Equivalence continuum

An equivalence continuum comprises all the temperaments where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is q₁, and the second interval, which we may call the targeted interval, is q₂, both in ratios, an equivalence continuum is formed by all the temperaments that satisfy q n
1  ~ q₂, where n is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified JI subgroup that are supported by a specified temperament of a lower rank (such as an equal temperament) on the same subgroup.

For example, in the syntonic–chromatic equivalence continuum, a number of syntonic commas is equated with the Pythagorean chromatic semitone: (81/80)ⁿ ~ 2187/2048, and this describes all temperaments supported by 7et since that is the unique temperament that tempers out both and hence all the combinations thereof. By specifying different values of n, we obtain temperaments such as porcupine, tetracot, amity, and so on.

The term was first used by Mike Battaglia in 2011.

Choice of basis

It can be shown that different choices of intervals can lead to essentially identical continua, where the related individual temperaments are the same. For instance, in the syntonic–chromatic equivalence continuum, if the stacked interval q₁ is the syntonic comma, it does not matter if the targeted interval q₂, a chromatic semitone, is Pythagorean (2187/2048), major (135/128), or classical (25/24), as they only differ by whole multiples of the syntonic comma. For consistency, the following scheme is established as the default choice for stacked and targeted intervals for equivalence continua of rank-2 temperaments:

• The stacked interval q₁ should have the least nonzero absolute value of order in the last formal prime. Typically it is ±1, but in case that is impossible, it is ±2, ±3, ….
• The targeted interval q₂ should have order 0 in the last formal prime. In particular, for continua of 2.3.5, 2.3.7, 2.3.11, …, it should be a 3-limit interval.
• The comma tempered out in the temperament corresponding to n = 1 should be smaller in size than q₂.

This guarantees that in the corresponding temperament, n equals the order of the last formal prime in the comma, and equals the number of steps to obtain the interval class of the second formal prime in the generator chain.

Inversion

A continuum can be inverted by setting m such that 1/m + 1/n = 1, with temperaments in it characterized by the relation (q₂/q₁)^m ~ q₂. Here the stacked interval is q₂/q₁, and the targeted interval remains q₂. For instance, the inversion of the syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)^m ~ 2187/2048.

This m-continuum, like the n-continuum, also meets the requirements for a possible default choice, and raises the question which one should be the n-continuum and which one should be the m-continuum. In principle, we take the n-continuum as the main continuum and the m-continuum supplementary. If one of the candidate stacked intervals is simpler and smaller, we set it to q₁ of the n-continuum so that more useful temperaments are included in it. However, the simpler interval is sometimes the larger one, in which case the choice could be made on a heuristic basis.

Geometric interpretation

This page or section may be difficult to understand to those unfamiliar with the mathematical concepts involved. A more accessible version will be worked on; in the meantime, feel free to ask questions in the Xenharmonic Alliance Discord server or Facebook group.

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 R^{n − r}, identifying R^{n − 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 = ∞, 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 u_x, u_y, u_z 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, and v₃ are relatively prime 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.] One coordinate v_i 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.

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 {u_x, u_y, u_z} = {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 = u_x = (1, 0, 0) and 126/125 = u_y = (0, 1, 0), thus corresponds to the plane z = 0. This corresponds to v = (0, 0, 1).
• Valentine tempers out 1029/1024 = u_z = (0, 0, 1) and 126/125 = u_y = (0, 1, 0). This corresponds to v = (1, 0, 0).
• Mohajira tempers out 81/80 = u_x = (1, 0, 0) and 6144/6125 = u_y − u_z = (0, 1, −1). This corresponds to v = (0, 1, 1).
• Hemithirds tempers out 1029/1024 = u_z = (0, 0, 1) and 3136/3125 = 2u_x + u_y = (2, 1, 0). This corresponds to v = (1, −2, 0).
• Miracle tempers out 1029/1024 = u_z = (0, 0, 1) and 225/224 = u_x − u_y = (1, −1, 0). This corresponds to v = (1, 1, 0).

List of equivalence continua

See also: Category: Equivalence continua

All equivalence continua currently on the wiki are rank-(n + 1) continua of rank-(n + 1) temperaments within a rank-(n + 2) subgroup that are supported by a rank-n system.

• 5-limit rank-2 continua include:
  • the father–3 equivalence continuum (3edo, a 3- and 5-limit record edo)
  • the syntonic–diatonic equivalence continuum (5edo, a 3- and 5-limit record edo)
  • the syntonic–chromatic equivalence continuum (7edo, a 3- and 5-limit record edo)
  • the schismic–Pythagorean equivalence continuum (12edo, a 3- and 5-limit record edo)
  • the syntonic–kleismic equivalence continuum (19edo, a 5-limit record edo)
  • the superpyth–22 equivalence continuum (22edo)
  • the syntonic–31 equivalence continuum (31edo, a 5-limit record edo)
  • the diaschismic–gothmic equivalence continuum (34edo, a 5-limit record edo)
  • the schismic–countercommatic equivalence continuum (41edo, a 3-limit record edo)
  • the schismic–Mercator equivalence continuum (53edo, a 3- and 5-limit record edo)
  • the ennealimmal–enneadecal equivalence continuum (171edo, a 5-limit record edo)
  • the tarot equivalence continuum (1848edo)

• 2.3.7-subgroup rank-2 continua include:
  • the Archytas–diatonic equivalence continuum (5edo, a 3-limit and 2.3.7-subgroup record edo)
  • the Archytas–chromatic equivalence continuum (7edo, a 3-limit record edo)

• 2.5.7-subgroup rank-2 continua include:
  • the jubilismic–augmented equivalence continuum (6edo, a 2.5.7-subgroup record edo)
  • the augmented–cloudy equivalence continuum (15edo, a 2.5.7-subgroup record edo)
  • the rainy–didacus equivalence continuum (31edo, a 2.5.7-subgroup record edo)

• 3.5.7-subgroup rank-2 continua include:
  • the sensamagic–gariboh equivalence continuum (13edt, a 3.5.7-subgroup record edo)

• 7-limit rank-3 continua include:
  • the syntonic–Archytas equivalence continuum (dominant)
  • the Marvel–syntonic equivalence continuum (septimal meantone)
  • the breedsmic–syntonic equivalence continuum (squares)

• 2.3.5.11-subgroup rank-3 continua include:
  • the syntonic–rastmic equivalence continuum (mohaha)