Equivalence continuum
An equivalence continuum is the space of all rank-k temperaments on a specified JI subgroup that is tempered out 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, ker(T)), or the space of n-k-dimensional subspaces of the kernel of T, the space 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 up/vq is tempered out by the temperament.