Aural Pure 5ths Equal Temperament

[Jerry Anderson]

Dear Jim,

Dr. Albert Sanderson published some fascinating data
in the June 1978 issue of the Piano Technicians' Journal.
Page 16 shows two charts which compare theoretical beat
rates to actual measured beat rates in a well tuned piano.
There are separate columns quantifying the 2:3 fifth and the 4:6
fifth beat rates.  These beat rates, and their rates of change
are significantly different.

The tuning this data was drawn from appears to be based on
a system where all octaves were tuned 0.5 bps sharp.  This
could, of course, have been done with any other temperament
system including a zero beating 2:3 fifth.  The data seem to
suggest two problems that may arise from a zero beating
fifth system.  First of all, both the 2:3 and 4:6 fifths will not be
zero beating at the same time.  This might be a problem in
in the middle , and certainly would be a difficulty in the lower
bass of smaller pianos where the first and second partials
are dominated by higher partials.

Secondly, the inharmonicity does not progress linearly as we
progress up the scale from the temperament, but clearly is
increasing in more like an exponential manner.  Regrettably,
Dr. Sanderson only gives data for two octaves, but his data
suggests what we all know intuitively, that the top octaves of
a piano are extremely inharmonic, and that inharmonicity
increases from note to note by progressively larger steps
as we go up the scale.

It seems to me, please correct me if I'm wrong, that this
non-linearity would make it impossible to apply a uniform
pure fifth at all points in the scale without making excessive
compromises with other intervals.  A pure fifth temperament
might well be desirable on a low inharmonicity instrument
in a large hall, but I have trouble imagining this system as
something universally applicable to all pianos, or even to
all 88 notes of any single piano.

I'd love to be wrong, and I'm very interested in your research.
If anyone can convince me otherwise it is certainly you!  Best
of luck with your project.

Jerry Anderson