View source for Regular temperament

A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that deforms JI intervals in a way consistent with stacking. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''. 

[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]). 

In order to simplify JI, [[Comma|commas]] are tempered out: small (or not-so-small) differences between intervals that the regular temperament represents with the unison. For example, in [[Meantone]], [[5/4]] (the classical major third) and [[Diatonic major third|81/64]] (the pythagorean major third) are equated. Their difference in just intonation is the ratio [[81/80]], so we say that Meantone '''tempers out''' 81/80. As a consequence, due to the rules of a regular temperament, 6561/6400, 531441/512000, etc (ratios obtained by stacking 81/80) are also tempered out, but knowing that Meantone tempers out 81/80 is enough to determine this information, so it is usually left unstated.

It turns out that tempering out a single comma reduces the dimensionality of an [[interval space]] by 1 (as long as that comma or any multiple of it can't be reached by stacking the other intervals you temper out); that is, if a JI subgroup can be reached by stacking any multiples of a minimum of three distinct intervals (such as 2, 3, and 5 in the [[5-limit]]), tempering out 1 comma leads to a system built out of two distinct intervals. This means that a [[rank-2 temperament]] (one with two generators, or a period and a generator) must always temper out ''p''-2 commas in a subgroup with ''p'' primes.

{{adv|Mathematically, regular temperaments are commonly treated as ''abstract'' tunings where the intervals have no concrete cent values until such are assigned to the generators. This is because abstract regular temperaments, such as "Magic", are meant to abstract features that concrete regular temperaments, such as 7-limit 19edo and 7-limit 22edo, have in common; see the "Temperament joining" section.}}

== Temperament joining ==
Any two [[equal temperament]]s have one specific rank-2 temperament that they both support, which can be seen as "what those equal temperaments have in common". The temperament shared by ''x''-ET and ''y''-ET is denoted by ''x'' & ''y''. For example, [[Porcupine]] is 15 & 22. This means [[15edo|15-ET]] and [[22edo|22-ET]] (and their sum, [[37edo|37-ET]]) have the characteristics of Porcupine in common (note 15 & 37 or 22 & 37 are other valid denotations of Porcupine). And they do: [[6/5]] is tuned sharp, [[4/3]] and [[10/9]] are tuned flat, (4/3)^2 is [[7/4]], ([[11/10]])^3 is 4/3, the soft tuning of the [[onyx]] scale (1L 6s MOS pattern) is supported, etc. Moreover, it means that you can take it the other way: Porcupine can be ''defined'' not by what commas it tempers out, but as "what 15-ET and 22-ET have in common". Similarly, a rank-3 temperament may be defined by three equal temperaments, and so on.

One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament's generator. This means that, while 12 & 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 & 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.

Note that this presents a currently rather idiosyncratic way of conceptualizing higher-rank temperaments as, analogously to equal temperaments, arbitrary in [[subgroup]].

== See also ==
* [[List of regular temperaments]]

{{Cat|regular temperaments{{!}}*}}