# 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.

When only two generators are needed, with one of them the octave, then it's called a "linear temperament.The best-known example of a linear temperaments is meantone temperament, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave. Other linear temperaments include the schismatic temperament of Hermann von Helmholtz and miracle temperament.

## Mathematical description

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 2^{19/12} to favour 2, or in quarter-comma meantone where 3 is tempered to 2·5^{1/4} to favor 2 and 5.

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. 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. Hence, meantone is a rank-2 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 one must determine 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, a harpsichord tuner it might think of quarter-comma meantone tuning 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 map's kernel (otherwise known as "nullspace") consists of *p*-limit intervals called commas, which are a property useful in describing temperaments.

## External links

- "Regular+Temperaments",
*Xenharmonic.Wikispaces.com*. - A. Milne, W. A. Sethares, and J. Plamondon,
*Isomorphic Controllers and Dynamic Tuning— Invariant Fingering Over a Tuning Continuum*, Computer Music Journal, Winter 2007 - Holmes, Rich,
*Microtonal scales: Rank-2 2-step (MOS) scales* - Smith, Gene Ward,
*Regular Temperaments* - Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice