Rank and codimension: Difference between revisions

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The '''rank''' of a [[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).

* [[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]] 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]]ses 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''', '''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]]).

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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.

==Rank-nullity theorem==

== Rank-nullity theorem ==

The [[wikipedia: Rank%E2%80%93nullity theorem|rank-nullity theorem]] states that <math>r + n = d</math>, where <math>r</math> is the rank, <math>n</math> is the nullity (or codimension, or corank), and <math>d</math> is the '''dimensionality'''. The dimensionality is the dimension of the system before it is tempered; it is the number of entries in the [[subgroup]]. For example, a 5-limit temperament is dimensionality-3, because it uses three primes: 2, 3, and 5 — a total of 3 primes. An 11-limit temperament is dimensionality-5, because it uses primes: 2, 3, 5, 7, and 11 — a total of 5 primes. If we temper one comma, we have a nullity-1 temperament; in a dimensionality-3 system, that would be a rank-2 temperament, because 3 - 1 = 2, but in a dimensionality-5 system, that would be a rank-4 temperament, because 5 - 1 = 4; this is of course because if <math>r + n = d</math>, then <math>d - r = n</math>.

The [~~https~~:~~//en.wikipedia.org/wiki/~~Rank%E2%80%~~93nullity_theorem the~~ rank-nullity theorem] states that <math>r + n = d</math>, where <math>r</math> is the rank, <math>n</math> is the nullity (or codimension, or corank), and <math>d</math> is the '''dimensionality'''. The dimensionality is the dimension of the system before it is tempered; it is the number of entries in the [[subgroup]]. For example, a 5-limit temperament is dimensionality-3, because it uses three primes: 2, 3, and 5 — a total of 3 primes. An 11-limit temperament is dimensionality-5, because it uses primes: 2, 3, 5, 7, and 11 — a total of 5 primes. If we temper one comma, we have a nullity-1 temperament; in a dimensionality-3 system, that would be a rank-2 temperament, because 3 - 1 = 2, but in a dimensionality-5 system, that would be a rank-4 temperament, because 5 - 1 = 4; this is of course because if <math>r + n = d</math>, then <math>d - r = n</math>.

All three of rank, nullity, and dimensionality are types of dimension:

@@ Line 2: / Line 2: @@
 The '''rank''' of a [[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).
+* [[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]] 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]]ses 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''', '''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]]).
@@ Line 20: / Line 20: @@
 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.
-==Rank-nullity theorem==
+== Rank-nullity theorem ==
+The [[wikipedia: Rank%E2%80%93nullity theorem|rank-nullity theorem]] states that <math>r + n = d</math>, where <math>r</math> is the rank, <math>n</math> is the nullity (or codimension, or corank), and <math>d</math> is the '''dimensionality'''. The dimensionality is the dimension of the system before it is tempered; it is the number of entries in the [[subgroup]]. For example, a 5-limit temperament is dimensionality-3, because it uses three primes: 2, 3, and 5 — a total of 3 primes. An 11-limit temperament is dimensionality-5, because it uses primes: 2, 3, 5, 7, and 11 — a total of 5 primes. If we temper one comma, we have a nullity-1 temperament; in a dimensionality-3 system, that would be a rank-2 temperament, because 3 - 1 = 2, but in a dimensionality-5 system, that would be a rank-4 temperament, because 5 - 1 = 4; this is of course because if <math>r + n = d</math>, then <math>d - r = n</math>.
-The [https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem the rank-nullity theorem] states that <math>r + n = d</math>, where <math>r</math> is the rank, <math>n</math> is the nullity (or codimension, or corank), and <math>d</math> is the '''dimensionality'''. The dimensionality is the dimension of the system before it is tempered; it is the number of entries in the [[subgroup]]. For example, a 5-limit temperament is dimensionality-3, because it uses three primes: 2, 3, and 5 — a total of 3 primes. An 11-limit temperament is dimensionality-5, because it uses primes: 2, 3, 5, 7, and 11 — a total of 5 primes. If we temper one comma, we have a nullity-1 temperament; in a dimensionality-3 system, that would be a rank-2 temperament, because 3 - 1 = 2, but in a dimensionality-5 system, that would be a rank-4 temperament, because 5 - 1 = 4; this is of course because if <math>r + n = d</math>, then <math>d - r = n</math>.
 All three of rank, nullity, and dimensionality are types of dimension: