Inharmonicity in other instruments

[Robert Scott]

Tom, in answer to your question:

>I'm having difficulty understanding what you mean by "any frequency can
>be excited in a resonant system." Don't the resonances of a string limit
>the vibrational possibilities? I've always believed that stiffness was a
>major contributing factor in inharmonicity; that, along with striking
>the strings with hammers as opposed to plucking or bowing. What is it
>exactly that produces the harmonics? Don't the hammers simply excite the
>harmonics that result from the strings own "resonant structure?"

That's a very good description of inharmonicity and resonances.
What I meant by "any frequency can be excited in a resonant system"
was that the impulse from the hammer itself contains energy at all
frequencies at the moment of impact.  But then, as you pointed out,
only the frequencies of the string's own resonances are actually
absorbed.

To understand what actually produces harmonics, it is helpful
to know a little bit of Fourier theory.  This theory says than any
periodic waveform can be decomposed into the sum of sine and cosine
waveforms whose frequencies are all exact multiples of the
frequency of the original periodic waveform (the fundamental).
Our ears respond to these sine and cosine components, so we hear
them as separate sounds.  The loudness of the various harmonics
is related to the shape of the original periodic waveform.  
Flutes, for example, have fundamental waveforms that are very
close to perfect sine waves.  Therefore the harmonics are
almost non-existent.  A kazoo, on the other hand, is rich in
harmonics, and the waveform looks nothing at all like a sine wave.
That is why its Fourier decomposition has lots of high-frequency
components which we hear as harmonics.

But Fourier theory does not apply to the piano because the waveform
of a piano is not periodic.  To be periodic, a waveform must be the
same from one cycle to the next. But if you look at a piano waveform
on an oscilloscope, you will see that the waveform is constantly 
changing its shape.  That is inharmonicity at work.  The waveform
still has sine and cosine components, but these components are not
exact multiples of one fundamental frequency.  So we call them
partials instead of harmonics.

It might be awkward, but imagine that you could somehow get a 
bow to scrape across a piano string.  What would happen to the
inharmonicities that you knew were there when the string was struck?
They would vannish!  The partials would be exact multiples.  Now
suppose that while you were bowing the piano string you gently
lifted the bow off the string and allowed the string to ring
free.  The resonances of the string would then be free to vibrate
at their own natural frequencies and the inharmonicity would come
right back!

Robert Scott
Detroit-Windsor Chapter PTG