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Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
COFT
The acronym COFT stands for Calculated Optimum Fixed
Tuning.
If the tuning of a piece is fixed, then two of the
three spring
types disappear: horizontal motion within a pitch is
gone, so the value of those springs is of no importance. Grounding
becomes a non-issue, because
drift is impossible with a fixed tuning,
and any tuning can be moved in absolute tuning space so that average
deflection is zero (or, notes are tuned only down from
12-tET, or
whatever...). So the only thing left to worry about is to sum the
values of all the vertical springs and then relax them to the least
pain
position.
[As an example:]
I now attach an actual table output for one of the pieces on my web site
(http://www.adaptune.com)
(a piece which works particularly well in COFT), Mozart's Andante
für Orgelwalze, K.616, sequence by Hans Jakob Heldstab. The piece
is in the key of F major. This is a
5-limit target tuning; COFT for
7-limit tunings is problematical.
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 some 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. Note that the interval
most strongly represented is C to E (0 to 4), the fifth degree and the
leading tone, with strength 451.285. That either is or is not
surprising...
A reformulation of the relationships among the columns:
[From John de Laubenfels,
Tuning
List posting]
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
Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 5.23 1 -4.94 5.832 96.923 89.831 -41.358 146.656
0 5.23 2 -2.23 46.949 200.941 192.544 -394.208 1654.979
0 5.23 3 9.09 37.199 311.141 303.866 -270.626 984.403
0 5.23 4 -7.74 451.285 386.241 387.032 356.875 141.108
0 5.23 5 5.80 226.761 498.038 500.571 574.442 727.603
0 5.23 6 -14.35 9.708 592.176 580.420 -114.132 670.889
0 5.23 7 4.54 359.893 701.937 699.319 -942.430 1233.942
0 5.23 8 15.45 41.169 813.253 810.225 -124.669 188.763
0 5.23 9 -6.33 253.599 885.373 888.445 778.823 1195.914
0 5.23 10 4.72 88.399 996.772 999.494 240.608 327.447
0 5.23 11 -9.23 9.339 1092.321 1085.540 -63.327 214.711
1 -4.94 0 5.23 5.832 1103.077 1110.169 41.358 146.656
1 -4.94 2 -2.23 3.059 99.584 102.713 9.572 14.976
1 -4.94 3 9.09 0.366 203.955 214.035 3.691 18.602
1 -4.94 4 -7.74 63.831 310.355 297.201 -839.596 5521.784
1 -4.94 5 5.80 16.209 387.663 410.740 374.065 4316.228
1 -4.94 6 -14.35 2.292 498.885 490.589 -19.014 78.867
1 -4.94 7 4.54 15.344 600.124 609.488 143.681 672.720
1 -4.94 8 15.45 0.732 701.977 720.394 13.485 124.180
1 -4.94 9 -6.33 24.272 813.326 798.614 -357.097 2626.886
1 -4.94 10 4.72 45.310 895.760 909.663 629.968 4379.358
1 -4.94 11 -9.23 0.346 996.045 995.709 -0.116 0.020
2 -2.23 0 5.23 46.949 999.059 1007.456 394.208 1654.979
2 -2.23 1 -4.94 3.059 1100.416 1097.287 -9.572 14.976
2 -2.23 3 9.09 2.323 111.876 111.322 -1.287 0.356
2 -2.23 4 -7.74 16.489 194.369 194.488 1.966 0.117
2 -2.23 5 5.80 138.697 313.790 308.027 -799.406 2303.756
2 -2.23 6 -14.35 52.589 386.459 387.876 74.502 52.773
2 -2.23 7 4.54 140.572 498.037 506.775 1228.231 5365.742
2 -2.23 8 15.45 5.110 603.257 617.681 73.711 531.601
2 -2.23 9 -6.33 159.585 701.941 695.900 -963.948 2911.292
2 -2.23 10 4.72 101.993 813.028 806.950 -619.897 1883.826
2 -2.23 11 -9.23 75.134 884.724 892.996 621.502 2570.492
3 9.09 0 5.23 37.199 888.859 896.134 270.626 984.403
3 9.09 1 -4.94 0.366 996.045 985.965 -3.691 18.602
3 9.09 2 -2.23 2.323 1088.124 1088.678 1.287 0.356
3 9.09 4 -7.74 0.566 92.079 83.166 -5.048 22.494
3 9.09 5 5.80 7.689 199.961 196.705 -25.035 40.756
3 9.09 6 -14.35 2.365 299.082 276.554 -53.284 600.189
3 9.09 7 4.54 25.943 386.837 395.453 223.525 962.928
3 9.09 8 15.45 7.423 498.023 506.359 61.878 257.914
3 9.09 9 -6.33 10.336 587.268 584.579 -27.800 37.383
3 9.09 10 4.72 10.261 701.977 695.628 -65.147 206.811
3 9.09 11 -9.23 12.282 812.395 781.674 -377.311 5795.754
4 -7.74 0 5.23 451.285 813.759 812.968 -356.875 141.108
4 -7.74 1 -4.94 63.831 889.645 902.799 839.596 5521.784
4 -7.74 2 -2.23 16.489 1005.631 1005.512 -1.966 0.117
4 -7.74 3 9.09 0.566 1107.921 1116.834 5.048 22.494
4 -7.74 5 5.80 18.108 111.736 113.539 32.642 29.421
4 -7.74 6 -14.35 1.586 202.709 193.388 -14.786 68.908
4 -7.74 7 4.54 376.137 315.014 312.287 -1025.750 1398.642
4 -7.74 8 15.45 2.173 427.706 423.193 -9.809 22.135
4 -7.74 9 -6.33 97.215 498.177 501.413 314.560 508.915
4 -7.74 10 4.72 46.695 606.077 612.462 298.140 951.786
4 -7.74 11 -9.23 23.293 701.977 698.508 -80.808 140.168
5 5.80 0 5.23 226.761 701.962 699.429 -574.442 727.603
5 5.80 1 -4.94 16.209 812.337 789.260 -374.065 4316.228
5 5.80 2 -2.23 138.697 886.210 891.973 799.406 2303.756
5 5.80 3 9.09 7.689 1000.039 1003.295 25.035 40.756
5 5.80 4 -7.74 18.108 1088.264 1086.461 -32.642 29.421
5 5.80 7 4.54 72.968 202.097 198.748 -244.348 409.120
5 5.80 8 15.45 37.645 313.133 309.654 -130.948 227.752
5 5.80 9 -6.33 335.699 386.204 387.874 560.530 467.970
5 5.80 10 4.72 63.070 498.076 498.923 53.415 22.619
5 5.80 11 -9.23 9.907 593.238 584.969 -81.922 338.712
6 -14.35 0 5.23 9.708 607.824 619.580 114.132 670.889
6 -14.35 1 -4.94 2.292 701.115 709.411 19.014 78.867
6 -14.35 2 -2.23 52.589 813.541 812.124 -74.502 52.773
6 -14.35 3 9.09 2.365 900.918 923.446 53.284 600.189
6 -14.35 4 -7.74 1.586 997.291 1006.612 14.786 68.908
6 -14.35 7 4.54 1.087 111.876 118.899 7.631 26.795
6 -14.35 9 -6.33 32.218 314.848 308.025 -219.841 750.059
6 -14.35 10 4.72 3.984 401.085 419.074 71.670 644.639
6 -14.35 11 -9.23 1.948 498.023 505.120 13.827 49.066
7 4.54 0 5.23 359.893 498.063 500.681 942.430 1233.942
7 4.54 1 -4.94 15.344 599.876 590.512 -143.681 672.720
7 4.54 2 -2.23 140.572 701.963 693.225 -1228.231 5365.742
7 4.54 3 9.09 25.943 813.163 804.547 -223.525 962.928
7 4.54 4 -7.74 376.137 884.986 887.713 1025.750 1398.642
7 4.54 5 5.80 72.968 997.903 1001.252 244.348 409.120
7 4.54 6 -14.35 1.087 1088.124 1081.101 -7.631 26.795
7 4.54 8 15.45 1.087 111.876 110.906 -1.054 0.511
7 4.54 9 -6.33 22.275 188.167 189.126 21.346 10.228
7 4.54 10 4.72 107.402 305.931 300.175 -618.188 1779.102
7 4.54 11 -9.23 99.459 386.338 386.221 -11.575 0.674
8 15.45 0 5.23 41.169 386.747 389.775 124.669 188.763
8 15.45 1 -4.94 0.732 498.023 479.606 -13.485 124.180
8 15.45 2 -2.23 5.110 596.743 582.319 -73.711 531.601
8 15.45 3 9.09 7.423 701.977 693.641 -61.878 257.914
8 15.45 4 -7.74 2.173 772.294 776.807 9.809 22.135
8 15.45 5 5.80 37.645 886.867 890.346 130.948 227.752
8 15.45 7 4.54 1.087 1088.124 1089.094 1.054 0.511
8 15.45 9 -6.33 3.005 92.079 78.220 -41.648 288.602
8 15.45 10 4.72 1.087 182.192 189.269 7.690 27.211
8 15.45 11 -9.23 6.564 288.028 275.315 -83.447 530.435
9 -6.33 0 5.23 253.599 314.627 311.555 -778.823 1195.914
9 -6.33 1 -4.94 24.272 386.674 401.386 357.097 2626.886
9 -6.33 2 -2.23 159.585 498.059 504.100 963.948 2911.292
9 -6.33 3 9.09 10.336 612.732 615.421 27.800 37.383
9 -6.33 4 -7.74 97.215 701.823 698.587 -314.560 508.915
9 -6.33 5 5.80 335.699 813.796 812.126 -560.530 467.970
9 -6.33 6 -14.35 32.218 885.152 891.975 219.841 750.059
9 -6.33 7 4.54 22.275 1011.833 1010.874 -21.346 10.228
9 -6.33 8 15.45 3.005 1107.921 1121.780 41.648 288.602
9 -6.33 10 4.72 12.072 110.641 111.050 4.930 1.007
9 -6.33 11 -9.23 10.725 191.502 197.096 59.991 167.785
10 4.72 0 5.23 88.399 203.228 200.506 -240.608 327.447
10 4.72 1 -4.94 45.310 304.240 290.337 -629.968 4379.358
10 4.72 2 -2.23 101.993 386.972 393.050 619.897 1883.826
10 4.72 3 9.09 10.261 498.023 504.372 65.147 206.811
10 4.72 4 -7.74 46.695 593.923 587.538 -298.140 951.786
10 4.72 5 5.80 63.070 701.924 701.077 -53.415 22.619
10 4.72 6 -14.35 3.984 798.915 780.926 -71.670 644.639
10 4.72 7 4.54 107.402 894.069 899.825 618.188 1779.102
10 4.72 8 15.45 1.087 1017.808 1010.731 -7.690 27.211
10 4.72 9 -6.33 12.072 1089.359 1088.950 -4.930 1.007
10 4.72 11 -9.23 3.839 85.216 86.046 3.187 1.323
11 -9.23 0 5.23 9.339 107.679 114.460 63.327 214.711
11 -9.23 1 -4.94 0.346 203.955 204.291 0.116 0.020
11 -9.23 2 -2.23 75.134 315.276 307.004 -621.502 2570.492
11 -9.23 3 9.09 12.282 387.605 418.326 377.311 5795.754
11 -9.23 4 -7.74 23.293 498.023 501.492 80.808 140.168
11 -9.23 5 5.80 9.907 606.762 615.031 81.922 338.712
11 -9.23 6 -14.35 1.948 701.977 694.880 -13.827 49.066
11 -9.23 7 4.54 99.459 813.662 813.779 11.575 0.674
11 -9.23 8 15.45 6.564 911.972 924.685 83.447 530.435
11 -9.23 9 -6.33 10.725 1008.498 1002.904 -59.991 167.785
11 -9.23 10 4.72 3.839 1114.784 1113.954 -3.187 1.323
---- ------ ---- ------ -------- -------- -------- ---------- ----------
painSum 57647.805
. . . . . . . . . . . . . . . . . . . . . . . . . .
From Paul Erlich, Tuning List posting:
COFT is useful (a) if you're going to perform the piece on an instrument with only 12 fixed pitches, such as an acoustic piano; or (b) if you're trying to understand the details that went into calculating the adaptive tuning.
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