Regular temperament

The syntonic tuning continuum (Milne 2007).

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.

The best-known example of a linear temperaments is meantone, but others 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 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 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.

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