Regular temperament

A regular temperament is a temperament (an approximation of just intonation) that is consistent. 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. 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 well temperaments) are considered irregular.

Equal temperaments are the regular temperament interpretations of equal tunings, meanwhile any just intonation subgroup itself can also be considered a regular temperament of itself (such as Pythagorean tuning).

In order to simplify JI, 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 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.

Temperament joining

Any rank-2 temperament may be represented by two equal temperaments, and can be seen as "what those equal temperaments have in common". For example, porcupine is 15 and 22. This means 15-ET and 22-ET (and their sum, 37-ET) have the characteristics of porcupine in common. 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 is supported, etc. And moreso, 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, etc.

See also: List of regular temperaments