Rank and codimension: Difference between revisions
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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 [[ | 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. | ||