Modulatory space

In equal temperament, twelve successive fifths equate to seven octaves exactly, and hence in terms of pitch classes closes back to itself, forming a circle.

By dividing the octave into n equal parts, and choosing an integer m

We may generalize immediately to any number of relatively prime factors, producing graphs can be drawn in a regular manner on an n-torus.

The pitch classes of any linear temperament can be represented as lying along an infinite chain of generators; in meantone for instance this would be -F-C-G-D-A- etc.

We may represent the modulatory space of such a temperament as n chains of generators in a circle, forming a cylinder.

The cylindrical appearance of this sort of modulatory space becomes more apparent when the period is a smaller fraction of an octave; for example, ennealimmal temperament has a modulatory space consisting of nine chains of minor thirds in a circle (where the thirds may be only 0.02 to 0.03 cents sharp.)

Five limit just intonation has a modulatory space based on the fact that its pitch classes can be represented by 3a 5b, where a and b are integers.

In many ways a more enlightening picture emerges if we represent it in terms of a hexagonal lattice instead; this is the Tonnetz of Hugo Riemann, discovered independently around the same time by Shohé Tanaka.