Tour of regular temperaments: Difference between revisions
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<span style="display: block; text-align: right;">[[de:Reguläre_Temperaturen]] | |||
[[レギュラーテンペラメントとランクrテンペラメント|日本語]]</span> | |||
=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|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. | ||