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]] | 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 which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals above a defined root note. For example, [[meantone]] temperament is generated by the [[2/1|octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a {{W|major second}} is equal to two perfect fifths minus an octave, and a {{W|major third}} is four perfect fifths minus two octaves. Regular temperaments theoretically have an infinite number of notes, 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 notes in between ''any two other notes''. | ||
In addition to unlimited modulation, regular temperaments by definition | In addition to unlimited modulation, regular temperaments are by definition thought of as being approximations of some 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). For example, the octave in meantone represents the just ratio [[2/1]], the perfect fifth [[3/2]], and the major third [[5/4]]. Certain intervals are tempered to the [[1/1|unison]], or [[tempering out|tempered out]]; in a regular temperament, these intervals are known as [[comma]]s. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that {{nowrap|(3/2)<sup>4</sup> / (2/1)<sup>2</sup> / (5/4) {{=}} [[81/80]]}} is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example [[9/8]] and [[10/9]] in meantone, are mapped to the same tempered interval in the temperament, in this case a major second. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: 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 | 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. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and only a single pitch is available. 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'']]. | ||
== History == | == History == | ||
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More comprehensive lists: | More comprehensive lists: | ||
* [[Bird's eye view of temperaments by accuracy]]: temperaments | * [[Bird's eye view of temperaments by accuracy]] (article): temperaments the Xen Wiki contributors find most useful for approximating JI - with edo tunings and note counts for the harmonies they target, and explanations of their structure | ||
* [[Survey of efficient temperaments by subgroup]]: temperaments | * [[Survey of efficient temperaments by subgroup]] (table): good general-purpose temperaments, sorted by size (notes per equave) and by JI subgroup | ||
* [[Map of rank-2 temperaments]]: temperaments by the size of their period and generator | * [[Map of rank-2 temperaments]] (table): temperaments (some general, some niche) sorted by the size of their period and generator | ||
* [[Temperaments for MOS shapes]]: | * [[Temperaments for MOS shapes]] (table): temperaments (some general, some niche) sorted by the scale shape they generate | ||
* [[Tour of regular temperaments]]: huge gallery of the dozens of families of temperaments that have been described | * [[Tour of regular temperaments]] (article): huge gallery of the dozens of families of temperaments that have been described; ''very technical - not for the faint of heart'' | ||
=== Other writings on temperaments === | === Other writings on temperaments === | ||
* [[Mike's lectures on regular temperament theory|Mike Battaglia's lectures on RTT]] | * [[Mike's lectures on regular temperament theory|Mike Battaglia's lectures on RTT]] | ||
== Notes == | |||
<references group="note"/> | |||
== External links == | == External links == | ||