Mathematical theory of regular temperaments: Difference between revisions

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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 Temperaments]].

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~~|JI~~]]"), 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.

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.

In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are "tempered together".

In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are ''tempered together''.

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.

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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".

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''.

== Dimensionality, or rank ==

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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 12et.

[[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 ===

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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 Temperaments]].
-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|JI]]"), 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.
+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.
-In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are "tempered together".
+In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are ''tempered together''.
 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.
@@ Line 9: / Line 9: @@
 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".
+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''.
 == Dimensionality, or rank ==
@@ 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 12et.
+[[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 ===