Definitions

# Regular temperament

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. The classic example of a regular temperament is meantone temperament, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave.

If the generators are all of the prime numbers up to a given prime p, we have what is called p-limit just intonation. Sometimes some irrational number close to one of these primes is substituted (an example of tempering) to favour other primes, as in twelve tone equal temperament where 3 is tempered to 219/12 to favour 2, or in quarter-comma meantone where 3 is tempered to 2·51/4 to favor 2 and 5.

In mathematical terminology, the products of these generators defines a free abelian group. The number of independent generators is the rank of an abelian group. The rank one tuning systems are equal temperaments, all of which can be spanned with only a single generator. A rank two temperament has two generators; if one of these generators is the octave, then in the terminology of Erv Wilson it is a linear temperament.

The best-known examples of linear temperaments are meantone and pythagorean tuning, but others include the schismatic temperament of Hermann von Helmholtz and miracle temperament. There exist regular temperaments of rank greater than two; just intonations with limits of five or greater provide one kind of example, but also the rank-three marvel temperament.

In studying regular temperaments, it can be useful to regard the temperament as having a map from p-limit just intonation (for some prime p) to the set of tempered intervals. To properly classify a temperament's dimensionality it must be determined how many of the given generators are independent, because its description may contain redundancies. Another way of considering this problem is that the rank of a temperament should be the rank of its image under this map.

For instance, for a harpsichord tuner it might be normal to think of meantone temperament as having three generators: the octave, the just major third (5/4) and the quarter-comma tempered fifth, but because four consecutive tempered fifths produces a just major third, the major third is redundant, reducing it to a rank-two temperament.

Other methods of linear and multilinear algebra can be applied to the map. For instance, a kernel of it would consist of p-limit intervals called commas, which are a property useful in describing temperaments.