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).

* ~~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.

* [[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|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|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 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]]).

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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).
-* 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.
+* [[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|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|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 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]]).