User:FloraC/Flora's 17-note well temperament

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The Canou XVII is a seventeen-note well temperament based on 17-et. 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 XII, but this time superpyth instead of meantone.

The result is pretty close to George Secor's well temperament, but my design is with 9/7 in mind instead of 14/11, basically the opposite of his. The distribution of the two tunings 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 tempered to three sizes: G1, G2, and G3. Their sizes and places are shown below.

  • 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

! canou17d.scl
!
Canou XVII, Flora Canou's well temperament based on 17-et, starting with D = 0
 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.

Sizes and occurrences of major third
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.

Sizes and occurrences of minor third
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#.