Editing Modulatory space (section)

== Toroidal modulatory spaces ==

If we divide the octave into n parts, where n = rs is the product of two relatively prime integers r and s, we may represent every element of the tone space as the product of a certain number of "r" generators times a certain number of "s" generators; in other words, as the [[direct sum of groups|direct sum]] of two cyclic groups of orders r and s. We may now define a graph with n vertices on which the group acts, by adding an edge between two pitch classes whenever they differ by either an
"r" generator or an "s" generator (the so-called [[Cayley graph]] of <math>\mathbb{Z}_{12}</math> with generators ''r'' and ''s''). The result is a graph of [[Glossary of graph theory#Genus|genus]] one, which is to say, a graph with a donut or [[torus]] shape. Such a graph is called a [[toroidal graph]].

An example is [[equal temperament]]; twelve is the product of 3 and 4, and we may represent any pitch class as a combination of thirds of an octave, or major thirds, and fourths of an octave, or minor thirds, and then draw a toroidal graph by drawing an edge whenever two pitch classes differ by a major or minor third.

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