Tour of regular temperaments: Difference between revisions

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[[de:Reguläre_Temperaturen]]

[[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]]

=Regular temperaments=

= Regular temperaments =

Regular temperaments are non-Just tunings in which the infinite number of intervals in [[~~Harmonic_Limit~~|p-limit]] [[~~Just_intonation|Just~~ intonation]], or any [[~~Just_intonation_subgroups~~|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[~~tempering_out~~|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. 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. 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.

'''Regular temperaments''' are non-Just tunings in which the infinite number of intervals in [[Harmonic Limit|''p''-limit]] [[just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. 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. 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.

A rank r ~~[http://en.wikipedia.org/wiki/Regular_temperament~~ regular temperament] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[~~Comma_pump_examples~~|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.

A rank ''r'' regular temperament in a particular tuning may be defined by giving ''r'' multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank ''r'' temperament will be defined by ''r'' generators, and thus ''r'' [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of ''r'' independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[Comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.

==Why would I want to use a regular temperament?==

== Why would I want to use a regular temperament? ==

Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.

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 [[de:Reguläre_Temperaturen]]
 [[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]]
-=Regular temperaments=
+= Regular temperaments =
-Regular temperaments are non-Just tunings in which the infinite number of intervals in [[Harmonic_Limit|p-limit]] [[Just_intonation|Just intonation]], or any [[Just_intonation_subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering_out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. 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. 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.
+'''Regular temperaments''' are non-Just tunings in which the infinite number of intervals in [[Harmonic Limit|''p''-limit]] [[just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. 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. 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.
-A rank r [http://en.wikipedia.org/wiki/Regular_temperament regular temperament] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[Comma_pump_examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
+A rank ''r'' regular temperament in a particular tuning may be defined by giving ''r'' multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank ''r'' temperament will be defined by ''r'' generators, and thus ''r'' [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of ''r'' independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[Comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
-==Why would I want to use a regular temperament?==
+== Why would I want to use a regular temperament? ==
 Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.