Rank and codimension: Difference between revisions

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The '''rank''' of a [[Regular_Temperaments|regular temperament]] is simply its dimension. For example, [[edo]]s are rank-1 (1-dimensional) because their pitches can be described with one number (the number of edo steps), and MOSes 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|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 '''[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 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. For example, [[5-limit]] (2.3.5 subgroup) meantone has codimension 1, since one comma (namely, [[81/80]]) needs to be tempered out. On the other hand, 7-limit meantone (5-limit meantone with C-A# seen as [[7/4]]) has codimension 2, since you need two commas ([[81/80]] and [[~~126~~/~~125~~]]).

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 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. For example, [[5-limit]] (2.3.5 subgroup) meantone has codimension 1, since one comma (namely, [[81/80]]) needs to be tempered out. On the other hand, 7-limit meantone (5-limit meantone with C-A# seen as [[7/4]]) has codimension 2, since you need two commas ([[81/80]] and [[225/224]]).

== Mathematical description ==

Mathematically, the rank of a regular temperament is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.

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 The '''rank''' of a [[Regular_Temperaments|regular temperament]] is simply its dimension. For example, [[edo]]s are rank-1 (1-dimensional) because their pitches can be described with one number (the number of edo steps), and MOSes 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|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 '''[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 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. For example, [[5-limit]] (2.3.5 subgroup) meantone has codimension 1, since one comma (namely, [[81/80]]) needs to be tempered out. On the other hand, 7-limit meantone (5-limit meantone with C-A# seen as [[7/4]]) has codimension 2, since you need two commas ([[81/80]] and [[126/125]]).
+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 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. For example, [[5-limit]] (2.3.5 subgroup) meantone has codimension 1, since one comma (namely, [[81/80]]) needs to be tempered out. On the other hand, 7-limit meantone (5-limit meantone with C-A# seen as [[7/4]]) has codimension 2, since you need two commas ([[81/80]] and [[225/224]]).
 == Mathematical description ==
 Mathematically, the rank of a regular temperament is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.