Regular temperament
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Regular temperament is a 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 flat 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. However, normally we aspire to adjust the tuning of one or more of these primes, to produce an actual "tempering" of the justly tuned ratios, as in meantone.
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 for historical reasons one less than this number is called the dimension of the temperament. A "0-dimensional" tuning system with a single generator is normally regarded as having approximated an octave after a certain number of steps, and therefore of constituting an equal temperament. A linear or one-dimensional temperament has two generators, one of which is usually taken to be an octave or some equal subdivision of an octave. By far the best-known example of a linear temperament is meantone, but others include the schismic temperament of Hermann von Helmholtz and miracle temperament. An example of a temperament which is neither equal nor linear is the tempering of the septimal kleisma of 225/224 to a unison with no other tempering, something which is often advantageous if we wish to introduce 7-limit harmony without detuning 5-limit just intonation very greatly.
In studying regular temperaments, it is generally advantageous to regard the temperament as having both a map from p-limit just intonation for some prime p, and a tuning map to particular values for the generators. From that point of view, it is no longer required that the generators of the temperament be independent, which is demoted to a mere matter of tuning; we are only interested in the rank of the group which is the image under the first map. Hence, for instance, an equal division into 31 parts is still to be regarded as a means of tuning a linear temperament. The first map can now be studied by the methods of linear and multilinear algebra. The kernel of the map, consisting of p-limit intervals called the commas of the temperament, can be used to gain insight into the properties of the temperament.