Editing Regular temperament (section)

== Mathematical description ==

If the generators are all the [[prime number]]s up to a given prime ''p'', we have what is called ''p''-[[limit (music)|limit]] [[just intonation]]. Sometimes some [[irrational number]] close to one of these primes is substituted (an example of [[Musical temperament|tempering]]) to favour other primes, as in twelve tone [[equal temperament]] where 3 is tempered to 2<sup>{{frac|19|12}}</sup> to favour 2, or in [[quarter-comma meantone]] where 3 is tempered to 2{{radic|5|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 temperament]]s, 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 <math>2</math> and <math> 5^{1/4}</math>.

In studying regular temperaments, it can be useful to regard the temperament as having a [[map (mathematics)|map]] from ''p''-limit just intonation (for some prime ''p'') to the [[Set (mathematics)|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 (mathematics)|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 algebra|linear]] and [[multilinear algebra]] can be applied to the map. For instance, a [[kernel (linear algebra)|map's kernel]] (otherwise known as "nullspace") consists of ''p''-limit intervals called [[comma (music)|commas]], which are a property useful in describing temperaments.