Yes, the difference between harmonics and overtones is important.
Well, I suppose there is some importance there, but before we
take too many steps down that road, we should realize that there
is a completely different road available to us, a well-trodden
highway, simultaneously easier and more fruitful.
I recommend speaking about /resonant modes/. This is of course
related to the general idea of /resonance/ and the general idea
of /mode/. In common parlance the noun "resonance" is sometimes
used as shorthand for "resonant mode", in some contexts.
The term "resonant modes" is the physics term, It has some huge
advantages, including:
-- It treats all modes on the same footing. (This is in
contrast to the "overtone" idea, which is problematic
because the fundamental is more-or-less universally *not*
considered one of the overtones.)
-- It does not presume that the system is harmonic, or that
there is any harmonic relationship between the various
modes.
-- The same term can be used, with no change in meaning, in
electrical engineering, QM, classical physics, *and* the
physics of music.
-- It works fine for higher-dimensional systems, such as
Chladni plates and drumheads, where you need more than one
number to describe the mode (1s, 2s, 2px, 2py, et cetera)
... and you don't expect anything resembling simple integer
relationships between the resonant frequencies.
-- We can talk about the width of each resonance; we don't
need to assume the resonances have infinite Q.
-- et cetera.
I don't know of any downside to the idea (or terminology) of
resonant modes.
Many of the uses of the word "harmonic" are self-contradictory.
We do harmonic analysis to find the amount of THD (total harmonic
distortion) in a stereo system. But THD arises because the system
is anharmonic; if the system were harmonic, there wouldn't be
harmonic distortion. HOWEVER ... I can't get too excited about
this. I always start from the assumption that all terminology
is misleading. That way, if I find some terminology that isn't
toooo misleading, I'm pleasantly surprised. Don't expect the name
of a thing to tell you the nature of a thing. A titmouse is not
a kind of mouse. As Voltaire remarked, the Holy Roman Empire was
neither holy, nor Roman, nor an empire.
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The terms "overtone" and "harmonic" are at present so entrenched
in the physics-of-music community that we have to mention them
in any course on the subject. But they can be relegated to a
subsidiary role. Resonant modes can be used as the fundamental
idea, and everything else can be defined in terms of resonant
modes. This is analogous to the way that (until recently) we
had to mention "condenser" as a deprecated synonym for capacitor
... or perhaps better, analogous to the way that imprecise
notions of "heat" become irrelevant when they are supplanted
by precise notions of energy and entropy.
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Having said all that, there is a huge piece of physics that has
not yet been mentioned. There are quite a number of musical
instruments, including brass instruments and (!) the human
voice, where there are *two* things going on, and taking about
harmonics _or_ overtones _or_ resonances doesn't begin to paint
a correct picture of the physics.
To a useful approximation, when playing a long, constant note,
such systems can be modeled as
a) a periodic excitation, which is then acted on by
b) a set of acoustical resonant filters.
These two elements play by different rules:
a) The periodic excitation guarantees, by Floquet's theorem,
that *whatever* comes out will be periodic, with the same
period. As a corollary, we can apply Fourier methods. The
terms in the Fourier series will be related by simple integer
ratios.
a') Because the excitation is impulsive, more like a Dirac
comb than like a simple sinusoid, there will be many, many
terms in the Fourier series.
b) It is very common to find that the principal acoustical
resonances are not related in any simple way.
c) [See below.]
A lot of people get completely buffaloed by this; they have
trouble keeping both ideas (a) and (b) in their head at the
same time.
Voice is the best way I know to illustrate the relationship
between (a) and (b). On a single note (constant pitch) sing
ah eeee owe ooo. The formants (acoustical resonances) move
around, while the excitation remains essentially unchanged.
This looks good on a spectrogram. An not-very-pretty example
is: http://static.flickr.com/91/241586415_4ff3540df8.jpg
I suppose _muting_ a horn would be another way of illustrating
the idea, but I've never actually done that experiment.
There are various software packages out there for computing
spectrograms in near-real time. I haven't evaluated them.
By the way:
c) The excitation (a) is not independent of the acoustical
resonances (b). Typically one of the acoustical resonances
(/not/ necessarily the fundamental) plays a dominant role
in setting the frequency of the excitation. In a brass
instrument, you can "lip" the excitation away from where
the acoustic resonance would naturally put it, but you
can't lip it very far.