Regular temperament: Difference between revisions

A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that consists of all intervals reachable by arbitrarily [[stacking]] a set of intervals and their inverses above a defined root note. As such, regular temperaments theoretically have an infinite number of intervals, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of intervals in between ''any two other intervals''.  

In addition, a regular temperament is by definition an approximation of some system of pure or target intervals, very often a [[just intonation subgroup]], in a perfectly consistent way: each abstract interval is interpreted as a tempered, or detuned, version of a set of target intervals, and the product of two tempered intervals must always be the tempered version of the product of the JI intervals. For example, if the ratios [[3/2]] and [[5/4]] are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].