Regular temperament: Difference between revisions

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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|31et]]

* {{W|Nicola Vicentino}} (1511–1576): [[adaptive JI]], [[31edo|31et]]

* Leonhard Euler (1707–1783): ~~tonespace (~~[[5-limit]])

* {{W|Leonhard Euler}} (1707–1783): [[5-limit]] tonespace

* Hermann von Helmholtz (1821–1894): psychoacoustics

* {{W|Hermann von Helmholtz}} (1821–1894): psychoacoustics

* ~~RHM~~ Bosanquet (1841–1913): regular mapping, generalized keyboard

* {{W|R. H. M. Bosanquet}} (1841–1913): regular mapping, generalized keyboard

* Shohe Tanaka (1862–1945): 5-limit tonespace (triangular projection)

* {{W|Shohe Tanaka}} (1862–1945): 5-limit tonespace (triangular projection)

* [[Adriaan Fokker]] (1887–1972): [[Fokker ~~blocks~~|periodicity blocks]]

* [[Adriaan Fokker]] (1887–1972): [[Fokker block|periodicity blocks]]

* [[Harry Partch]] (1901–1974): [[JI|extended JI]]

* [[Erv Wilson]] (1928–2016): extended tonespace (and projections), [[mos]], scale tree

* [[Easley Blackwood]] (1933–2023): ~~blackwood~~[10], syntonic comma vanishing relation as equation

* [[Easley Blackwood]] (1933–2023): Blackwood[10], syntonic comma vanishing relation as equation

* [[George Secor]] (1943–2020): miracle temperament

A significant amount of this theory's early development occurred online via the {{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 {{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.

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The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s 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 ~~two~~ 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.

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.

Each temperament has two names: a traditional name and a [[color notation|color name]]. The traditional names are diverse in [[temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|Wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}}. See also [[Color notation/Temperament names]].

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== External links ==

* [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"]

* [https://youtu.be/ZoAuVgndmbU John Moriarty ~~–~~ Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture

* [https://youtu.be/ZoAuVgndmbU John Moriarty – Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture

[[Category:Regular temperament theory| ]]

@@ Line 17: / Line 17: @@
 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|31et]]
+* {{W|Nicola Vicentino}} (1511–1576): [[adaptive JI]], [[31edo|31et]]
-* Leonhard Euler (1707–1783): tonespace ([[5-limit]])
+* {{W|Leonhard Euler}} (1707–1783): [[5-limit]] tonespace
-* Hermann von Helmholtz (1821–1894): psychoacoustics
+* {{W|Hermann von Helmholtz}} (1821–1894): psychoacoustics
-* RHM Bosanquet (1841–1913): regular mapping, generalized keyboard
+* {{W|R. H. M.  Bosanquet}} (1841–1913): regular mapping, generalized keyboard
-* Shohe Tanaka (1862–1945): 5-limit tonespace (triangular projection)
+* {{W|Shohe Tanaka}} (1862–1945): 5-limit tonespace (triangular projection)
-* [[Adriaan Fokker]] (1887–1972): [[Fokker blocks|periodicity blocks]]
+* [[Adriaan Fokker]] (1887–1972): [[Fokker block|periodicity blocks]]
 * [[Harry Partch]] (1901–1974): [[JI|extended JI]]
 * [[Erv Wilson]] (1928–2016): extended tonespace (and projections), [[mos]], scale tree
-* [[Easley Blackwood]] (1933–2023): blackwood[10], syntonic comma vanishing relation as equation
+* [[Easley Blackwood]] (1933–2023): Blackwood[10], syntonic comma vanishing relation as equation
 * [[George Secor]] (1943–2020): miracle temperament
-A significant amount of this theory's early development occurred online via the {{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 {{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.
@@ Line 46: / Line 46: @@
 The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s 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 two 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.
+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.
 Each temperament has two names: a traditional name and a [[color notation|color name]]. The traditional names are diverse in [[temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|Wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}}. See also [[Color notation/Temperament names]].
@@ Line 86: / Line 88: @@
 == External links ==
 * [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"]
-* [https://youtu.be/ZoAuVgndmbU John Moriarty &ndash; Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture
+* [https://youtu.be/ZoAuVgndmbU John Moriarty – Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture
 [[Category:Regular temperament theory| ]] <!-- Main article -->