Regular temperament

A regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios.

In mathematical terminology, the products of these generators define a free abelian group.

The number of independent generators is the rank of an abelian group.

To properly classify a temperament's dimensionality one must determine how many of the given generators are independent, because its description may contain redundancies.

For instance, a map's kernel (otherwise known as "nullspace") consists of p-limit intervals called commas, which are a property useful in describing temperaments.

Some example linear temperaments with the generator close to a fifth. "linear temperaments" are regular temperaments of rank two, with one generator as shown, and the other generator the octave. (Milne 2007).