- Regular temperament
**Regular temperament**is any tempered system ofmusical 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 ismeantone 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 toneequal 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 defines a

free abelian group . The number of independent generators is therank of an abelian group . The rank one tuning systems areequal temperament s, 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 ofErv Wilson it is a "linear temperament".The best-known examples of linear temperaments are meantone and

pythagorean tuning , but others include theschismatic temperament ofHermann von Helmholtz andmiracle 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-threemarvel 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.**External links***A. Milne, W. A. Sethares, and J. Plamondon, [

*http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 "Isomorphic Controllers and Dynamic Tuning— Invariant Fingering Over a Tuning Continuum"*] , Computer Music Journal, Winter 2007

* [*http://web.syr.edu/~rsholmes/music/xen/scale_mos.html Holmes, Rich, "Microtonal scales: Rank-2 2-step (MOS) scales"*]

* [*http://66.98.148.43/~xenharmo/regular.html Smith, Gene Ward, "Regular Temperaments"*]

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