Mathematical theory of regular temperaments: Difference between revisions
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This article focuses on the mathematical tools used to describe a regular temperament. For an introduction to regular temperaments, see [[Regular | | de = Verallgemeinerte reguläre Temperatur | ||
| en = Mathematical theory of regular temperaments | |||
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| ja = レギュラーテンペラメントとランクrテンペラメント | |||
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:''This article focuses on the mathematical tools used to describe a regular temperament. For an introduction to regular temperaments, see [[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/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. | ||
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== See also == | == See also == | ||
* [[Tour of | * [[Tour of regular temperaments]] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
[[Category:Math]] | [[Category:Math]] | ||
{{todo|reduce mathslang|improve definition|cleanup}} | {{todo|reduce mathslang|improve definition|cleanup}} | ||