Rank and codimension: Difference between revisions

Line 5:

* [[Mos]]ses and temperaments based on them are rank 2 (2-dimensional), because the two dimensions are the number of [[period]]s and the number of [[generator]]s. For instance, every interval of [[meantone]] can be obtained as a combination of a certain number of octaves (the period) up or down, plus a certain number of flattened meantone fifths (the generator) up or down.

The '''codimension''', '''co-rank''', or '''nullity''' of a temperament is the number of [[comma]]s needed to completely define the temperament as a temperament of a given [[JI subgroup]] (for example the ''p''-[[prime limit]]). For a rank 2 temperament such as meantone, this depends on the dimension of the JI subgroup it is a temperament of: namely, you need to ~~temper out~~ ''n'' – 2 commas to get a rank 2 temperament from a JI subgroup of dimension ''n''. For example, [[5-limit]] meantone has codimension 1: since 2.3.5 is a 3-dimensional JI subgroup, one comma (namely, [[81/80]]) needs to be ~~tempered out~~. On the other hand, 7-limit meantone (i.e. 5-limit meantone with C-A# seen as [[7/4]]) has codimension 2: since 2.3.5.7 is a 4-dimensional JI subgroup, you need two commas ([[81/80]] and [[225/224]]).

The '''codimension''', '''co-rank''', or '''nullity''' of a temperament is the number of [[comma]]s needed to completely define the temperament as a temperament of a given [[JI subgroup]] (for example the ''p''-[[prime limit]]). For a rank 2 temperament such as meantone, this depends on the dimension of the JI subgroup it is a temperament of: namely, you need to make ''n'' – 2 commas [[vanish]] to get a rank 2 temperament from a JI subgroup of dimension ''n''. For example, [[5-limit]] meantone has codimension 1: since 2.3.5 is a 3-dimensional JI subgroup, one comma (namely, [[81/80]]) needs to be made to vanish. On the other hand, 7-limit meantone (i.e. 5-limit meantone with C-A# seen as [[7/4]]) has codimension 2: since 2.3.5.7 is a 4-dimensional JI subgroup, you need two commas ([[81/80]] and [[225/224]]).

== Mathematical description ==

Line 12:

In the parlance of group theory, the intervals of a regular temperament comprise a [[wikipedia: Free abelian group #Rank|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.

The [[wikipedia: Codimension|codimension]] or [[wikipedia: Free abelian group #Rank|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it ~~tempers out~~ ''n'' - ''r'' independent commas, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.

The [[wikipedia: Codimension|codimension]] or [[wikipedia: Free abelian group #Rank|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes ''n'' - ''r'' independent commas vanish, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.

Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists|normal val list]] of ''r'' vals, or equivalently by an ''r''-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.

@@ Line 5: / Line 5: @@
 * [[Mos]]ses and temperaments based on them are rank 2 (2-dimensional), because the two dimensions are the number of [[period]]s and the number of [[generator]]s. For instance, every interval of [[meantone]] can be obtained as a combination of a certain number of octaves (the period) up or down, plus a certain number of flattened meantone fifths (the generator) up or down.
-The '''codimension''', '''co-rank''', or '''nullity''' of a temperament is the number of [[comma]]s needed to completely define the temperament as a temperament of a given [[JI subgroup]] (for example the ''p''-[[prime limit]]). For a rank 2 temperament such as meantone, this depends on the dimension of the JI subgroup it is a temperament of: namely, you need to temper out ''n'' – 2 commas to get a rank 2 temperament from a JI subgroup of dimension ''n''. For example, [[5-limit]] meantone has codimension 1: since 2.3.5 is a 3-dimensional JI subgroup, one comma (namely, [[81/80]]) needs to be tempered out. On the other hand, 7-limit meantone (i.e. 5-limit meantone with C-A# seen as [[7/4]]) has codimension 2: since 2.3.5.7 is a 4-dimensional JI subgroup, you need two commas ([[81/80]] and [[225/224]]).
+The '''codimension''', '''co-rank''', or '''nullity''' of a temperament is the number of [[comma]]s needed to completely define the temperament as a temperament of a given [[JI subgroup]] (for example the ''p''-[[prime limit]]). For a rank 2 temperament such as meantone, this depends on the dimension of the JI subgroup it is a temperament of: namely, you need to make ''n'' – 2 commas [[vanish]] to get a rank 2 temperament from a JI subgroup of dimension ''n''. For example, [[5-limit]] meantone has codimension 1: since 2.3.5 is a 3-dimensional JI subgroup, one comma (namely, [[81/80]]) needs to be made to vanish. On the other hand, 7-limit meantone (i.e. 5-limit meantone with C-A# seen as [[7/4]]) has codimension 2: since 2.3.5.7 is a 4-dimensional JI subgroup, you need two commas ([[81/80]] and [[225/224]]).
 == Mathematical description ==
@@ Line 12: / Line 12: @@
 In the parlance of group theory, the intervals of a regular temperament comprise a [[wikipedia: Free abelian group #Rank|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.
-The [[wikipedia: Codimension|codimension]] or [[wikipedia: Free abelian group #Rank|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it tempers out ''n'' - ''r'' independent commas, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.
+The [[wikipedia: Codimension|codimension]] or [[wikipedia: Free abelian group #Rank|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes ''n'' - ''r'' independent commas vanish, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.
 Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists|normal val list]] of ''r'' vals, or equivalently by an ''r''-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.