Regular temperament: Difference between revisions

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A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that looks the same no matter which pitch you start from (or consider the [[tonic]]). In other words, unlimited free modulation is possible: any [[interval]] can be stacked as many times as you like. A regular temperament is [[generate]]d by a set of [[generator|generating intervals]] (usually one of which is considered the [[period]]), and any note that can be reached by stacking any number of these generating intervals is considered to be part of the regular temperament. Regular temperaments generally have an infinite number of notes; and other than [[equal temperament]]s, every regular temperament actually has an infinite number of notes in between ''any two other notes''.

In addition to unlimited modulation, regular temperaments by definition are thought of as being approximations of some more complicated system of pure or target intervals, very often a [[just intonation]] (JI) [[subgroup]]. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: the sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. Multiple pure intervals may be represented by the same tempered interval (so they are tempered together), but a single pure interval must never be represented by different tempered intervals. Certain intervals are tempered to the [[1/1|unison]]; in a regular temperament, these intervals are known as [[comma]]s. Any two just intervals seperated by a comma of a temperament are mapped to the same tempered interval in that temperament. For example, [[meantone]] temperament, the [[5-limit]] temperament tempering out [[81/80]], maps [[81/64]] and [[5/4]] to the same interval, as 81/64 and 5/4 differ by 81/80 in JI.

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a [[trivial temperament]] where no tempering is happening: no commas are tempered out, but all are preserved as small pitch differences. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and there is only a single pitch. In between 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'']].

* [[Erv Wilson]] (1928–2016): extended tonespace (and projections), [[mos]], scale tree

* [[Bird's eye view of temperaments by accuracy]]: temperaments that people think are most valuable for approximating JI, with edo tunings and note counts for the harmonies they target, and explanations of their structure (article)

* [[Survey of efficient temperaments by subgroup]]: temperaments based on notes needed per equave and JI subgroup (table)

* [[Map of rank-2 temperaments]]: temperaments by the size of their period and generator (table)

* [[Temperaments for MOS shapes]]: list of temperaments by the scale shape they generate (table, ''work-in-progress'')

* [[Tour of regular temperaments]]: huge gallery of the dozens of families of temperaments that have been described (article, ''very technical - not for the faint of heart'')

* [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"]

* [https://youtu.be/ZoAuVgndmbU John Moriarty – Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture