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 [[JI]] pitches that the pitch represents (called a [[transversal]]). Specifically, a '''detempered scale''' or a '''detemperament''' 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 map. If exactly one JI interpretation is used for each pitch, then the detempering is called a '''one-to-one detempering'''. Ideally the resultant JI pitch system will have a compact lattice. | In [[regular temperament theory]], '''detempering''' is the process of taking a tempered pitch system and replacing each of its pitches with one or more [[JI]] pitches that the pitch represents (called a [[transversal]]). Specifically, a '''detempered scale''' or a '''detemperament''' 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 map. If exactly one JI interpretation is used for each pitch, then the detempering is called a '''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. | ||
The term ''detempering'' is often also used in an RTT-agnostic way to refer to a scale which, when some subset of its step sizes are equated, "tempers" to a given simpler scale structure (with fewer step sizes). In this sense, [[diasem]] (LMLSLMLSL) is a detempering of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S. Alternatively, the math term ''preimage'' could be used to avoid RTT connotations (although it has the same ambiguity as to one-to-one-ness as ''detempering''). | The term ''detempering'' is often also used in an RTT-agnostic way to refer to a scale which, when some subset of its step sizes are equated, "tempers" to a given simpler scale structure (with fewer step sizes). In this sense, [[diasem]] (LMLSLMLSL) is a detempering of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S. Alternatively, the math term ''preimage'' could be used to avoid RTT connotations (although it has the same ambiguity as to one-to-one-ness as ''detempering''). | ||
It is a distinct concept from a [[Neji|quasi-equal rational tuning, or neji]], the main difference being that a neji replaces an equal temperament's pitches with pitches from a single harmonic series. | It is a distinct concept from a [[Neji|quasi-equal rational tuning, or neji]], the main difference being that a neji replaces an equal temperament's pitches with pitches from a single harmonic series. | ||
== Examples == | == Examples == | ||
Revision as of 11:58, 3 September 2021
In regular temperament theory, detempering is the process of taking a tempered pitch system and replacing each of its pitches with one or more JI pitches that the pitch represents (called a transversal). Specifically, a detempered scale or a detemperament 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 map. If exactly one JI interpretation is used for each pitch, then the detempering is called a 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.
The term detempering is often also used in an RTT-agnostic way to refer to a scale which, when some subset of its step sizes are equated, "tempers" to a given simpler scale structure (with fewer step sizes). In this sense, diasem (LMLSLMLSL) is a detempering of semiquartal (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S. Alternatively, the math term preimage could be used to avoid RTT connotations (although it has the same ambiguity as to one-to-one-ness as detempering).
It is a distinct concept from a quasi-equal rational tuning, or neji, the main difference being that a neji replaces an equal temperament's pitches with pitches from a single harmonic series.