Calculated Optimum Fixed Tuning (COFT)

Grounding to COFT



OK, found the time to try that idea Paul E. and I have discussed: doing
adaptive JI but grounding the notes not to 12-tET, but to the values
from Calculated Optimum Fixed Tuning (COFT).

Paul, you probably won't be surprised to hear that, in the Bach/Busoni
piece, the total pain is cut by more than half when the grounding is
moved in this fashion.

Here's something that's slightly surprising to me: COFT grounding is
not quite the ideal grounding for adaptive JI.  I realized that, in
relaxing the spring matrix, I could add a step into the process to move
the grounding points as I'm relaxing the tuning at each moment in time.
For the Bach/Busoni, the COFT values are:

   E0 34 42  (For pitch  0, we have bend   7.6228)
   E1 68 3C  (For pitch  1, we have bend -10.0961)
   E2 04 40  (For pitch  2, we have bend   0.1052)
   E3 17 42  (For pitch  3, we have bend   6.9058)
   E4 14 40  (For pitch  4, we have bend   0.5031)
   E5 25 44  (For pitch  5, we have bend  13.5871)
   E6 0D 3C  (For pitch  6, we have bend -12.3436)
   E7 42 40  (For pitch  7, we have bend   1.6553)
   E8 17 3C  (For pitch  8, we have bend -12.0965)
   EA 2E 40  (For pitch  9, we have bend   1.1569)
   EB 2F 43  (For pitch 10, we have bend  10.6647)
   EC 4A 3D  (For pitch 11, we have bend  -7.6646)

(These are slightly different than what I reported before; I fixed a
bug in the COFT routine...).

After adaptive relaxation, the ideal grounding points move to:

   E0 00 42  (For pitch  0, we have target bend   6.3493)
   E1 08 3D  (For pitch  1, we have target bend  -9.2995)
   E2 01 40  (For pitch  2, we have target bend   0.0276)
   E3 6B 41  (For pitch  3, we have target bend   5.8163)
   E4 20 40  (For pitch  4, we have target bend   0.8013)
   E5 0A 44  (For pitch  5, we have target bend  12.9117)
   E6 28 3C  (For pitch  6, we have target bend -11.6671)
   E7 2E 40  (For pitch  7, we have target bend   1.1474)
   E8 75 3C  (For pitch  8, we have target bend  -9.7780)
   EA 33 40  (For pitch  9, we have target bend   1.2646)
   EB 0A 43  (For pitch 10, we have target bend   9.7413)
   EC 6E 3D  (For pitch 11, we have target bend  -6.7823)

I don't know why the values pull back like this, but they do.  Of course
the differences may not be worth thinking much about, being just a cent
or two on each pitch.