Regular temperament: Difference between revisions

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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.

The [[rank]] of a temperament is its dimension. It equals the number of ~~primes~~ in the subgroup minus the number of independent commas that are tempered out.

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.

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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.
-The [[rank]] of a temperament is its dimension. It equals the number of primes in the subgroup minus the number of independent commas that are tempered out.
+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.