Equivalence continuum
An equivalence continuum is the space of all rank-k temperaments on a specified JI subgroup that are 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 associated with an edo), as then G = Gr(1, 2) = RP1 (the real projective line), 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 up/vq is tempered out by the temperament.
A higher-dimensional example: When r = 1, n = 4 (e.g. when S is the 7-limit), and k = 2, our Grassmannian becomes Gr(2, 3), which can be identified with RP2 (the real projective plane, the space of lines through the origin in 3-dimensional space) by taking the unique line Rv perpendicular to the plane of commas tempered out for each temperament. Say that this vector v has components (v1, v2, v3), so that the plane has equation v1x + v2y + v3z = 0. [More to come...]