This is posting-1 of two different comments I have on this topic. The
second posting will follow, as soon as I write it.
First, John Denker is correct that piano octaves are stretched. But the
greater departure from the scientific scale of small-integer ratios is
the use of the equal-tempered scale. Literally nothing on the piano is
exactly small-integer ratios.
However, John's comment that small integer ratios has little to do with
harmony, consonance, and dissonance is too strong in my opinion. We
eventually ended up with the equal-tempered scale, but this was the
outgrowth of much experimentation that took place trying to develop a
scale that was mostly consonant, and that work did indeed center on
figuring ways to come as close as possible to small-integer ratios.
John already alluded to the main component when he mentioned that the
second harmonic of the lower note in an octave needs to "sound in tune"
with the first harmonic of the higher note in an octave. But John
didn't mention what in-tune or out-of-tune means, or more explicitly, in
what ways do in-tune and out-of-tune sound different? The key component
is beats. Are beats present; how loud are they; what is the beat
frequency?
Audible beats when an octave is played is a clear indicator that the
octave is out-of-tune. A perfect 2:1 ratio for the first harmonics of
both notes of an octave on a piano is not the best tuning because the
dominant beat, if it is present, is usually between the 2nd harmonic of
the lower note with the 1st harmonic of the higher note. These
harmonics need to be 1:1 exactly, or we get beats and it sounds out of
tune. Note that on some strings and/or on some pianos the beat of the
4th harmonic of the lower note with the 2nd harmonic of the higher note
can be pronounced. It doesn't really matter whether you are doing 2:1
or 4:2 or 6:3 tuning of the octaves, the goal is to eliminate the
audible beats. If this cannot be done, get the beats as soft as
possible (which means deciding whether 2:1 beating or 4:2 beating is
more objectionable) and also get them as slow as possible.
For the musical interval of a just-scale fifth, the frequencies are
indeed in the ratio of 2:3 if stretching is not considered. However,
the beating we hear is again not the fundamentals beating, but the third
harmonic of the lower note beating with the second harmonic of the
higher note. To get these matched, some stretching is needed, but the
stretch is not as much between the 2nd and 3rd harmonic as between 1st
and 2nd, so a musical fifth sounds reasonably nice even when tuned
nearly exactly to the 2:3 ratio. Of course switching to the
equal-tempered scale, we do not use the exact 2:3 ratio but
equal-tempered sticks very close to 2;3 because this interval "gets ugly
fast" if it isn't pretty close. In equal temperament the beats are
audible, but as long as the beat frequency is pretty slow (which means
close to perfect) the beats are not perceived as objectionable.
It would take too much time to analyze all the notes in the scale, but I
think it is an overstatement to say that small whole-number ratios are
not involved. In general, it has been determined that consonance means
lack of objectionable beats. Beats cannot be eliminated, but can be
minimized. To minimize them, first choose a scale in which the notes
have small whole-number ratios, then adjust them for minimal beating.
Also, if you are musical or a critical listener, and you listen to
simultaneous pure tones from sine-wave generators, you will probably
still come to the conclusion that small whole-number ratios sound good
together. Here we don't have harmonics beating, so it is the phase that
is shifting and is noticed by some. It is somewhat interesting to
determine if musicians like a fifth (when listening to pure tones)
because they are used to it, or because they notice the phase. If they
habitually select an equal-tempered fifth as best sounding, it is
probably because they are used to it. If they choose the true 2:3 ratio
they are probably listening for the phase change.
When you switch to more natural sources of sound, such as strings, where
harmonics are involved, the harmonics need to sound good together as
well as, or even more than, the fundamentals. This again would point to
small whole-number ratios if the harmonics did not suffer inharmonic
problems. Also note that when speaking of the inharmonicity of strings
due to string stiffness, the piano is the worst-case scenario because it
utilizes the stiffest strings, by far. A similar instrument (because it
has full scales of strings) but does not have nearly as stiff strings
(and doesn't get stretched-tuning as much), is the harpsichord. And the
harpsichord is mostly what J.S. Bach and his contemporaries used when
they were experimenting with various scales. Bach also experimented
with organ tuning where different stops have different harmonic
content... a very complicated prospect.
BTW, when I tune pianos, I use a strobotuner because for me it takes too
long to count equal-tempered beats and/or I don't do it enough to be
good at it. But I am pretty effective with the electronic tuner, and I
let each piano tell me how to stretch it by recalibrating the tuner by
watching the harmonics. Some people feel that the "bible" for tuning
(and rebuilding) pianos is "Piano Servicing, Tuning, and Rebuilding" by
Arthur A. Reblitz. He has a pretty extensive section devoted to
teaching the reader to tune by ear by counting beat frequencies of the
harmonics.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu