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 infinite set of intervals in to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be some [[Harmonic Limit|''p''-limit]] [[just intonation]] — or any [[Just intonation subgroups|subgroup]] thereof — AKA rational numbers, 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".

For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.

As another example, if [[meantone]] temperament is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".

A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament "regular".

== Dimensionality, or rank ==

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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 infinite set of intervals in to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. Typically, the source set is assumed to be some [[Harmonic Limit|''p''-limit]] [[just intonation]] — or any [[Just intonation subgroups|subgroup]] thereof — AKA rational numbers, 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".
+For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
+As another example, if [[meantone]] temperament is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
+A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament "regular".
 == Dimensionality, or rank ==