Rank and codimension: Difference between revisions

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* [[MOS]]es 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''' or '''co-rank''' 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 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]]).

== Mathematical description ==

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 * [[MOS]]es 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''' or '''co-rank''' 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 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]]).
 == Mathematical description ==