Rank and codimension
The rank of a regular temperament is simply its dimension. For example:
- Edos are rank-1 (1-dimensional) because their pitches can be described with one number (the number of edo steps).
- Mosses and temperaments based on them are rank 2 (2-dimensional), because the two dimensions are the number of periods and the number of generators. 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 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.
In the parlance of group theory, the intervals of a regular temperament comprise a 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 codimension or co-rank of a temperament is the number of commas needed to completely define the temperament. If the temperament tempers the 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 temperaments; an abstract regular temperament is of rank r if it is defined by a 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, 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.
Rank-nullity theorem
The rank–nullity theorem states that 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 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 r + n = d, then d − r = n.
All three of rank, nullity, and dimensionality are types of dimension:
- Rank is the dimension of the image (or range).
- 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".
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.