Scientists use these definitions of resonant systems:

    Fundamental: the lowest natural resonant frequency

    Overtones:   the remaining frequencies above the fundamental

    Harmonics:   Overtones which are an integral multiple of the
fundamental frequency (2,3,4, etc).

I think in the piano trade "Partials" is used like "Overtones".

An air column resonator, as in an organ pipe, has overtones which are
truly harmonic, due to the coupling between the modes.  Wurlitzer
"violin" pipes featured the "Gavioli Frein", a blade of brass
intercepting the wind sheet near the mouth of the pipe.  ("Frein" is
French for "brake".)  When the little frein is properly adjusted the
pipe will always produce the fundamental tone, and increased pressure
just enriches the harmonics.  Without the frein the pipe would cease
oscillating at the fundamental.  Another open pipe design intentionally
suppresses the fundamental because of a tiny hole half-way up the
resonator.  (David Wasson, can you elaborate on these statements?)

The vibrating string in the piano has non-harmonic overtones.  When the
hammer strikes the string all of the possible modes are excited.  There
is inadequate coupling to synchronize the different frequencies, and so
they are "out-of-tune" with each other.  Moreover, the sound amplitude
of each overtone can decay differently from all the others.

A common experiment in the college physics lab uses a thin string of
music wire stretched across a heavy cast-iron base, as in a piano.  The
students measure the frequency of each of the resonance modes, aided by
an oscilloscope.  The instructor waits patiently for the question, "Why
aren't the overtones harmonic?"  Then he explains that the cast-iron
frame, seemingly stationary, is also vibrating with the string,
especially at the higher frequencies.  The result of this undesirable
motion is that the overtones resonate at frequencies a little less than
an integral harmonic.  In radio antennas this is called the "end
effect".

Scientists "build a model" in order to validate a theory.  The
mathematical model of the vibration of a struck piano string is quite
complex -- but the more complex it is, the better it approximates the
real-world action of the piano.  By studying the model (and perhaps
using it as the basis of a computer simulation) scientists have been
able to achieve better pianos, and faster than if only "cut and try"
experiments were employed.

I suspect that the strides in piano technology made by Steinway & Sons
a century ago were due to both analytical studies and empirical
trials.  _Scientific American_ magazine frequently publishes
interesting articles, adapted from the scientific tomes, about musical
instrument science and technology.

Observed natural phenomena may be explained using a mathematical
model, but there is surely nothing wrong in expressing the conclusions
in the everyday words of the musical world.  Thus the "warm tone" of a
given piano or violin is explained quantitatively by the mathematical
model, which shows that certain overtones dominate, or whatever. ...

Aren't pianos wonderful?  The sound of the violin may tug at your
heart, but the hammered piano hits you in the belly!

-- Robbie Rhodes  <rrhodes@foxtail.com>