {{nowrap|''r'' − 1}} basis elements (i.e., the wedge product of {{nowrap|''r'' − 1}} elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an ''r''-fold wedge product with sign ±1. Take the corresponding element of the wedgie times the ±1 sign (which is computed from the parity of the permutation of the ''r'' elements.) This gives a val; do this for every combination of {{nowrap|''r'' − 1}} basis elements to obtain ''n'' choose {{nowrap|''r'' − 1}} vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.
-An alternative explanation of this process is provided here: [[Intro to exterior algebra for RTT#Converting varianced multivectors to matrices]]
 === Frobenius projection matrices ===
@@ Line 85: / Line 83: @@
 == Geometry of regular temperaments ==
-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'',&nbsp;''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'',&nbsp;''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'',&nbsp;''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'',&nbsp;''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.
+{{Main|Plucker coordinates}}
-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,&nbsp;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.