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 choice 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.

A higher-dimensional example: Say that r = 1, n = 4 (e.g. when S is the 7-limit), and k = 2, for example the set of rank-2 7-limit temperaments supported by 31edo. Then 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 the vector v defining the unique line has components (v₁, v₂, v₃), so that the plane associated with the rank-2 temperament has equation v₁x + v₂y + v₃z = 0. [We may further assume that v₁, v₂, v₃ are integers, since the condition of being perpendicular to two integer vectors is defined by a system of linear equations with integer coefficients, thus has a unique rational solution up to scaling.] If we (can) scale v by 1/v₁, then the resulting vector v/v₁ = (1, v₂/v₁, v₃/v₁) = (1, s, t) points in the same direction as v and describes two rational (or infinite) parameters s and t which defines any temperament on 31edo's 7-limit rank-2 continuum uniquely. Note that this discussion assumes specific coordinates (x, y, z) using some comma basis u_x, u_y, u_z for 7-limit 31edo.