Equivalence continuum

An equivalence continuum is the space of all rank-k temperaments on a specified JI subgroup that is supported by a specified temperament of a lower rank on the same subgroup (such as an edo viewed as a temperament on said subgroup).

Examples:

• The syntonic-chromatic equivalence continuum is the 5-limit rank-2 equivalence continuum of 7edo.

Mathematical theory

Mathematically, the rank-k equivalence continuum associated with a rank-r temperament T on a rank-n subgroup S can be described as the set of rational points on the Grassmannian G = Gr(n-k, n-r) = Gr(n-k, ker(T)). This is the space of n-k-dimensional sublattices of the kernel of T, the rank-(n-r) lattice of commas tempered out by T.

This has a particularly simple description when r = 1 (i.e. when T is an edo), n = 3 (for example, when S is the 5-limit, 2.3.7 or 2.5.7) and k = 2 (so that we're considering the equivalence continua of rank-2 temperaments), as then G = Gr(1, 2) = RP¹ (real projective space of dimension 1), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational slope passing through the origin on the Cartesian plane R² where the lattice of ker(T) lives. The lattice of ker(T) is generated by a basis of some pair of two commas u and v in S tempered out by the edo; view the plane as having two perpendicular axes corresponding to u and v directions. A rational point, i.e. a temperament on the continuum, then corresponds to a rational ratio p/q, where u^p/v^q is tempered out by the temperament.