Well temperament: Difference between revisions

Line 38:

If the main generator is different from a fifth, then there are multiple wolf fifths which are evenly distributed along the circle of fifths. Each wolf fifth is typically more in tune than the single wolf fifths of the fifth-generated cases, since the total mistuning is spread out over multiple intervals, but that also means that wolf fifths are more likely to be used frequently in such well temperaments.

Well temperaments based on rank-2 temperaments can be designed to follow the structure of a [[moment of symmetry]] (mos) scale. In that case, each generic interval comes in two sizes, which ensures that there will be exactly two kinds of fifths even if the generator is not a tempered perfect fifth~~. This kind of well temperament could also be arrived at from an equal tuning where the steps are [[comma]]<nowiki/>s~~.

Well temperaments based on rank-2 temperaments can be designed to follow the structure of a [[moment of symmetry]] (mos) scale. In that case, each generic interval comes in two sizes, which ensures that there will be exactly two kinds of fifths even if the generator is not a tempered perfect fifth.

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.

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.

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

@@ Line 38: / Line 38: @@
 If the main generator is different from a fifth, then there are multiple wolf fifths which are evenly distributed along the circle of fifths. Each wolf fifth is typically more in tune than the single wolf fifths of the fifth-generated cases, since the total mistuning is spread out over multiple intervals, but that also means that wolf fifths are more likely to be used frequently in such well temperaments.
-Well temperaments based on rank-2 temperaments can be designed to follow the structure of a [[moment of symmetry]] (mos) scale. In that case, each generic interval comes in two sizes, which ensures that there will be exactly two kinds of fifths even if the generator is not a tempered perfect fifth.  This kind of well temperament could also be arrived at from an equal tuning where the steps are [[comma]]<nowiki/>s.
+Well temperaments based on rank-2 temperaments can be designed to follow the structure of a [[moment of symmetry]] (mos) scale. In that case, each generic interval comes in two sizes, which ensures that there will be exactly two kinds of fifths even if the generator is not a tempered perfect fifth.
 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.
+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.
 Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves.