Inharmonicity in other instruments

[Robert Scott]

In writing about accordians, Bill Bremmer stated that "The reeds do have
some inharmonicity."  I have heard similar statements about pipe organs
and other instruments.  However, the fact is that inharmonicity is
a phenomenon that is unique to struck instruments, like the piano,
the harpsichord, the xylophone, bells, dulcimer, guitar, etc.  There is
no inharmonicity in instruments that produce a continuous sound
from continuous excitation, like the violin, the pipe organ, horns,
woodwinds, and accordians.

This can be seen if you examine what inharmonicity really is
and where it comes from.  Inharmonicity is the condition of
having partials that are not locked to the fundamental by
whole-number ratios.  In a piano and in other struck or plucked 
instruments, inharmonicity is possible because of the way in
which the energy is converted into periodic oscillations.  In
these instruments, an initial pulse of energy is applied by
striking or plucking something that has resonances.  Since the
initial pulse is not periodic, any frequency can be excited in a
resonant system (like a string).  In a piano string or in a bell,
the resonances that support the different partials are essentially
independent.  They superimpose on each other without really
affecting each other. That is why the second partial can "ring"
at a frequency that is unrelated to the fundamental.  What is
surprising about piano strings is not that they have inharmonicity,
but that the paritials are as close to true harmonics as they are.
In large bells, for example, the partials are so unrelated that
it takes careful machining of different portions of the bell to
separately adjust different partials so that they approach
harmonic alignment.

The situation is quite different with continuously excited
instruments.  In these instruments, the very mechanism of
injecting energy into the system is periodic.  In the case of
a reed instrument, the air forced over the reed causes it to 
vibrate at one frequency.  The reed cannot vibrate at several
frequencies at once.  So it vibrates at the fundamental frequency.
So where do the partials come from?  They come from the
fact that the vibrations of the reed are not sine waves,
and so by normal spectral analysis, they have harmonics.  These
are true harmonics that are locked to the fundamental by whole-
number ratios.  One interesting consequence of this analysis is
that a violin cannot exhibit inharmoncity when it is bowed, but
it can when it is plucked.

Now if you want to verify this with your ETDs, be very careful
about measurement error.  Many of these musical instruments are not
nearly as stable as pianos, so what you think is inharmonicity may
actually be variations in the fundamental.  As Bill Bremmer also
pointed out "When you play the bellows forcefully, the pitch will
flatten a good 4¢ or more."  If you have access to an oscilloscope,
you can also verify the total lack of inharmonicity by looking at
the waveform. If there is inharmonicity, you would see a waveform
that undulated as the partials ride up and down the fundamental.
Look at a piano waveform to see what I mean.  Then look at the
waveform of a continuously excited instrument.  If the waveform
is constant, then there is no inharmonicity.

Robert Scott
Detroit-Windsor Chapter, PTG