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From: zilla@nyit.UUCP (John Lewis)
Newsgroups: net.music.synth
Subject: Re: piano sound and stretched harmonics
Message-ID: <202@nyit.UUCP>
Date: Thu, 13-Feb-86 01:40:41 EST
Article-I.D.: nyit.202
Posted: Thu Feb 13 01:40:41 1986
Date-Received: Sat, 15-Feb-86 02:46:05 EST
Distribution: net
Organization: NYIT Computer Graphics Lab., Old Westbury, N.Y.
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[void]
The stretched harmonics in the piano have their origin in the
wave equation, modified to include stiffness and damping.
Both stiffness and damping contribute to the stretching in different
ways.  The effect of stiffness was studied empirically by Young
in JASA (52?).  The amount of deviation increases non-linearly with
the harmonic and if I recall it becomes more than a semitone above
about the 12th harmonic.  The stretching is greater in the non-wound
strings and probably accounts for some of the tinny piano sound
in the highest octave.  The effect of an assumed linear damping
factor is to linearly shift the whole spectrum down slightly,
which also stretches the harmonics.  A number of other deviations
from the ideal wave eq model are known which probably contribute
to the piano sound, for example, modes which are not present in
the original 'excitation' (hammering of the string) appear later
in a nonlinear model (transfer of energy between modes).  Also
the pairs or triplets of piano strings transfer energy and this
apparently is responsible for the piano envelope (articles in
Scientific American on the piano sound).

What is interesting is that stretched harmonics are found in any instrument
described by the wave equation, though they are more audible 
in struck-string instruments.  The non-harmonic spectrum may account
for the warmth or harshness of real vs. synthesized musical sounds;
most of the existing synthesis techniques produce a strictly harmonic
spectrum (including sampled wavetable techniques).

			j.p.lewis
			nyit computer graphics lab