Well temperament: Difference between revisions

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=== Detempering ===

Well temperaments can be obtained by [[detempering]] 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 ~~a wolf interval~~ 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 ~~that wolf interval~~.

Well temperaments can be obtained by [[detempering]] 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.

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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A similar process is to pick an 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.

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

=== Neji ===

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 === Detempering ===
-Well temperaments can be obtained by [[detempering]] 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 a wolf interval 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 that wolf interval.
+Well temperaments can be obtained by [[detempering]] 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.
 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 45: / Line 45: @@
 A similar process is to pick an 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.
+Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves.
 === Neji ===