Rank and codimension - Revision history

VectorGraphics at 05:01, 12 June 2025

2025-06-12T05:01:13Z

← Older revision		Revision as of 05:01, 12 June 2025
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	The {{w\|codimension}} or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes {{nowrap\|''n'' − ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap\|''n'' − ''r''}}. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.		The {{w\|codimension}} or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes {{nowrap\|''n'' − ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap\|''n'' − ''r''}}. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.

	Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists\|normal val list]] of ''r'' vals~~, or equivalently by an ''r''-multival~~. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.		Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists\|normal val list]] of ''r'' vals. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.

	Although the term "rank" as used here is exactly the same as used in group theory and linear algebra, it is important to note that the term "co-rank" is being used slightly differently. In both cases, the co-rank is the dimension of the cokernel (the quotient of codomain by image), and hence can be thought of as measuring the degree to which a homomorphism fails to be surjective. However, for any so-called temperament, if the group-theoretic co-rank is not 0, it is not a temperament at all, but is [[contorted]]. And if the linear-algebraic co-rank is not 0, that is even worse—it means you have a completely free generator with no mapping specified at any point along the chain. So the both the group-theoretic co-rank and the linear-algebraic co-rank are useless for a temperament, as they are always 0.		Although the term "rank" as used here is exactly the same as used in group theory and linear algebra, it is important to note that the term "co-rank" is being used slightly differently. In both cases, the co-rank is the dimension of the cokernel (the quotient of codomain by image), and hence can be thought of as measuring the degree to which a homomorphism fails to be surjective. However, for any so-called temperament, if the group-theoretic co-rank is not 0, it is not a temperament at all, but is [[contorted]]. And if the linear-algebraic co-rank is not 0, that is even worse—it means you have a completely free generator with no mapping specified at any point along the chain. So the both the group-theoretic co-rank and the linear-algebraic co-rank are useless for a temperament, as they are always 0.

VectorGraphics at 03:39, 21 February 2025

2025-02-21T03:39:56Z

← Older revision		Revision as of 03:39, 21 February 2025
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	The '''rank''' of a [[regular temperament]] is simply its dimension. For example:		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]]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 perfect 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 perfect 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]] {{nowrap\|''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]] {{nowrap\|''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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	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.		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.

	In the parlance of group theory, the intervals of a regular temperament comprise a [[~~wikipedia:Free abelian group #Rank~~\|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.		In the parlance of group theory, the intervals of a regular temperament comprise a [[Stacking\|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.

	The {{w\|codimension}} or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes {{nowrap\|''n'' − ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap\|''n'' − ''r''}}. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.		The {{w\|codimension}} or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes {{nowrap\|''n'' − ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap\|''n'' − ''r''}}. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.

ArrowHead294 at 20:58, 3 January 2025

2025-01-03T20:58:11Z

← Older revision		Revision as of 20:58, 3 January 2025
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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.		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 {{w\|rank–nullity theorem}} states that {{nowrap\|''r'' + ''n'' {{=}} ''d''}}, where ''r'' is the rank, ''n'' is the nullity (or codimension, or corank), and ''d'' 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 {{nowrap\|3 − 1 {{=}} 2}}, but in a dimensionality-5 system, that would be a rank-4 temperament, because {{nowrap\|5 − 1 {{=}} 4}}; this is of course because if {{nowrap\|''r'' + ''n'' {{=}} ''d''}}, then {{nowrap\|''d'' − ''r'' {{=}} ''n''}}.		The {{w\|rank–nullity theorem}} states that {{nowrap\|''r'' + ''n'' {{=}} ''d''}}, where ''r'' is the rank, ''n'' is the nullity (or codimension, or corank), and ''d'' 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 {{nowrap\|3 − 1 {{=}} 2}}, but in a dimensionality-5 system, that would be a rank-4 temperament, because {{nowrap\|5 − 1 {{=}} 4}}; this is of course because if {{nowrap\|''r'' + ''n'' {{=}} ''d''}}, then {{nowrap\|''d'' − ''r'' {{=}} ''n''}}.

ArrowHead294: Formatting

2024-11-09T16:01:31Z

Formatting

Cmloegcmluin: prefer "vanish" to "temper out"

2022-12-27T18:13:17Z

prefer "vanish" to "temper out"

← Older revision		Revision as of 18:13, 27 December 2022
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	* [[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.		* [[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]]).		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 make ''n'' – 2 commas [[vanish]] 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 made to vanish. 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 ==		== Mathematical description ==
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	In the parlance of group theory, the intervals of a regular temperament comprise a [[wikipedia: Free abelian group #Rank\|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.		In the parlance of group theory, the intervals of a regular temperament comprise a [[wikipedia: Free abelian group #Rank\|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.

	The [[wikipedia: Codimension\|codimension]] or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it ~~tempers out~~ ''n'' - ''r'' independent commas, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.		The [[wikipedia: Codimension\|codimension]] or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes ''n'' - ''r'' independent commas vanish, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.

	Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists\|normal val list]] of ''r'' vals, or equivalently by an ''r''-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.		Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists\|normal val list]] of ''r'' vals, or equivalently by an ''r''-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.

Cmloegcmluin: prefer nullspace to null-space

2022-12-15T00:17:42Z

prefer nullspace to null-space

← Older revision		Revision as of 00:17, 15 December 2022
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	All three of rank, nullity, and dimensionality are types of dimension:		All three of rank, nullity, and dimensionality are types of dimension:
	* Rank is the dimension of the [[image]] (or range).		* Rank is the dimension of the [[image]] (or range).
	* Nullity is the dimension of the [[~~null-space~~]] (or kernel).		* Nullity is the dimension of the [[nullspace]] (or kernel).
	* Dimensionality is the dimension of the (sub)group (or domain). It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".		* Dimensionality is the dimension of the (sub)group (or domain). It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".

FloraC: Style

2022-08-18T18:03:52Z

Style

← Older revision		Revision as of 18:03, 18 August 2022
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	The '''rank''' of a [[regular temperament]] is simply its dimension. For example:		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]]).		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.		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:		All three of rank, nullity, and dimensionality are types of dimension:

Cmloegcmluin: /* Rank-nullity theorem */ add links

2022-01-24T22:58:15Z

Rank-nullity theorem: add links

← Older revision		Revision as of 22:58, 24 January 2022
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	All three of rank, nullity, and dimensionality are types of dimension:		All three of rank, nullity, and dimensionality are types of dimension:
	* Rank is the dimension of the image (or range).		* Rank is the dimension of the [[image]] (or range).
	* Nullity is the dimension of the null-space (or kernel).		* Nullity is the dimension of the [[null-space]] (or kernel).
	* Dimensionality is the dimension of the (sub)group (or domain). It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".		* Dimensionality is the dimension of the (sub)group (or domain). It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".

Cmloegcmluin: consolidate categories

2021-12-25T13:37:55Z

consolidate categories

← Older revision		Revision as of 13:37, 25 December 2021
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	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	~~[[Category:Temperament]]~~
	~~[[Category:Theory]]~~
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]

Cmloegcmluin: /* Rank-nullity theorem */

2021-12-11T20:41:20Z

Rank-nullity theorem

← Older revision		Revision as of 20:41, 11 December 2021
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	* Rank is the dimension of the image (or range).		* Rank is the dimension of the image (or range).
	* Nullity is the dimension of the null-space (or kernel).		* Nullity is the dimension of the null-space (or kernel).
	* Dimensionality is the dimension of the domain. It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".		* Dimensionality is the dimension of the (sub)group (or domain). It is a bit confusing that, unlike the other two, "dimensionality" actually contains the word "dimension".

	Collectively we could refer to these as a temperament's '''dimensions'''.		Collectively we could refer to these as a temperament's '''dimensions'''.

← Older revision		Revision as of 16:01, 9 November 2024
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	The '''rank''' of a [[regular temperament]] is simply its dimension. For example:		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]]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.		* [[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 perfect 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 ~~make~~ ''n'' – 2 commas ~~[[vanish]]~~ 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 ~~made to vanish~~. 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]] {{nowrap\|''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 ==		== 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.		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.

	In the parlance of group theory, the intervals of a regular temperament comprise a [[wikipedia: Free abelian group #Rank\|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.		In the parlance of group theory, the intervals of a regular temperament comprise a [[wikipedia:Free abelian group #Rank\|finitely generated free abelian group]] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.

	The ~~[[wikipedia: Codimension~~\|codimension]] or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes ''n'' - ''r'' independent commas vanish, it will be of rank ''r'' and codimension ''n'' - ''r''. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.		The {{w\|codimension}} or [[wikipedia: Free abelian group #Rank\|co-rank]] of a temperament is the number of [[comma]]s needed to completely define the temperament. If the temperament tempers the [[Harmonic limit\|''p''-limit]] just intonation group generated by the first ''n'' primes, then if it makes {{nowrap\|''n'' − ''r''}} independent commas vanish, it will be of rank ''r'' and codimension {{nowrap\|''n'' − ''r''}}. The terminology can also be applied to [[just intonation subgroups]]. In all cases care must be taken to specify the exact just intonation group which is being tempered by the tempering out of a set of commas.

	Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists\|normal val list]] of ''r'' vals, or equivalently by an ''r''-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.		Looking only at the number of independent generators of a tuning can obscure its real nature, at least as it is being applied. For instance, a 31et tuning of meantone temperament, with a meantone fifth of 18\31 octaves, is of rank one in the sense that all the intervals in the tuning are generated from 1\31; however, it is being used as a rank two tuning. This issue can be gotten around by means of [[abstract regular temperament]]s; an abstract regular temperament is of rank ''r'' if it is defined by a [[Normal lists\|normal val list]] of ''r'' vals, or equivalently by an ''r''-multival. The abstractly characterized intervals of the abstract temperament can then be mapped to a tuning; if the mapping is to a rank one tuning such as 31et, that does not affect the rank of the temperament.

	Although the term "rank" as used here is exactly the same as used in group theory and linear algebra, it is important to note that the term "co-rank" is being used slightly differently. In both cases, the co-rank is the dimension of the cokernel (the quotient of codomain by image), and hence can be thought of as measuring the degree to which a homomorphism fails to be surjective. However, for any so-called temperament, if the group-theoretic co-rank is not 0, it is not a temperament at all ~~– it~~ is [[contorted]]. And if the linear-algebraic co-rank is not 0, that is even worse – it means you have a completely free generator with no mapping specified at any point along the chain. So the both the group-theoretic co-rank and the linear-algebraic co-rank are useless ~~pieces of information~~ for a temperament – they are always 0.		Although the term "rank" as used here is exactly the same as used in group theory and linear algebra, it is important to note that the term "co-rank" is being used slightly differently. In both cases, the co-rank is the dimension of the cokernel (the quotient of codomain by image), and hence can be thought of as measuring the degree to which a homomorphism fails to be surjective. However, for any so-called temperament, if the group-theoretic co-rank is not 0, it is not a temperament at all, but is [[contorted]]. And if the linear-algebraic co-rank is not 0, that is even worse—it means you have a completely free generator with no mapping specified at any point along the chain. So the both the group-theoretic co-rank and the linear-algebraic co-rank are useless for a temperament, as they are always 0.

	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.		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 {{w\|rank–nullity theorem}} states that {{nowrap\|''r'' + ''n'' {{=}} ''d''}}, where ''r'' is the rank, ''n'' is the nullity (or codimension, or corank), and ''d'' 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 {{nowrap\|3 − 1 {{=}} 2}}, but in a dimensionality-5 system, that would be a rank-4 temperament, because {{nowrap\|5 − 1 {{=}} 4}}; this is of course because if {{nowrap\|''r'' + ''n'' {{=}} ''d''}}, then {{nowrap\|''d'' − ''r'' {{=}} ''n''}}.

	All three of rank, nullity, and dimensionality are types of dimension:		All three of rank, nullity, and dimensionality are types of dimension:
Line 30:		Line 30:
	Collectively we could refer to these as a temperament's '''dimensions'''.		Collectively we could refer to these as a temperament's '''dimensions'''.

	The rank-nullity theorem is also discussed in [[Mike's Lecture on the First Fundamental Law of Tempering]].		The rank–nullity theorem is also discussed in [[Mike's Lecture on the First Fundamental Law of Tempering]].

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]