Equivalence continuum

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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). This term was first used by Flora Canou.


Geometric interpretation

Mathematically, the rank-k equivalence continuum C(k, T) associated with a rank-r temperament T on a rank-n subgroup S is the space of saturated (n − k)-dimensional sublattices of the kernel (set of all intervals tempered out) of T, the rank-(n − r) lattice of commas tempered out by T. This is a set of rational points on the Grassmannian G = Gr(n − k, n − r) of (n − k)-dimensional vector subspaces of Rn−r, identifying Rn−r with the R-vector space ker(T) ⊗ R.

1-dimensional continua

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 or infinite 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 x and y axes corresponding to u and v directions. A rational point, i.e. a temperament on the continuum, then corresponds to a line with equation py = qx, of rational or infinite slope t = q/p, where the temperament is defined by the identification pu ~ qv (written additively). When t = 0, this corresponds to the temperament tempering out v. When t = (unsigned) infinity, this corresponds to the temperament tempering out u.

2-dimensional continua

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 G becomes Gr(2, 3). Define a coordinate system (x, y, z) for ker(T) using some fixed comma basis ux, uy, uz for ker(T). Then our Grassmannian can be identified with RP2 (the real projective plane, the 2-dimensional space of lines through the origin in 3-dimensional space) by taking the unique line Rv perpendicular (according to the dot product given by the given coordinates) to the plane of commas tempered out for each temperament. Note that the real projective plane RP2 can be visualized as a sphere with diametrically opposite points viewed as the same point.

Say that the vector v (which depends on T) defining this 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 with gcd 1, 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.] One coordinate vi is always guaranteed to be nonzero, for any temperament. Assuming v1 ≠ 0, 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 with v1 ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Two-dimensional coordinates can similarly be assigned for the set of all temperaments such that v2 ≠ 0 and the set of all temperaments such that v3 ≠ 0.

NB: Not all equivalence continua are projective spaces, because not all Grassmannians are; for example, C(4, 13-limit orwell), the equivalence continuum of rank-4 13-limit temperaments supported by 13-limit orwell, embeds into Gr(2, 4), which is not a projective space.

Example (7-limit rank-2 temperaments in 31edo)

Let's look at where some well-known 7-limit rank-2 temperaments supported by 31edo live in the 2-dimensional equivalence continuum C(2, 7-limit 31edo). Choose the basis ux, uy, uz = 81/80, 126/125, 1029/1024 to define (x, y, z) coordinates on the kernel of 7-limit 31edo. Then:

  • septimal meantone tempers out 81/80 = ux = (1, 0, 0) and 126/125 = uy = (0, 1, 0), thus corresponds to the plane z = 0. This corresponds to v = (0, 0, 1).
  • valentine tempers out 1029/1024 = uz = (0, 0, 1) and 126/125 = uy = (0, 1, 0). This corresponds to v = (1, 0, 0).
  • mohajira tempers out 81/80 = ux = (1, 0, 0) and 6144/6125 = uyuz = (0, 1, -1). This corresponds to v = (0, 1, 1).
  • hemithirds tempers out 1029/1024 = uz = (0, 0, 1) and 3136/3125 = 2ux + uy = (2, 1, 0). This corresponds to v = (1, −2, 0).
  • miracle tempers out 1029/1024 = uz = (0, 0, 1) and 225/224 = uxuy = (1, −1, 0). This corresponds to v = (1, 1, 0).