Interval analysis of Haydn Sonata

Ed Foote's latest CD, "Six Degrees of Tonality", contains, among many
pieces, a Haydn Piano Sonata (no 10 in Eb). I thought it might be fun
to compare the historical temperament that Ed used (in this case, Prinz,
aka Kirnberger III) to a temperament suggested by the intervals
contained in the piece itself.

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

Kirnberger III tallied worse than 12-tET, and much worse than an unusual
meantone range (Cb to E was the best 12-note set), which was very close
to my COFT.

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

(meantone-like pitch degrees:)
E6 7C 43 (For pitch 6, we have bend 12.4157)
E1 66 43 (For pitch 1, we have bend 11.8702)
E8 30 42 (For pitch 8, we have bend 7.4302)
E3 23 41 (For pitch 3, we have bend 3.9831)
EB 4E 40 (For pitch 10, we have bend 1.9264)
E5 17 40 (For pitch 5, we have bend 0.5832)
E0 08 3F (For pitch 0, we have bend -2.9398)
E7 6D 3D (For pitch 7, we have bend -6.7235)
E2 38 3C (For pitch 2, we have bend -11.1407)
EA 6D 3B (For pitch 9, we have bend -12.9846)

(transitional pitch degrees:)
E4 25 3D (For pitch 4, we have bend -8.4883)
EC 26 41 (For pitch 11, we have bend 4.0680)

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

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 -2.94 4 -8.49 86.131 386.561 394.452 679.639 2681.422
7 -6.72 11 4.07 73.253 387.363 410.791 1716.182 20103.580
2 -11.14 6 12.42 2.357 397.448 423.556 61.542 803.382
9 -12.98 1 11.87 2.726 394.278 424.855 83.352 1274.301
4 -8.49 8 7.43 10.898 390.764 415.918 274.141 3447.875
11 4.07 3 3.98 85.794 386.848 399.915 1121.036 7324.096
6 12.42 10 1.93 235.739 386.381 389.511 737.801 1154.563
1 11.87 5 0.58 339.727 386.316 388.713 814.349 976.025
8 7.43 0 -2.94 385.967 386.427 389.630 1236.358 1980.199
3 3.98 7 -6.72 636.844 386.646 389.293 1686.194 2232.297
10 1.93 2 -11.14 809.829 386.360 386.933 464.323 133.112
5 0.58 9 -12.98 506.189 386.434 386.432 -0.996 0.001

Not surprisingly, the major third most represented is Bb to D, the
dominant and leading tone.

Here are overall numbers for several tunings:

12-tET Total spring pain: 729063.071
Werckmeister III spring pain: 859050.789
Kirnberger III Total spring pain: 888543.678
Thomas Young Total spring pain: 774346.300
31 from Cb to E : spring pain: 354791.067
31 from Gb to B : spring pain: 427660.954
31 from Db to F# : spring pain: 827362.457
31 from Ab to C# : spring pain: 1447712.949
31 from Eb to G# : spring pain: 2217852.100
31 from Bb to D# : spring pain: 3174777.136
31 from F to A# : spring pain: 4042854.356
COFT Total spring pain: 296211.282
After relaxing, Total spring pain: 156333.908
Final vertical spring pain: 94400.682
Final horizontal spring pain: 11702.854
Final grounding spring pain: 50230.372
Bend range applied: -15.5901 to 15.1134

Note that meantone scale Gb to B is almost as good as Cb to E, but
Db to F# is much worse, because the major third Gb-Bb is fairly well
represented. A more usual Eb to G# is terrible.

Kirnberger III's pain numbers are most heavily inflated by the major
triad Ab-C-Eb (subdominant to the tonic key). Of course, from one
perspective this gives the piece more color.

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. As
expected, the newer alternative (so-called "tuning file free"), which
targets 7th degree at 9/10 of root above, gives a somewhat more extreme
range of tuning deviations from 12-tET.

This analysis is based upon the only MIDI sequence of the piece that I
could find, performed by Bunji Hisamori. It's fair to say that Bunji is
no Enid Katahn, but I hope that the interval set is reasonably well
represented.

JdL