Brian Whatcott mentions that nothing was explicitly said about the
tuning of the lower octaves of a piano. He wondered if those octaves
might be shrunk as opposed to stretched. Here is some insight on that.
(1) The harmonic series "imperfections" are not as bad in the long
strings as in the short strings because: the effective length of the
string becomes shorter for higher harmonics, but the amount of
shortening is a smaller percentage of the overall length of the bass
strings simply because they are longer. Hence, the corrections needed
in the lower octaves of the piano are not as severe as in the higher
octaves. On an inexpensive piano (i.e. not worth too much tuning time)
the tuner might not bother to adjust the lower octaves. After all, on
a spinet piano (small upright) you don't get much bass sound anyway.
But on an expensive studio upright (tall), or good grand piano, the
lower octaves are adjusted. (A person used to playing on a spinet once
played my 7'4" grand and suddenly remarked, "does this have some sort
of bass amplification?" Answer: "Well, yes. It has a sound board
driven by nice long bass strings.")
(2) The lower octaves are not shrunk, they are also stretched. In the
wording I was using earlier, you have to tune the lower octaves more
and more flat as you go down the piano.
By the way, John Denker is correct... it's usually called "stretching"
rather than sharping the treble and flatting the bass. But when I am
addressing people who might not understand what stretching means, I
speak of tuning the upper note's fundamental sharp with respect to the
lower note's fundamental (or the lower note is tuned flat with respect
to the upper note.) When I teach this to music students at the
college, speaking in terms of sharp tuning or flat tuning instantly
conveys what is happening whereas stretching is a bit harder for them
to grasp.
When I responded to Denker and said I could call it sharp if I wanted
to, I was just being mischievous.
(3) It is interesting that there are published "formulas" for how much
a piano should be stretched, and some piano tuners adhere to a formula,
sharping their tuning instrument by a specific amount as they go up the
scale and flatting it a specific amount as they go down. (And, as
mentioned above, most formulas involve less flatting going down than
sharping going up.)
However, I don't use a specific formula and I don't recommend a
formula. I let the piano "tell me" how it "wants to be tuned."
Here's how you "talk to" your piano.
Using an electronic tuner, or by ear if you can and want to, tune the
middle octave to a perfect equal-tempered scale. When you want to
begin stretching (on the way up), use the already-tuned note an octave
below the note you're about ready to tune, to tell you how to tune the
upper note.
(1) Get the lower (already tuned) note to play it's second harmonic.
Do this by slowly pressing the key to raise its damper, but not fast
enough to make the hammer hit. That is, get its damper raised without
making a sound. Holding the lower note that way, hit a percussive blow
on the note one octave above that you are getting ready to tune
(immediately releasing the higher key, but keeping the lower key
depressed). The burst of vibration on the sound-board/bridge caused by
the upper note will cause the lower note to vibrate predominantly in
its second harmonic. The frequency you hear emanating from that lower
string(s) vibrating in its second harmonic is the frequency you want
for the fundamental of the upper note.
Since I have an electronic tuner, I don't tune the upper note to the
second harmonic of the lower note by ear. Rather, while the lower note
is sounding (in its second harmonic) I adjust the tuner so that it
detects that frequency as being "in tune." Then I use the recalibrated
tuner to tune the upper note.
It would take a long time to do this for every note. But once you do
this for a few notes all the way up (and down) to see how much stretch
this particular piano "wants," you can note the calibration settings
for those experimentally determined notes, then extrapolate the
calibration for the notes in between. If you take a good piano and
tune it this way, it sounds beautiful.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817