Mathematical theory of regular temperaments: Difference between revisions

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A '''regular temperament''' is a homomorphism that maps an abelian group of target/~~pure~~ intervals to another abelian group of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]), ~~and~~ tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.

A '''regular temperament''' is a homomorphism that maps an abelian group of target/just intervals to another abelian group of [[tempering out|tempered]] intervals. In other words, it is a function from a group of just intervals to another simpler group of intervals that preserves the operation of [[stacking]]. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]). Musically, tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.

In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same [[mapped interval|tempered interval]] — in this case we say that the two JI intervals are ''tempered together''.

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 {{Expert|Regular temperament}}
-A '''regular temperament''' is a homomorphism that maps an abelian group of target/pure intervals to another abelian group of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]), and tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.
+A '''regular temperament''' is a homomorphism that maps an abelian group of target/just intervals to another abelian group of [[tempering out|tempered]] intervals. In other words, it is a function from a group of just intervals to another simpler group of intervals that preserves the operation of [[stacking]]. Typically, the source set is assumed to be a multiplicative subgroup of the rational numbers (aka [[just intonation]]). Musically, tempering is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison (it is ''tempered out'' in the temperament). The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the dimensionality of JI, thereby simplifying the pitch relationships.
 In mathematical terms, it is a function whose domain is a target tuning we wish to approximate, and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same [[mapped interval|tempered interval]] — in this case we say that the two JI intervals are ''tempered together''.