Suppose we have two vals u and v which together with two intervals e

and f define a tempered notation. We want to take e to be an interval

of equivalence, and we want conditions on the vals; the frst is that

the matrix

[u, v] = [u(e) v(e)]

[u(f) v(f)]

be unimodular; that is that it have determinant +-1, so that

[u,v]^(-1) is an integral matrix. The second is that [u, v] is to

have positive values, the third that u(e) > v(e), the fourth that

gcd(u(e), v(e)) = 1, and finally that gcd(u(e), u(f)) = 1. We want to

construct a tempered block of u(e) notes in the temperament defined

by this notation, hence we want a complete set of residues mod u(e)

from the first coordinates of our block, which is to be sorted

according to the second coordinate, and all of it to satisfy the

conditions for a semiblock.

We first shift coordinates to the notation [u, w], where

w = u(e) v - v(e) u. Since u(e) and v(e) are relatively prime, we can

without loss of generality take a generator to be of the form [n 1],

where n is relatively prime to u(e). However, not all such choices

give us a note, since the determinant of [u, w] =

u(e) (u(e)v(f) - v(e) u(f)) = +- u(e) because of the unimodularity

condition. The sublattice of actual notes is generated by

(e, f) = [u, v]^(-1) (in the u, v notation), transforming this to the

[u, w] notation by multiplying by [u, w] gives us [u, v]^(-1) [u, w],

which is

s [1 -v(e)]

[0 u(e)],

where s = det([u,v]) = +-1. This tells us that notes are of the form

[n, m u(e) - n v(e)], so that modulo u(e) they are [n, - n v(e)].

Choosing a generator of the form [n 1] therefore means solving the

congruence n = -1/v(e) (mod u(e)); since v(e) is prime to u(e), we

may solve this congruence for a unique positive n less than u(e),

giving us a generator. If we transform this generator back to the

[u, v] notation by [a, b] = [n 1] [u, w]^(-1), we get

[ n 1] [1/u(e) 0] = [p -s],

[-s u(f)/u(e) -s]

where the value of p is an integer, and s of course is +-1. Hence

[a, b] corresponds to an interval r = e^p f^(-s), so that modulo our

interval of equivalence e it is f or 1/f. The canonical example would

be where e is 2, f is 3, and the generator we get is, modulo octaves,

a fifth or a fourth.

This argument is too convoluted and I should try to improve it, but

the theorem is more widely applicable than it at first may seem.