I wasn't able to meet face-to-face with Fritz Gellerman when I visited
in Florida, but we had a pleasant telephone conversation about this
subject, and the tuning scale in Digest 961122.  I have added the
column "Equal Temperment Tuning", and in the last column is shown the
difference when compared to the "de Caus" Just Tuning Scale:

•   note   Equal  deCaus  diff.
          cents  cents   cents

     C       0      0       0
     C#    100     71    - 29
     D     200    204    +  4
     D#    300
     E     400    386    - 14
     F     500    498    -  2
     F#    600    590    - 10
     G     700    702    +  2
     G#    800
     A     900    884    - 16
     A#   1000
     B    1100   1088    - 12

     C    1200   1200       0
                         ____
                   sum:  - 77

•There are 9 notes per octave in this organ scale, therefore the average
difference is -77 / 9 = -8.56 cents.

If this tuning scale were applied to a piano, the piano tuner would
have to consider the effect upon the net tension of the piano strings,
which would be reduced in tuning the piano "flat" from its normal
(Concert) pitch.  With organ pipes this means that most of the tuning
stoppers or slides must pulled out, and so new problems might arise.

The net effect can be minimized if 8.5 cents is added to the de Caus
scale:

•   note   Equal  deCaus      diff.
          cents  cents      cents

     C       0      8.5    +  8.5
     C#    100     79.5    - 21.5
     D     200    212.5    + 12.5
     D#    300
     E     400    394.5    -  5.5
     F     500    506.5    +  6.5
     F#    600    598.5    -  1.5
     G     700    710.5    + 10.5
     G#    800
     A     900    892.5    -  7.5
     A#   1000
     B    1100   1096.5    -  3.5

     C    1200   1208.5    +  8.5
                             ____
                      sum:  - 1.5

•average difference = -1.5 / 9 = -0.17 cents.

The "tables of cents" given above would be used with a modern
electronic tuning device, while the old-time organbuilder would
probably "count the beats" in the time-honoured tradition.  Thus, for
completeness, the tables should be expanded to also show the absolute
frequencies, and the resulting beat frequencies.

I'll do this later -- I just got home after spending a long sleepless
night in the Dallas-Ft. Worth airport, where the first storm of the
winter canceled my flight home!

• ----------------------------------
|         Robbie Rhodes            |
| Return-Path: rrhodes@foxtail.com |
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