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.
Read more about this topic: Regular Temperament
Famous quotes containing the words mathematical and/or description:
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“The Sage of Toronto ... spent several decades marveling at the numerous freedoms created by a “global village” instantly and effortlessly accessible to all. Villages, unlike towns, have always been ruled by conformism, isolation, petty surveillance, boredom and repetitive malicious gossip about the same families. Which is a precise enough description of the global spectacle’s present vulgarity.”
—Guy Debord (b. 1931)