Curve or direct measuring?

[Jim Coleman, Sr.]

To Renaud and Farrell especially:

I have done many direct measurement tunings and my general feeling is 
that the direct method is a waste of time for equal temperament tunings.
The problem is that you don't know what the interval widths should be 
until you have tuned the temperament to discover its preferred stretch.
With the SAT FAC curve method, you can tell in advance what the stretch 
needs to be and then ALL the intervals are adjusted accordingly.

The FAC program was designed from much experimental data using a Concert 
Grand. The choice of using 4th partials for tuning in the temperament 
region was a wise decision because it helps the M3rds to be really smooth 
in graduation. The 4ths also are quite smooth. The 5ths may waver a 
little because of the wire size changes which may be encountered, but 
with high quality pianos this is no problem. With spinet pianos that have 
wound strings in the treble bridge, there is no perfect answer anyway. In 
these, keeping the 3rds, 4ths and octaves smooth is the best that can be 
done. Even on spinets there is no problem in the upper part of the 
temperament octave. As one approaches the lower part of the temperament 
tuning downward from A4, one can sense if the temperament needs to be 
stretched out even more than at first contemplated. Any such change can 
be made gradually using the DOB feature of the SAT III. This however is 
not needed with normally scaled pianos (medium to large pianos).

As I have indicated in previous posts, my favorite method for tuning 
involves first satisfying the temperament octave stretch from A3 to A4
before tuning anything but the A4. A mere test of the relationship of the
tuned A4 in respect to the FAC settings for A3 will show immediately 
whether one needs to stretch the temperament octave more to satisfy one's
personal preferences for octave widths. The Double Octave Beat control
is used to satisfy this basic need of the piano (more details can be 
supplied for those who are interested). From here on down the chromatic 
tuning process, the M3rds can be played at the same time one is watching
the machine or program. In tuning downward from A4, the first M3rd 
available for testing is the F4-A4 M3rd. Playing both notes will not 
interfere with the F4 display since the A4 has no partials which are on 
any of the partials of the F4. Here is the opportunity to use the best of 
both worlds - tuning aurally and electronically at the same time. The 
same can be done for E4-G#4, D#4-G4, D4-F#4. However, at the note D4, one 
can also use the aural interval of a 5th with A4. Having first learned to 
tune aurally back in the 1940s using only 4ths and 5ths, I like to use 
the 5ths from here on down the scale with occasional checking of the 
M3rds. This provides a better temperament than I was ever able to do 
using direct interval measurements.

The problem with direct interval measurements is that it is difficult to 
figure out what the 4ths need to be if the 5ths need to be narrowed by,
say -1.8 cents. The next problem is that to provide a smooth continuum of 
M3rds, the 3rds will usually be graduated smaller and smaller cents-wise
as one procedes up the scale. The F3-A3 M3rd may be 13.8 cents wide, but 
the F4-A4 3rd may be only 12 cents wide. I have written programs in Qbasic
which tried to do all this direct interval measuring given the 
inharmonicity of each partial of each note of the scale. This is when I 
found out how difficult or impossible it is to do direct interval tuning.
That convinced me of the superiority of curve fit tuning. And when that 
is coupled with the complement of listening to the intervals at the same 
time, this is the best of both worlds. Try it, you'll like it!

Jim Coleman, Sr.