User:FloraC/Flora's 17-note well temperament
The Canou II is a seventeen-tone well temperament based on 17et. In summary, it is 1/5-comma superpyth in the natural keys and close to Neo-Gothic tuning in the remote keys. The goal is the same as that of Canou I, but this time superpyth instead of meantone.
The result is pretty close to George Secor's well temperament, but this design prioritizes targeting 9/7 instead of 14/11, basically the opposite of Secor's. The distribution of the generators is also quite different.
Definition
The commas in consideration are the 17-comma ~ 66.7650¢, denoted p, and the septimal comma ~ 27.2641¢, denoted c.
The generator is a perfect fifth, which comes in three sizes, with eight 1/5-comma superpyth fifths, a 17edo fifth, and eight neogothic fifths. Specifically,
- g1 = P5 + c/5 ~ 707.4078¢
- at Bb–F, F–C, C–G, G–D, D–A, A–E, E–B, B–F#.
- g2 = P5 + p/17 ~ 705.8824¢ = 10\17
- at F#–C#.
- g3 = P5 + 2p/17 - c/5 ~ 704.3569¢
- at C#–G#, G#–D#, D#–A#, A#–Gb, Gb–Db, Db–Ab, Ab–Eb, Eb–Bb.
As Scala file
! canou2d.scl ! Canou II, Flora Canou's second well temperament, D = 0, single circle edition 17 ! 66.0118364414 144.227403157 214.815638451 277.776542324 352.941176471 429.631276902 492.592180775 561.654949784 639.870516500 707.407819225 770.368723098 848.584289814 922.223457676 985.184361549 1057.29806313 1135.51362984 1200.
Intervals
The major third gradually shifts between 9/7 in the natural keys and 14/11 in the remote keys.
| Cents | Error | Occurrences | |
|---|---|---|---|
| from 9/7 | from 14/11 | ||
| 429.631 | -5.453 | +12.123 | Bb–D, F–A, C–E, G–B, D–F# |
| 428.106 | -6.978 | +10.598 | A–C# |
| 426.580 | -8.504 | +9.072 | Eb–G |
| 425.055 | -10.029 | +7.547 | E–G# |
| 423.529 | -11.555 | +6.021 | Ab–C |
| 422.004 | -13.080 | +4.496 | B–D# |
| 420.478 | -14.606 | +2.971 | Db–F |
| 418.953 | -16.131 | +1.445 | F#–A# |
| 417.428 | -17.657 | -0.080 | C#–E#, G#–B#, Fb–Ab, Cb–Eb, Gb–Bb |
The minor third gradually shifts between 7/6 in the natural keys and 13/11 in the remote keys.
| Cents | Error | Occurrences | |
|---|---|---|---|
| from 7/6 | from 13/11 | ||
| 277.777 | +10.906 | -11.433 | G–Bb, D–F, A–C, E–G, B–D, F#–A |
| 279.302 | +12.431 | -9.908 | C#–E |
| 280.827 | +13.957 | -8.382 | C–Eb |
| 282.353 | +15.482 | -6.857 | G#–B |
| 283.878 | +17.008 | -5.331 | F–Ab |
| 285.404 | +18.533 | -3.806 | D#–F# |
| 286.929 | +20.058 | -2.280 | A#–C#, E#–G#, B#/Db–D#/Fb, Ab–Cb, Eb–Gb, Bb–Db |
In addition, the neutral second (augmented unison) ranges between 12/11, 13/12 and 14/13, with the largest interval of 151.855¢ at F–F# and Bb–B, and the smallest interval of 130.498¢ at G#–Bb and C#–Eb.
Similarly, the neutral third (diminished fourth) ranges mostly between 11/9 and 16/13, with the smallest interval of 340.737¢ at F#–Bb and the largest interval of 365.145¢ at Bb–C#.