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).

* [[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.

* [[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''' 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]]).

== 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 thing called "codimension" above can be interpreted in linear algebra terms as the codimension of the subspace of supporting vals, relative to the ambient space of all vals. For any mapping matrix mapping from monzos to tmonzos, it's also the co-rank of the dual transformation from tvals back to vals.

[[Category:~~math~~]]

[[Category:~~rank~~]]

[[Category:Math]]

[[Category:~~rank_1~~]]

[[Category:Rank]]

[[Category:~~theory~~]]

[[Category:Rank 1]]

[[Category:Theory]]

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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).
-* [[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.
+* [[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''' 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]]).
 == 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 thing called "codimension" above can be interpreted in linear algebra terms as the codimension of the subspace of supporting vals, relative to the ambient space of all vals. For any mapping matrix mapping from monzos to tmonzos, it's also the co-rank of the dual transformation from tvals back to vals.
-[[Category:math]]
-[[Category:rank]]
+[[Category:Math]]
-[[Category:rank_1]]
+[[Category:Rank]]
-[[Category:theory]]
+[[Category:Rank 1]]
+[[Category:Theory]]