Regular temperament: Difference between revisions
Like mos scale, this is a broad topic, so an example section is good to have |
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{{Wikipedia}} | {{Wikipedia}} | ||
A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that | 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, and that approximates some system of pure or target intervals, very often a [[just intonation subgroup]], in a perfectly consistent way. | ||
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''. 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. | |||
As a result, a regular temperament 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. | |||
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'']]. | 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'']]. | ||
== Examples == | == Examples == | ||
[[Meantone]] is the most historically significant regular temperament. It is generated by the [[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 [[major second]] is equal to two perfect fifths minus an octave, and a [[major third]] is four perfect fifths minus two octaves. | [[Meantone]] is the most historically significant regular temperament. It is generated by the [[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 [[major second]] is equal to two perfect fifths minus an octave, and a [[major third]] is four perfect fifths minus two octaves. | ||
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 [[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. | 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 [[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. | ||