Rank and codimension: Difference between revisions

Line 14:

The {{w|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 {{nowrap|''n'' − ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap|''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.

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. 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.

Although the term "rank" as used here is exactly the same as used in group theory and linear algebra, it is important to note that the term "co-rank" is being used slightly differently. In both cases, the co-rank is the dimension of the cokernel (the quotient of codomain by image), and hence can be thought of as measuring the degree to which a homomorphism fails to be surjective. However, for any so-called temperament, if the group-theoretic co-rank is not 0, it is not a temperament at all, but is [[contorted]]. And if the linear-algebraic co-rank is not 0, that is even worse—it means you have a completely free generator with no mapping specified at any point along the chain. So the both the group-theoretic co-rank and the linear-algebraic co-rank are useless for a temperament, as they are always 0.

@@ Line 14: / Line 14: @@
 The {{w|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 {{nowrap|''n'' &minus; ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap|''n'' &minus; ''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.
+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. 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.
 Although the term "rank" as used here is exactly the same as used in group theory and linear algebra, it is important to note that the term "co-rank" is being used slightly differently. In both cases, the co-rank is the dimension of the cokernel (the quotient of codomain by image), and hence can be thought of as measuring the degree to which a homomorphism fails to be surjective. However, for any so-called temperament, if the group-theoretic co-rank is not 0, it is not a temperament at all, but is [[contorted]]. And if the linear-algebraic co-rank is not 0, that is even worse&mdash;it means you have a completely free generator with no mapping specified at any point along the chain. So the both the group-theoretic co-rank and the linear-algebraic co-rank are useless for a temperament, as they are always 0.