Regular temperament: Difference between revisions
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{{Wikipedia}} | {{Wikipedia}} | ||
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. 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''. | 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 | 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'']]. | |||
One particularly simple kind of regular temperaments is | |||
== History == | == History == | ||
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The [[rank]] of a temperament is its dimension. It equals the number of generators in the [[Just intonation subgroup|subgroup]] being used minus the number of independent commas that are tempered out. | The [[rank]] of a temperament is its dimension. It equals the number of generators in the [[Just intonation subgroup|subgroup]] being used minus the number of independent 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 most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney–Euclidean"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[ | 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 most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney–Euclidean"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CWE]] ("Constained Weil–Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. 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 and CTE generators. In addition, for each temperament there is a [[optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. | ||
The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[ | The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[Sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CWE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups. | ||
Usually, temperaments have names coming from a wide array of [[temperament names|sources]], but they can also have systematic and rigorous names, | Usually, temperaments have names coming from a wide array of [[temperament names|sources]], but they can also have systematic and rigorous names, from which the comma(s) can be deduced. The most common systematic temperament naming system on the wiki is [[Kite's color notation]]: {{nowrap|wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}} (see also [[Kite's color notation/Temperament names]]). | ||
Yet another recent development is the concept of a [[pergen]], appearing in our [[ | Yet another recent development is the concept of a [[pergen]], appearing in our [[Tour of regular temperaments]] as (P8, P5/2) or somesuch, which classifies temperaments by their period and the generator(s), giving ideas of how to notate these temperaments. For rank-2 temperaments we have developed a similar classification system called [[ploidacot]]. | ||
== Further reading == | == Further reading == | ||
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* ''[[A Middle Path]]'': this is [[Paul Erlich]]'s guide to RTT (regular temperament theory) | * ''[[A Middle Path]]'': this is [[Paul Erlich]]'s guide to RTT (regular temperament theory) | ||
* [[Dave Keenan & Douglas Blumeyer's guide to RTT]] | * [[Dave Keenan & Douglas Blumeyer's guide to RTT]] | ||
* [[ | * [[Keenan Pepper's explanation of vals]] | ||
=== Key regular temperament concepts === | === Key regular temperament concepts === | ||
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Temperaments that approximate important harmonies relatively well with a small number of notes: | Temperaments that approximate important harmonies relatively well with a small number of notes: | ||
* [[Low harmonic entropy linear temperaments]] | * [[Low harmonic entropy linear temperaments]] | ||
* [[Middle Path table of | * [[Middle Path table of 5-limit rank-2 temperaments]] | ||
* [[Middle Path table of | * [[Middle Path table of 7-limit rank-2 temperaments]] | ||
* [[Middle Path table of | * [[Middle Path table of 11-limit rank-2 temperaments]] | ||
More comprehensive lists: | More comprehensive lists: | ||
* [[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 | |||
* [[Bird's eye view of temperaments by accuracy]]: | * [[Survey of efficient temperaments by subgroup]] (table): good general-purpose temperaments, sorted by size (notes per equave) and by JI subgroup | ||
* [[Survey of efficient temperaments by subgroup]]: | * [[Map of rank-2 temperaments]] (table): temperaments (some general, some niche) sorted by the size of their period and generator | ||
* [[Tour of regular temperaments]]: | * [[Temperaments for MOS shapes]] (table): temperaments (some general, some niche) sorted by the scale shape they generate | ||
* [[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 | * [[Mike's lectures on regular temperament theory|Mike Battaglia's lectures on RTT]] | ||
== External links == | == External links == | ||