Regular temperament

A regular temperament is any

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 2⁴√5 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, though they don't have to be integer-based equal temperaments. The non-octave scales of Wendy Carlos, such as the Alpha scale, use one generator that does not stack up to the octave. A rank-two temperament has two generators; hence, meantone is a rank-2 temperament. For the case of quarter-comma meantone, these may be chosen as ${\displaystyle 2}$ and ${\displaystyle 5^{1/4}}$.

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 Temperament", on Xenharmonic Wiki.
• 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