(1) It seems strange to me for someone to say that musical scales and
their temperaments are tangential to the discussion of small
whole-number ratios when small whole-number ratios provide the
historical context for the development of the musical scale. The
original topic we were asked to comment on was why we don't have ratios
involving seven (8:7, 9:7, 10:7) in our scale. That means the orignal
question was, in fact, a question about scales and temperaments. It was
not a question about octaves, stretched or otherwise.
The simple answer to the question is that the 7th harmonic is way out of
tune with the nearest note one would choose based upon the lower
harmonics, which predominate and are more important.
For musical note C, the 7th harmonic ends up considerably flat of A# and
way sharp of A. It just doesn't fit in a musical scale based upon 2:3,
5:4, 4:3 ratios because new notes based on 8:7, 9:7, etc. are going to
beat objectionably when played with other notes in the established
scale.
So the problem is not how musical notes based on the seventh harmonic
will sound when played with the fundamental or with eath other, but how
they would sound when played with other notes in a scale consisting of
notes determined by 2:3, 5:4, etc. ratios.
If you fix musical note G as being the 3rd harmonic of C, and fix
musical note E as being the 5th harmonic of C, then if you base B upon
the 3rd harmonic of E and also the 5th harmonic of G, there is no
discrepancy. You get the same frequency for B regardless of whether you
figure it based upon E or based upon G. That's pretty amazing and is
the type of thing that led to the development of the scale we use today
even though we have ended up with a compromised scale.
(2) I repeat that the piano is the worst-case stringed intrument to pick
in terms of "inharmonic harmonics" because of its very stiff strings.
The historical context for the early scales was based upon harpsichords
and organs which were not "stretched" because they don't suffer from
anywhere near as bad of inharmonicities. Even the early pianoforte did
not use as stiff of strings as today's pianos.
There are even worse cases than the piano, but not involving strings.
One that comes to mind is carillon bells. The typical bell-shaped bell
is of course very stiff and has a very inharmonic 5th harmonic (which is
the basis for the third note on the musical scale). The 5th harmonic is
very flat in this case, so a music-scale third based upon this harmonic
is so flat it is nearly a minor third. Carillon players of traditional
carillons have to be very careful to avoid note parings that beat badly.
I said "traditional carillons" because a bell foundary in the
Netherlands is making carillons with bells newly designed by music
scientists at Eindhoven University of Technology. The bells have a
modified shape and have a harmonic series that is much closer to integer
multiples of the fundamental. These were featured in a Nova program
called "What is Music" that originally aired in 1989. I have an
off-the-air video tape of it, but it seems it is no longer availaable
from WBGH (Boston). Too bad, because it is a good show.
(3) Some musicians, perhaps especially violin players and singers, do
not like stretched tuning for a piano. They seem to hear and prefer
octaves as not stetched and this can lead to conflicts when they are
accompanied by a piano that is stretched. This means that some tuners
will not stretch a piano (or not stretch it as much) if it is not being
used as a solo instrument. In settings where pianos and organs are both
present and played together, it can be particularly tough for two
reason... (a) the organ is more true to the equal-tempered scale (not
stretched), (2) the air temperature changes the overall tuning of the
organ much more than it affects the tuning of the piano.
(4) When tuning a piano, you first tune the "temperament octave" (often
F below middle-C to F above middle-C) or sometimes a temperament tenth
(F to A) and this is tuned true, that is, not stretched. Thus, this
central octave is tuned true to the equal-tempered scale (or whatever
temperament you are using). Therefore this "termperment octave" on the
piano is indeed based upon the small whole-number ratios which have then
been altered to produce the desired temperament. After this middle
section is tuned first, the notes above and below are tuned by octave
relationships to the temperament octave, and this is when the stretching
occurs.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu