Detempering: Difference between revisions

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In [[regular temperament theory]], '''detempering''' is the process of taking a tempered ~~pitch~~ system and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the ~~(typically [[JI]])~~ pitches that the pitch represents. Specifically, a '''detempered ~~scale~~''' or a '''detempering''' has each pitch of a tempered ~~scale~~ (according to a fixed regular temperament) replaced with some set of JI interpretations of the pitch under the temperament mapping. If exactly one JI interpretation is used for each pitch, then the detempering is called a '''one-to-one detempering~~''', or a '''[[transversal scale]]'~~''. Ideally the resultant JI pitch system will have a compact lattice.

In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one JI interpretation is used for each pitch, then the detempering is called a [[transversal]], or ''one-to-one detempering''. Ideally the resultant JI pitch system will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.

A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.

The term ''deregularization'' can be used as a [[JI-agnostic]] alternative to ''detempering''. In this sense, [[diasem]] (LMLSLMLSL) is a deregularization of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.

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-In [[regular temperament theory]], '''detempering''' is the process of taking a tempered pitch system and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the (typically [[JI]]) pitches that the pitch represents. Specifically, a '''detempered scale''' or a '''detempering''' has each pitch of a tempered scale (according to a fixed regular temperament) replaced with some set of JI interpretations of the pitch under the temperament mapping. If exactly one JI interpretation is used for each pitch, then the detempering is called a '''one-to-one detempering''', or a '''[[transversal scale]]'''. Ideally the resultant JI pitch system will have a compact lattice.
+In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one JI interpretation is used for each pitch, then the detempering is called a [[transversal]], or ''one-to-one detempering''. Ideally the resultant JI pitch system will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.
-A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.
 The term ''deregularization'' can be used as a [[JI-agnostic]] alternative to ''detempering''. In this sense, [[diasem]] (LMLSLMLSL) is a deregularization of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.