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 Temperaments]].

A '''regular temperament''' is a ~~mathematical object~~ 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|JI]]"), 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|JI]]"), 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.

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 tempered interval — in this case we say that the two JI intervals are "tempered together".

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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 Temperaments]].
-A '''regular temperament''' is a mathematical object 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|JI]]"), 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|JI]]"), 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.
 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 tempered interval — in this case we say that the two JI intervals are "tempered together".