Regular Temperament

If the generators are all of the prime numbers up to a given prime ''p'', we have what is called ''p''- Limit Just Intonation . Often some irrational number close to one of these primes is substituted (this is called "tempering") to favour another prime, as in twelve tone Equal Temperament 3 is tempered to favour 2, or in quarter-comma meantone where 3 is tempered to favour 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 , but one less than this number is sometimes called the dimension of the temperament. "0-dimensional" tuning systems are Equal Temperament s, all of which can spanned with only a single generator. A linear (or one-dimensional temperament) has two generators, one of which is usually assumed to be the octave; Erv Wilson reserved the word ''linear temperament'' for only those temperaments in which one of the generators is the octave, but any temperament with two generators can at least be called a "rank two" temperament.

The best-known examples of a linear temperament are meantone and Pythagorean Tuning , but others include the Schismatic Temperament of Hermann Von Helmholtz and Miracle Temperament . There exist temperaments which are neither equal nor linear. Just intonations with limits of 5 or greater, for 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 independant, 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 propert useful in describing temperaments.

EXTERNAL LINKS

Holmes, Rich, ''Microtonal scales: Rank-2 2-step (MOS) scales''
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Smith, Gene Ward, ''Regular Temperaments''
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Regular Temperament