Well temperament: Difference between revisions

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The same idea could also be applied to other equal temperaments, using circles of other intervals, possibly with other equaves. For example: [[George Secor]]'s [[secor29htt|29-tone high tolerance temperament]].

=== Detempering ~~(if the philosophy is [[RTT]]-based)~~ or ~~''Deregularizing/Detuning'' (RTT-agnostic)~~ ===

=== Detempering or deregularizing ===

Well temperaments can be obtained by [[detempering]] ~~(if the philosophy is [[RTT]]-based) or ''deregularizing/detuning'' (RTT-agnostic)~~ an equal tuning. This implies going from a [[rank]]-1 temperament to a ~~rank-2 (or higher)~~ temperament by adding one (or more) extra generator(s) — a common choice is to add a pure [[octave]] —, which creates an imperfect generator at the end of the generator chain. Whereas historical well temperaments often make use of irregular patterns of fifth sizes around the circle of fifths, detemperaments have identical generators all along the circle except for the imperfect generator.

Well temperaments can be obtained by [[Detempering|detempering or deregularizing]] an equal tuning. This implies going from a [[rank]]-1 temperament to a multirank temperament by adding one (or more) extra generator(s) – a common choice is to add a pure [[octave]] –, which creates an imperfect generator at the end of the generator chain. Whereas historical well temperaments often make use of irregular patterns of fifth sizes around the circle of fifths, detemperaments have identical generators all along the circle except for the imperfect generator.

If the main generator is a fifth, then there is only one wolf fifth that closes the circle of fifths, a feature which is often associated to tunings such as [[quarter-comma meantone]]. However, these tunings are not always considered as well temperaments because they may not preserve transposability due to their higher mistunings.

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For examples: [http://lumma.org/tuning/gws/duowell.htm Duowell], a well-tuning of [[Duodene]]

A similar process is to pick a mos scale with the desired number of tones and a [[step ratio]] close to 1. If the step ratio is [[superparticular]], then it is also a [[maximally even]] scale. In that particular case, the resulting well temperament is not only a detemperament, but also a subset of a finer equal tuning, where individual steps are usually [[comma]]-sized. If the superset of the particular detemperament ~~(if the philosophy is [[RTT]]-based)~~ or ''deregularization~~/detuning'' (RTT-agnostic)~~ is a fine enough equal tuning, it has uneven sisters.

A similar process is to pick a mos scale with the desired number of tones and a [[step ratio]] close to 1. If the step ratio is [[superparticular]], then it is also a [[maximally even]] scale. In that particular case, the resulting well temperament is not only a detemperament, but also a subset of a finer equal tuning, where individual steps are usually [[comma]]-sized. If the superset of the particular detemperament or deregularization is a fine enough equal tuning, it has uneven sisters{{clarify}}.

Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves.

@@ Line 33: / Line 33: @@
 The same idea could also be applied to other equal temperaments, using circles of other intervals, possibly with other equaves. For example: [[George Secor]]'s [[secor29htt|29-tone high tolerance temperament]].
-=== Detempering (if the philosophy is [[RTT]]-based) or ''Deregularizing/Detuning'' (RTT-agnostic) ===
+=== Detempering or deregularizing ===
-Well temperaments can be obtained by [[detempering]] (if the philosophy is [[RTT]]-based) or ''deregularizing/detuning'' (RTT-agnostic) an equal tuning. This implies going from a [[rank]]-1 temperament to a rank-2 (or higher) temperament by adding one (or more) extra generator(s) — a common choice is to add a pure [[octave]] —, which creates an imperfect generator at the end of the generator chain. Whereas historical well temperaments often make use of irregular patterns of fifth sizes around the circle of fifths, detemperaments have identical generators all along the circle except for the imperfect generator.
+Well temperaments can be obtained by [[Detempering|detempering or deregularizing]] an equal tuning. This implies going from a [[rank]]-1 temperament to a multirank temperament by adding one (or more) extra generator(s) – a common choice is to add a pure [[octave]] –, which creates an imperfect generator at the end of the generator chain. Whereas historical well temperaments often make use of irregular patterns of fifth sizes around the circle of fifths, detemperaments have identical generators all along the circle except for the imperfect generator.
 If the main generator is a fifth, then there is only one wolf fifth that closes the circle of fifths, a feature which is often associated to tunings such as [[quarter-comma meantone]]. However, these tunings are not always considered as well temperaments because they may not preserve transposability due to their higher mistunings.
@@ Line 44: / Line 44: @@
 For examples: [http://lumma.org/tuning/gws/duowell.htm Duowell], a well-tuning of [[Duodene]]
-A similar process is to pick a mos scale with the desired number of tones and a [[step ratio]] close to 1. If the step ratio is [[superparticular]], then it is also a [[maximally even]] scale. In that particular case, the resulting well temperament is not only a detemperament, but also a subset of a finer equal tuning, where individual steps are usually [[comma]]-sized. If the superset of the particular detemperament (if the philosophy is [[RTT]]-based) or ''deregularization/detuning'' (RTT-agnostic) is a fine enough equal tuning, it has uneven sisters.
+A similar process is to pick a mos scale with the desired number of tones and a [[step ratio]] close to 1. If the step ratio is [[superparticular]], then it is also a [[maximally even]] scale. In that particular case, the resulting well temperament is not only a detemperament, but also a subset of a finer equal tuning, where individual steps are usually [[comma]]-sized. If the superset of the particular detemperament or deregularization is a fine enough equal tuning, it has uneven sisters{{clarify}}.
 Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves.