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") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). 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 generators. In addition, for each temperament there is a [[Optimal ~~GPV~~ sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better accuracy.

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") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). 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 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.

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") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). 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 generators. In addition, for each temperament there is a [[Optimal GPV sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better accuracy.
+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") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). 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 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.
 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]].