Regular temperament
A regular temperament is a mathematical object that maps an infinite set of intervals in to a smaller, though still infinite, set of tempered intervals. Typically, the source set is assumed to be some p-limit just intonation — or any subgroup thereof — AKA rational numbers, and tempering is done by deliberately mistuning some of the ratios such that a comma or set of commas vanishes by becoming a unison (it is "tempered out" in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships.
In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are "tempered together".
For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
As another example, if meantone temperament is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "syntonic comma", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament "regular".
Introductions to regular temperament theory
Why would I want to use a regular temperament?
Regular temperaments are of most use to musicians who want their music to sound as much as possible like low-overtone just intonation, but without the difficulties normally associated with low-overtone JI, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.
What do I need to know to understand all the numbers on the pages for individual regular temperaments?
Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are vals (aka mappings) and tempering out commas, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.
The rank of a temperament equals the number of primes in the subgroup minus the number of linearly independent (i.e. non-redundant) commas that are tempered out.
Another recent contribution to the field of temperament is the concept of optimization, which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The two most frequently used forms of optimization are POTE ("Pure-Octave Tenney-Euclidean") and TOP ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE generators. In addition, for each temperament there is a list of EDOs showing possible EDO tunings in the order of better accuracy.
Yet another recent development is the concept of a pergen, appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a 5th or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every strong extension of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but don't uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
Each temperament has two names: a traditional name and a color name. The traditional names are arbitrary, but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also Color Notation/Temperament Names.