Rank and codimension: Difference between revisions

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The ''rank'' of a [[Regular_Temperaments|regular temperament]] is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.

The '''rank''' of a [[Regular_Temperaments|regular temperament]] is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.

In the parlance of group theory, the intervals of a regular temperament comprise a [http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Codimension codimension] or [http://en.wikipedia.org/wiki/Free_abelian_group#Rank co-rank] of a temperament is the number of [[Comma|commas]] 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|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 '''[http://en.wikipedia.org/wiki/Codimension codimension]''' or [http://en.wikipedia.org/wiki/Free_abelian_group#Rank co-rank] of a temperament is the number of [[Comma|commas]] 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|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|abstract regular temperaments]]; 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.

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-The ''rank'' of a [[Regular_Temperaments|regular temperament]] is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.
+The '''rank''' of a [[Regular_Temperaments|regular temperament]] is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.
 In the parlance of group theory, the intervals of a regular temperament comprise a [http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Codimension codimension] or [http://en.wikipedia.org/wiki/Free_abelian_group#Rank co-rank] of a temperament is the number of [[Comma|commas]] 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|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 '''[http://en.wikipedia.org/wiki/Codimension codimension]''' or [http://en.wikipedia.org/wiki/Free_abelian_group#Rank co-rank] of a temperament is the number of [[Comma|commas]] 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|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|abstract regular temperaments]]; 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.