# 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). This term was first used by Flora Canou.

Examples:

- The syntonic-chromatic equivalence continuum is the 5-limit rank-2 equivalence continuum of 7edo.

## 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 **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) = **R**P^{1} (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 **R**^{2} 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 *p***u** ~ *q***v** (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 **u**_{x}, **u**_{y}, **u**_{z} for ker(T). Then our Grassmannian can be identified with **R**P^{2} (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 **R**P^{2} 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 (*v*_{1}, *v*_{2}, *v*_{3}), so that the plane associated with the rank-2 temperament has equation *v*_{1}*x* + *v*_{2}*y* + *v*_{3}*z* = 0. [We may further assume that *v*_{1}, *v*_{2}, *v*_{3} 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 *v*_{i} is always guaranteed to be nonzero, for any temperament. Assuming *v*_{1} ≠ 0, we can scale **v** by 1/*v*_{1}, then the resulting vector **v**/*v*_{1} = (1, *v*_{2}/*v*_{1}, v_{3}/*v*_{1}) = (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 *v*_{1} ≠ 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 *v*_{2} ≠ 0 and the set of all temperaments such that *v*_{3} ≠ 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 **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 = 2**u**_{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).