Regular temperament: Difference between revisions
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{{Wikipedia}} | {{Wikipedia}} | ||
A '''regular temperament''' 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. 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. | ||
One particularly simple kind of regular temperaments is the equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a | One particularly simple kind of regular temperaments is the equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a {{w|Triviality (mathematics)|trivial}} temperament where no tempering is happening: no [[comma]]s are tempered out, but all are preserved as small pitch differences. 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]]''. | ||
== History == | == History == | ||
The roots of '''regular temperament theory''' ('''RTT''') can be traced back for centuries. The practice far predates the theory, and in particular [[meantone]] temperament has been known since the 15th century. Many early pioneers set the stage for the general theory to come: | The roots of '''regular temperament theory''' ('''RTT''') can be traced back for centuries. The practice far predates the theory, and in particular [[meantone]] temperament has been known since the 15th century. Many early pioneers set the stage for the general theory to come: | ||
* Nicola Vicentino ( | * {{W|Nicola Vicentino}} (1511–1576): [[adaptive JI]], [[31edo|31et]] | ||
* Leonhard Euler ( | * {{W|Leonhard Euler}} (1707–1783): [[5-limit]] tonespace | ||
* Hermann von Helmholtz ( | * {{W|Hermann von Helmholtz}} (1821–1894): psychoacoustics | ||
* | * {{W|R. H. M. Bosanquet}} (1841–1913): regular mapping, generalized keyboard | ||
* Shohe Tanaka ( | * {{W|Shohe Tanaka}} (1862–1945): 5-limit tonespace (triangular projection) | ||
* [[Adriaan Fokker]] ( | * [[Adriaan Fokker]] (1887–1972): [[Fokker block|periodicity blocks]] | ||
* [[Harry Partch]] ( | * [[Harry Partch]] (1901–1974): [[JI|extended JI]] | ||
* [[Erv Wilson]] ( | * [[Erv Wilson]] (1928–2016): extended tonespace (and projections), [[mos]], scale tree | ||
* [[Easley Blackwood]] ( | * [[Easley Blackwood]] (1933–2023): Blackwood[10], syntonic comma vanishing relation as equation | ||
* [[George Secor]] ( | * [[George Secor]] (1943–2020): miracle temperament | ||
A significant amount of this theory's early development occurred online via the | A significant amount of this theory's early development occurred online via the {{w|Yahoo! Groups}} service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's. | ||
In 2001 [[Gene Ward Smith]] joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as [[Mike Battaglia]] also did much of the work to document RTT on this wiki. | In 2001 [[Gene Ward Smith]] joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as [[Mike Battaglia]] also did much of the work to document RTT on this wiki. | ||
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Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s ([[mapping]]s), [[monzo]]s and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand. | Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s ([[mapping]]s), [[monzo]]s and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand. | ||
The [[rank]] of a temperament is its dimension. It equals the number of [[ | 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 | 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 [[CTE]] ("Constained Tenney–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 [[User:Sintel|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 CTE 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, from which the comma(s) can be deduced. The most common systematic temperament naming system on the wiki is [[color notation]]: {{nowrap|wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}} (see also [[Color notation/Temperament names]]). | |||
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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More comprehensive lists: | More comprehensive lists: | ||
* [[Map of rank-2 temperaments]]: a visual map of temperaments based on the size of their period and generator | * [[Map of rank-2 temperaments]]: a visual map of temperaments based on the size of their period and generator | ||
* [[Bird's eye view of temperaments by accuracy]]: a compilation of rank 2 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 | |||
* [[Survey of efficient temperaments by subgroup]]: a visual map of temperaments based on notes needed per equave and JI subgroup | * [[Survey of efficient temperaments by subgroup]]: a visual map of temperaments based on notes needed per equave and JI subgroup | ||
* [[Tour of regular temperaments]]: a huge gallery of the dozens of families of temperaments that have been described; not for the faint of heart | * [[Tour of regular temperaments]]: a huge gallery of the dozens of families of temperaments that have been described; not for the faint of heart | ||
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== External links == | == External links == | ||
* [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"] | * [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"] | ||
* [https://youtu.be/ZoAuVgndmbU John Moriarty | * [https://youtu.be/ZoAuVgndmbU John Moriarty – Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture | ||
[[Category:Regular temperament theory| ]] <!-- Main article --> | [[Category:Regular temperament theory| ]] <!-- Main article --> | ||