Regular temperament: Difference between revisions

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The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s in the subgroup 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 ~~due to a subtle error in the definition of~~ POTE ~~where POTE simply compresses/stretches all generators~~ to ~~make octaves pure while CTE optimizes with the new implied tunings of intervals in mind~~. 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 accuracy ~~(with respect~~ to ~~absolute error, not [[consistency]])~~. The most common browser tools used for finding optimal tunings (~~which are~~ useful ~~if you wish to investigate~~ temperaments ~~you've designed yourself~~) are [http://x31eq.com/temper/ ~~x31eq~~]~~, by~~ [[~~Graham Breed~~]]~~, and~~ [https://sintel.pythonanywhere.com/ ~~sintel's temp finder~~]; the former gives ~~temp~~ 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 ~~a different~~ notation ~~instead of~~ [[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 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.

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

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 The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s in the subgroup 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 due to a subtle error in the definition of POTE where POTE simply compresses/stretches all generators to make octaves pure while CTE optimizes with the new implied tunings of intervals in mind. 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 accuracy (with respect to absolute error, not [[consistency]]).  The most common browser tools used for finding optimal tunings (which are useful if you wish to investigate temperaments you've designed yourself) are [http://x31eq.com/temper/ x31eq], by [[Graham Breed]], and [https://sintel.pythonanywhere.com/ sintel's temp finder]; the former gives temp 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 a different notation instead of [[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 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.
 Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are diverse in terms of [[Temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]].