Editing Regular temperament

[[Image:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg|right|250px|thumb|Some example linear temperaments with the generator close to a fifth. "linear temperaments" are regular temperaments of rank two, with one generator as shown, and the other generator the octave. (Milne 2007).]]

A '''regular temperament''' is any [[Musical temperament|tempered]] 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. For instance, in [[12-TET]], the system of music most commonly used in the Western world, the generator is a tempered fifth (700 cents), which is the basis behind the [[circle of fifths]].

When only two generators are needed, with one of them the octave, this is sometimes called a "linear temperament". The best-known example of linear temperaments are [[meantone temperament|meantone temperaments]], where the generating intervals are usually given in terms of a slightly flattened fifth and the octave. Other linear temperaments include the [[schismatic temperament]] of [[Hermann von Helmholtz]] and [[miracle temperament]].

== 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.

== External links ==
* {{Xenharmonic wiki|Regular_temperament|Regular Temperament}} <!--This is an open wiki, which seems to me to be too lacking in citations to good sources to qualify for exemption from Wikipia's proscription against linking to such sites in [[WP:ELNO]] -->
* 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://lumma.org/tuning/gws/regular.html Smith, Gene Ward, ''Regular Temperaments'']
* Barbieri, Patrizio. [http://www.patriziobarbieri.it/1.htm Enharmonic instruments and music, 1470-1900]. (2008) Latina, Il Levante Libreria Editrice

{{Musical tuning}}

[[Category:Musical temperaments]]