Regular temperament: Difference between revisions
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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''' 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 | 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. For example, 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; if so, the temperament is irregular. | ||
One particularly simple kind of regular temperaments is the equal temperaments, which represent all intervals by multiples of a single smallest step. | 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 [[Wikipedia: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]]''. | ||
At the other extreme, JI itself can be considered a [[Wikipedia: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 (1511–1576): [[adaptive JI]], [[31edo| | * Nicola Vicentino (1511–1576): [[adaptive JI]], [[31edo|31et]] | ||
* Leonhard Euler (1707-1783): tonespace ([[5-limit]]) | * Leonhard Euler (1707-1783): tonespace ([[5-limit]]) | ||
* Hermann von Helmholtz (1821-1894): psychoacoustics | * Hermann von Helmholtz (1821-1894): psychoacoustics | ||
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* [[Adriaan Fokker]] (1887-1972): [[Fokker blocks|periodicity blocks]] | * [[Adriaan Fokker]] (1887-1972): [[Fokker blocks|periodicity blocks]] | ||
* [[Harry Partch]] (1901-1974): [[JI|extended JI]] | * [[Harry Partch]] (1901-1974): [[JI|extended JI]] | ||
* [[Erv Wilson]] (1928-2016): extended tonespace (and projections), [[ | * [[Erv Wilson]] (1928-2016): extended tonespace (and projections), [[mos]], scale tree | ||
* [[Easley Blackwood]] (1933-): blackwood[10], syntonic comma vanishing relation as equation | * [[Easley Blackwood]] (1933-): blackwood[10], syntonic comma vanishing relation as equation | ||
* [[George Secor]] (1943-2020): miracle temperament | * [[George Secor]] (1943-2020): miracle temperament | ||
A significant amount of this theory's early development occurred online via the [[Wikipedia: Yahoo! Groups]] service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and Paul Hahn in the late 1990's. | A significant amount of this theory's early development occurred online via the [[Wikipedia: Yahoo! Groups|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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== See also == | == See also == | ||
; Introductory materials | |||
* [[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]] | ||
* [[Keenan's explanation of vals|Keenan Pepper's explanation of vals]] | * [[Keenan's explanation of vals|Keenan Pepper's explanation of vals]] | ||
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* [[Douglas Blumeyer's RTT How-To]] | * [[Douglas Blumeyer's RTT How-To]] | ||
; Lists | |||
* [[Tour of regular temperaments]] | * [[Tour of regular temperaments]] | ||
; Related topics | |||
* [[Monzo]] | * [[Monzo]] | ||
* [[Val]] | * [[Val]] | ||