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:

    
       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 John de Laubenfels, Tuning List posting]

. . . . . . . . . . . . . . . . . . . . . . . . . .

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.


(to download a zip file of the entire Dictionary, click here)

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