Well temperament: Difference between revisions
→Historical well temperaments: improve and expand on the descriptions |
→Generalizations: -> classification by approaches. Cleanup |
||
| Line 23: | Line 23: | ||
# four pure fifths (F#–C#, C#–G#, G#–Eb and Eb–Bb) | # four pure fifths (F#–C#, C#–G#, G#–Eb and Eb–Bb) | ||
== | == Classification by approaches == | ||
There are several | There are several approaches to well temperaments. These are not strictly mutually exclusive, but they provide different frameworks that cater to various goals. | ||
=== Circle of fifths === | === Circle of fifths === | ||
Well temperaments can be structured around the usual uneven distribution of differently-sized fifths, but with a wider palette of fifths, such as [[superpyth]] fifths (approx. 702{{cent}}-720{{cent}}) and [[flattone]] fifths (approx. 691{{cent}}-695{{cent}}). Consequently, major thirds also come in various sizes, sometimes approximating other intervals than the usual [[5/4]], such as [[9/7]] and [[14/11]]. | Well temperaments can be structured around the usual uneven distribution of differently-sized fifths, but with a wider palette of fifths, such as [[superpyth]] fifths (approx. 702{{cent}}-720{{cent}}) and [[flattone]] fifths (approx. 691{{cent}}-695{{cent}}). Consequently, major thirds also come in various sizes, sometimes approximating other intervals than the usual [[5/4]], such as [[9/7]] and [[14/11]]. For example: [[Carl Lumma]]'s [[Cauldron]]. | ||
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]]. | |||
The | |||
=== Detempering === | === Detempering === | ||
| Line 44: | 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. | 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]] ( | 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 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. | ||
Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves. | Again, well temperaments designed through detempering could eventually be generalized to any circle of intervals with any equaves. | ||