Interval analysis of Chopin's "Fantaisie-Impromptu"

Ed Foote's latest CD, "Six Degrees of Tonality" also contains Chopin's
"Fantaisie-Impromptu", op 66. For this piece, Ed has chosen the
DeMorgan temperament, a kind of reverse well temperament. It's easy to
see right away that this is a good choice for consonant intervals: the
piece moves between C# minor and C# major (Db major? I don't have a
score), many fifths away from C major. Again, I thought it might be fun
to see what tuning might be implied by the intervals contained in the
piece itself.

As always, my analysis presupposes that intervals (major and minor
thirds, fourths, and inversions) should best be as close to Just as
possible. This supposition might be contrary to the original composer's
practice, and certainly, wishes.

I don't have the complete numbers on the DeMorgan temperament. Ed's
very nice graph shows major thirds, and these can't be mapped back to
the circle of fifths, because there are four independent rings of three
major thirds each.

I found the ideal fixed tuning deviations in cents from 12-tET to be
(note: 0 == C, 1 == C#/Db, etc.):

EA 24 43 (For pitch 9, we have bend 10.2563)
E4 5D 44 (For pitch 4, we have bend 14.7708)
EC 4D 43 (For pitch 11, we have bend 11.2585)
E6 5A 40 (For pitch 6, we have bend 2.2069)
E1 28 40 (For pitch 1, we have bend 0.9876)
E8 67 40 (For pitch 8, we have bend 2.5318)
E3 65 40 (For pitch 3, we have bend 2.4739)
EB 4B 3E (For pitch 10, we have bend -4.4407)
E5 6F 3C (For pitch 5, we have bend -9.8065)
E0 3B 3D (For pitch 0, we have bend -7.9490)
E7 21 3D (For pitch 7, we have bend -8.5817)
E2 4F 3B (For pitch 2, we have bend -13.7080)

It looks like a crazy meantone, Bbb through D, or A through C##.

I found the major thirds to be distributed as follows (note the Strength
column):

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 -7.95 4 14.77 17.984 396.485 422.720 471.810 6188.845
7 -8.58 11 11.26 2.680 409.470 419.840 27.796 144.132
2 -13.71 6 2.21 6.018 386.316 415.915 178.141 2636.398
9 10.26 1 0.99 260.604 386.398 390.731 1129.332 2446.990
4 14.77 8 2.53 436.938 386.371 387.761 607.175 421.869
11 11.26 3 2.47 80.399 386.514 391.215 377.968 888.442
6 2.21 10 -4.44 141.774 386.625 393.352 953.778 3208.252
1 0.99 5 -9.81 413.161 386.493 389.206 1121.072 1520.960
8 2.53 0 -7.95 325.603 386.974 389.519 828.684 1054.532
3 2.47 7 -8.58 50.153 386.660 388.944 114.581 130.888
10 -4.44 2 -13.71 53.976 386.530 390.733 226.867 476.772
5 -9.81 9 10.26 39.900 386.716 420.063 1330.546 22184.770

The most represented thirds are E-G# (from C# minor) and C#-E# (from C#
major). The worst tuned third with any significant representation is
F-A: it crosses the "wolf" of the implied scale.

Here are overall numbers for several tunings:

12-tET spring pain: 412773.685
Werckmeister III spring pain: 655849.733
Kirnberger III spring pain: 717610.855
Thomas Young spring pain: 606498.189
31 from Abb to C : spring pain: 369492.569
31 from Ebb to G : spring pain: 296282.321
31 from Bbb to D : spring pain: 228561.220
31 from Fb to A : spring pain: 664412.815
31 from Cb to E : spring pain: 1437109.584
31 from Gb to B : spring pain: 1577641.020
31 from Db to F# : spring pain: 1840655.512
31 from Ab to C# : spring pain: 2043629.162
31 from Eb to G# : spring pain: 2032731.011
31 from Bb to D# : spring pain: 1699991.837
31 from F to A# : spring pain: 1552912.778
31 from C to E# : spring pain: 838201.692
31 from G to B# : spring pain: 369492.569
31 from D to F##: spring pain: 296282.321
31 from A to C##: spring pain: 228561.220
COFT Total spring pain: 156552.767
After relaxing, Total spring pain: 78610.590
Final vertical spring pain: 44545.009
Final horizontal spring pain: 6911.803
Final grounding spring pain: 27153.779
Bend range applied: -16.0062 to 16.6416

Notes: in this analysis I used my older 5-limit tuning targets, in which
dom 7th chords tend to have a 7th degree 8/9 of root above.

The analysis is based upon one of several very good MIDI sequences I
was able to find on prs.net, performed by Larry Ellis.

JdL