# calculated optimum fixed tuning (COFT)

[Joe Monzo]

The acronym COFT stands for Calculated Optimum Fixed Tuning, was coined by John de Laubenfels (see below), and concerns adaptive-tuning and adaptive-JI.

. . . . . . . . .
[John deLaubenfels, Tuning List posting]

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 fÃ¼r 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.

• 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:

```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
```
. . . . . . . . .
[Paul Erlich, Tuning List posting]

COFT is useful:

1. (a) if you're going to perform the piece on an instrument with only 12 fixed pitches, such as an acoustic piano; or
2. (b) if you're trying to understand the details that went into calculating the adaptive tuning.
. . . . . . . . .

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