Mathematical theory of regular temperaments: Difference between revisions

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For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.

To use a concrete example, if [[Meantone family#~~septimal~~ meantone|7-limit meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".

To use a concrete example, if [[Meantone family #Septimal meantone|7-limit meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".

A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament "regular".

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=== Rank-1 (equal) temperaments ===

[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated ~~EDO~~ or ~~ED2~~) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-~~EDO~~ is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An ~~EDO~~ can be treated as an ET by applying a temperament mapping to the intervals of the ~~EDO~~, typically by using a val for a temperament supported by that ~~EDO~~, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by ~~12-ET~~.

[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated edo or ed2) are similar concepts, although there are distinctions in the way these terms are used. A ''p''-limit ET is simply a ''p''-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of ''n''-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an ''n''-edo is a division of the octave into ''n'' equal parts, with no consideration given to mapping of JI intervals. An edo can be treated as an ET by applying a temperament mapping to the intervals of the edo, typically by using a val for a temperament supported by that edo, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12et.

=== Rank-2 (including linear) temperaments ===

A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to ~~12-ET~~ by tempering out the Pythagorean comma.

A ''p''-limit rank-2 temperament maps all intervals of ''p''-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12et by tempering out the Pythagorean comma.

Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].

== Characterizing a regular temperament ==

=== Normal val ~~list~~ ===

=== Normal val lists ===

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To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.

=== ~~The normal~~ val ~~list~~ ===

=== Normal val lists ===

To translate from the normal val list to the RREF, simply reduce the normal val list. To obtain the normal val list from the RREF, clear denominators from the rows of the RREF, saturate the result, reduce that to Hermite normal form and make the adjustment to normal val form.

=== ~~The normal~~ comma ~~list~~ ===

=== Normal comma lists ===

To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection matrix by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.

== ~~The~~ Geometry of ~~Regular Temperaments~~ ==

== Geometry of regular temperaments ==

Abstract regular temperaments can be identified with [~~http~~:~~//en.wikipedia.org/wiki/Rational_point~~ rational points] on an [~~http~~:~~//en.wikipedia.org/wiki/Algebraic_variety~~ algebraic variety] known as a [~~http~~:~~//en.wikipedia.org/wiki/~~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'''^n. This has an embedding into a real vector space known as the [~~http~~:~~//en.wikipedia.org/wiki/~~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 [[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.

Grassmannians have the structure of a smooth, homogenous [~~http~~:~~//en.wikipedia.org/wiki/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 [[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").

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

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.

[[File:dualzoom.gif|alt=dualzoom.gif|dualzoom.gif]]

== See also ==

* [[Tour of Regular Temperaments]]

* [[Wikipedia: Regular temperament]]

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[[Category:Math]]

[[Category:Temperament]]

~~[[Category:Theory]]~~

@@ Line 7: / Line 7: @@
 For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
-To use a concrete example, if [[Meantone family#septimal meantone|7-limit meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
+To use a concrete example, if [[Meantone family #Septimal meantone|7-limit meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
 A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament "regular".
@@ Line 16: / Line 16: @@
 === Rank-1 (equal) temperaments ===
-[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET.
+[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated edo or ed2) are similar concepts, although there are distinctions in the way these terms are used. A ''p''-limit ET is simply a ''p''-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of ''n''-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an ''n''-edo is a division of the octave into ''n'' equal parts, with no consideration given to mapping of JI intervals. An edo can be treated as an ET by applying a temperament mapping to the intervals of the edo, typically by using a val for a temperament supported by that edo, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12et.
 === Rank-2 (including linear) temperaments ===
-A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.
+A ''p''-limit rank-2 temperament maps all intervals of ''p''-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12et by tempering out the Pythagorean comma.
 Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].
 == Characterizing a regular temperament ==
-=== Normal val list ===
+=== Normal val lists ===
 {{Main| Normal lists #Normal val lists }}
@@ Line 75: / Line 75: @@
 To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.
-=== The normal val list ===
+=== Normal val lists ===
 To translate from the normal val list to the RREF, simply reduce the normal val list. To obtain the normal val list from the RREF, clear denominators from the rows of the RREF, saturate the result, reduce that to Hermite normal form and make the adjustment to normal val form.
-=== The normal comma list ===
+=== Normal comma lists ===
 To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection matrix by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.
-== The Geometry of Regular Temperaments ==
+== Geometry of regular temperaments ==
-Abstract regular temperaments can be identified with [http://en.wikipedia.org/wiki/Rational_point rational points] on an [http://en.wikipedia.org/wiki/Algebraic_variety algebraic variety] known as a [http://en.wikipedia.org/wiki/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'''^n. This has an embedding into a real vector space known as the [http://en.wikipedia.org/wiki/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 [[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.
-Grassmannians have the structure of a smooth, homogenous [http://en.wikipedia.org/wiki/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 [[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").
-See also [[equivalence continuum]] for a description of the space of rank-''r'' temperaments supported by a given temperament, such as an edo (rank-1 temperament), as an algebraic variety.
+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.
 [[File:dualzoom.gif|alt=dualzoom.gif|dualzoom.gif]]
 == See also ==
 * [[Tour of Regular Temperaments]]
 * [[Wikipedia: Regular temperament]]
@@ Line 101: / Line 100: @@
 [[Category:Math]]
 [[Category:Temperament]]
-[[Category:Theory]]
 {{todo|reduce mathslang|improve definition|cleanup}}