Interval analysis of Alison's piece.

[I wrote (on the big list):]
>>I'm terrible at reading scores, but if you had a midi file of this,
>>I could recommend a fixed tuning that would tend to optimize
>>consonance.

[Alison wrote:]
>Thanks, John. Here's a midi file attached. This should be interesting.
>It'll take me a while to listen to the results because I'll have to
>convert the .wav files to something a Mac can use.

The values I recommended to Alison on the big list are:

Bb: -7.6
F : -12.4
C : -2.3
G : +15.5
D : +12.8
A : +4.4
E : +4.3
B : +5.7
F#: -2.6
C#: -2.0
G#: -4.9
D#: -11.1

This is based upon a 5-limit tuning-file free analysis of the intervals.
The analysis I'm posting here is slightly different, due to small
melodic springs, but let's not quibble about fractions of a cent, ok?

Interval strengths shown in the table are reduced from "nominal" by
the following factors:

1: 1/32
2: 1/16
3: nominal (1/1)
4: nominal (1/1)
5: nominal (1/1)
6: 1/8
7: same as 5
8: same as 4
9: same as 3
10: same as 2
11: same as 1

How to read the table: each record shows a pair of pitches, along with
their final tuning, in cents relative to 12-tET. The Strength field
is an integral of loudness over time of that pair of pitches sounding
in the sequence (with adjustment for less important intervals).
Ideal should tend to show a quasi-JI tuning for this interval (quasi
only because sometimes different interpretations of the interval
conflict to some extent in the composite shown). Actual reflects the
tunings chosen for the two notes. Force is the means of communicating
urgency of request, and is Strength times the difference of Actual and
Ideal; the force for all intervals of each note adds to zero because the
spring set has been relaxed to a state of minimum energy ("pain"). The
Pain column is proportional to (Ideal - Actual) squared times strength.

In this table 0 == C, 1 == C# and/or Db, etc.

A reformulation of the relationships among the columns:

Force = Strength * (Actual - Ideal)
Pain = 0.5 * Strength * (Actual - Ideal)^2

Which implies,

Pain / Force = 0.5 * (Actual - Ideal)

Pain = Force * 0.5 * (Actual - Ideal)
Pain = Force * (Actual - Ideal) / 2.0

The piece is in the key of E, and the interval most represented is E
to B:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 -2.33 1 -1.91 0.045 97.303 100.415 0.140 0.217
0 -2.33 2 12.78 0.030 197.316 215.105 0.525 4.671
0 -2.33 3 -10.93 1.627 315.641 291.397 -39.445 478.155
0 -2.33 4 4.20 6.779 386.314 406.524 137.005 1384.442
0 -2.33 5 -12.30 12.573 498.045 490.020 -100.893 404.821
0 -2.33 6 -2.55 0.183 600.000 599.776 -0.041 0.005
0 -2.33 7 15.36 3.921 701.955 717.689 61.687 485.296
0 -2.33 8 -4.91 10.874 813.686 797.415 -176.933 1439.460
0 -2.33 9 4.43 5.231 884.359 906.751 117.140 1311.516
0 -2.33 10 -7.48 0.071 1002.684 994.845 -0.555 2.176
0 -2.33 11 5.64 0.210 1101.381 1107.963 1.382 4.547
1 -1.91 0 -2.33 0.045 1102.697 1099.585 -0.140 0.217
1 -1.91 2 12.78 0.087 100.000 114.690 1.282 9.417
1 -1.91 3 -10.93 0.056 197.316 190.982 -0.357 1.130
1 -1.91 4 4.20 4.138 315.641 306.109 -39.444 187.999
1 -1.91 6 -2.55 36.919 498.045 499.362 48.606 31.996
1 -1.91 7 15.36 0.241 600.000 617.274 4.158 35.910
1 -1.91 9 4.43 9.219 813.686 806.336 -67.760 249.027
1 -1.91 10 -7.48 5.242 884.359 894.430 52.792 265.854
1 -1.91 11 5.64 0.178 1002.684 1007.548 0.864 2.100
2 12.78 0 -2.33 0.030 1002.684 984.895 -0.525 4.671
2 12.78 1 -1.91 0.087 1100.000 1085.310 -1.282 9.417
2 12.78 3 -10.93 0.014 97.303 76.292 -0.296 3.106
2 12.78 4 4.20 0.192 199.046 191.419 -1.463 5.578
2 12.78 6 -2.55 11.294 386.314 384.671 -18.550 15.233
2 12.78 9 4.43 3.871 701.955 691.646 -39.908 205.710
2 12.78 11 5.64 7.298 884.359 892.858 62.027 263.587
3 -10.93 0 -2.33 1.627 884.359 908.603 39.445 478.155
3 -10.93 1 -1.91 0.056 1002.684 1009.018 0.357 1.130
3 -10.93 2 12.78 0.014 1102.697 1123.708 0.296 3.106
3 -10.93 4 4.20 0.502 99.416 115.127 7.881 61.912
3 -10.93 5 -12.30 0.548 199.721 198.623 -0.602 0.330
3 -10.93 6 -2.55 17.353 315.641 308.379 -126.017 457.574
3 -10.93 8 -4.91 6.390 498.045 506.018 50.945 203.088
3 -10.93 10 -7.48 7.995 701.955 703.448 11.937 8.912
3 -10.93 11 5.64 5.472 813.686 816.566 15.757 22.686
4 4.20 0 -2.33 6.779 813.686 793.476 -137.005 1384.442
4 4.20 1 -1.91 4.138 884.359 893.891 39.444 187.999
4 4.20 2 12.78 0.192 1000.954 1008.581 1.463 5.578
4 4.20 3 -10.93 0.502 1100.584 1084.873 -7.881 61.912
4 4.20 5 -12.30 0.461 98.730 83.496 -7.024 53.504
4 4.20 6 -2.55 1.282 199.184 193.253 -7.602 22.544
4 4.20 7 15.36 25.083 315.641 311.165 -112.273 251.270
4 4.20 8 -4.91 54.842 386.314 390.891 251.043 574.581
4 4.20 9 4.43 17.403 498.045 500.227 37.975 41.432
4 4.20 10 -7.48 1.138 600.000 588.322 -13.287 77.588
4 4.20 11 5.64 86.825 701.955 701.439 -44.792 11.554
5 -12.30 0 -2.33 12.573 701.955 709.980 100.893 404.821
5 -12.30 3 -10.93 0.548 1000.279 1001.377 0.602 0.330
5 -12.30 4 4.20 0.461 1101.270 1116.504 7.024 53.504
5 -12.30 7 15.36 0.048 197.316 227.669 1.445 21.933
5 -12.30 8 -4.91 31.236 315.641 307.395 -257.587 1062.086
5 -12.30 9 4.43 3.998 386.314 416.731 121.592 1849.238
5 -12.30 10 -7.48 3.844 498.045 504.825 26.065 88.362
6 -2.55 0 -2.33 0.183 600.000 600.224 0.041 0.005
6 -2.55 1 -1.91 36.919 701.955 700.638 -48.606 31.996
6 -2.55 2 12.78 11.294 813.686 815.329 18.550 15.233
6 -2.55 3 -10.93 17.353 884.359 891.621 126.017 457.574
6 -2.55 4 4.20 1.282 1000.816 1006.747 7.602 22.544
6 -2.55 7 15.36 0.015 97.303 117.913 0.314 3.232
6 -2.55 8 -4.91 0.149 197.316 197.639 0.048 0.008
6 -2.55 9 4.43 19.364 315.641 306.974 -167.825 727.254
6 -2.55 11 5.64 6.289 498.045 508.187 63.784 323.435
7 15.36 0 -2.33 3.921 498.045 482.311 -61.687 485.296
7 15.36 1 -1.91 0.241 600.000 582.726 -4.158 35.910
7 15.36 4 4.20 25.083 884.359 888.835 112.273 251.270
7 15.36 5 -12.30 0.048 1002.684 972.331 -1.445 21.933
7 15.36 6 -2.55 0.015 1102.697 1082.087 -0.314 3.232
7 15.36 8 -4.91 0.031 97.303 79.726 -0.553 4.859
7 15.36 9 4.43 0.032 197.316 189.062 -0.263 1.087
7 15.36 10 -7.48 1.951 315.641 277.156 -75.083 1444.787
7 15.36 11 5.64 7.886 386.314 390.274 31.231 61.839
8 -4.91 0 -2.33 10.874 386.314 402.585 176.933 1439.460
8 -4.91 3 -10.93 6.390 701.955 693.982 -50.945 203.088
8 -4.91 4 4.20 54.842 813.686 809.109 -251.043 574.581
8 -4.91 5 -12.30 31.236 884.359 892.605 257.587 1062.086
8 -4.91 6 -2.55 0.149 1002.684 1002.361 -0.048 0.008
8 -4.91 7 15.36 0.031 1102.697 1120.274 0.553 4.859
8 -4.91 9 4.43 0.012 97.303 109.336 0.150 0.903
8 -4.91 10 -7.48 0.206 197.316 197.430 0.024 0.001
8 -4.91 11 5.64 26.156 315.641 310.548 -133.225 339.286
9 4.43 0 -2.33 5.231 315.641 293.249 -117.140 1311.516
9 4.43 1 -1.91 9.219 386.314 393.664 67.760 249.027
9 4.43 2 12.78 3.871 498.045 508.354 39.908 205.710
9 4.43 4 4.20 17.403 701.955 699.773 -37.975 41.432
9 4.43 5 -12.30 3.998 813.686 783.269 -121.592 1849.238
9 4.43 6 -2.55 19.364 884.359 893.026 167.825 727.254
9 4.43 7 15.36 0.032 1002.684 1010.938 0.263 1.087
9 4.43 8 -4.91 0.012 1102.697 1090.664 -0.150 0.903
9 4.43 11 5.64 0.424 198.705 201.212 1.064 1.333
10 -7.48 0 -2.33 0.071 197.316 205.155 0.555 2.176
10 -7.48 1 -1.91 5.242 315.641 305.570 -52.792 265.854
10 -7.48 3 -10.93 7.995 498.045 496.552 -11.937 8.912
10 -7.48 4 4.20 1.138 600.000 611.678 13.287 77.588
10 -7.48 5 -12.30 3.844 701.955 695.175 -26.065 88.362
10 -7.48 7 15.36 1.951 884.359 922.844 75.083 1444.787
10 -7.48 8 -4.91 0.206 1002.684 1002.570 -0.024 0.001
10 -7.48 11 5.64 0.118 97.303 113.118 1.871 14.798
11 5.64 0 -2.33 0.210 98.619 92.037 -1.382 4.547
11 5.64 1 -1.91 0.178 197.316 192.452 -0.864 2.100
11 5.64 2 12.78 7.298 315.641 307.142 -62.027 263.587
11 5.64 3 -10.93 5.472 386.314 383.434 -15.757 22.686
11 5.64 4 4.20 86.825 498.045 498.561 44.792 11.554
11 5.64 6 -2.55 6.289 701.955 691.813 -63.784 323.435
11 5.64 7 15.36 7.886 813.686 809.726 -31.231 61.839
11 5.64 8 -4.91 26.156 884.359 889.452 133.225 339.286
11 5.64 9 4.43 0.424 1001.295 998.788 -1.064 1.333
11 5.64 10 -7.48 0.118 1102.697 1086.882 -1.871 14.798
---- ------ ---- ------ -------- -------- -------- ---------- ----------
painSum 14523.370

JdL