integral overtones

Natural systems have all sorts of wierd overtone sets. Nice uniform strings
will generate integer overtones. Nonlinear systems - systems with distortion,
like overdriven loudspeakers - also seem to generate integer overtones.

Systems with just one degree of freedom seem to be constrained to move
periodically, which forces integer overtones, as has already been pointed
out. One degree of freedom means a two dimensional phase space (the space of
combinations of position and velocity). In three or more dimensions one can
embed arbitrary graphs, but in two dimensions the possibilities are cut back
severely. In a similar way it seems like the trajectory of a simple
oscillator just doesn't have the topological freedom to fold around in
strange ways. I might well be wrong about this. (I should hunt around in the
literature but most of that stuff goes way over my head!)

But the tendency of nonlinearity to create integral overtones seems to go
beyond one dimensional systems. In a relatively calm ocean, waves of all
different frequencies run over each other freely. But once waves start
cresting, then distinct shapes start to appear and move through space
together. This sort of phase coherence is again a periodic motion that
implies integer overtones.

It does seem that nature has a certain prejudice in favor of integers!

Actually our common use of integers to count distinct objects seems related
to the kind of phase coherence created by nonlinearity. Nonlinearity makes
things clump together and split apart, and that's why there are things to
count. (And to do the counting!) Huh, and that's really the root of the
distinction between digital and analog circuitry - digital circuits use
distortion to make nice distinct pulses. (It's getting too late at night!)

Jim

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> I am pleased to read that others are at last contemplating that
> "harmonics" could be mapped at other ratios.
> Now let's have some viable mathematical models!

Bill Sethares has been studying that question in great detail, and
certainly has a very powerful mathematical model.

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There is also an article that provides a quantitative analysis of the
inharmonicity of piano strings:

AUTHOR: RASCH, Rudolf A.; HEETVELT, Vincent
TITLE: String inharmonicity and piano tuning.
SOURCE: Music perception, USA Vol. III/2 (winter 1985) 171-89.
LANGUAGE: English.
DOC TYPE: ap -- Article in a periodical
CLASS: MUSIC AND RELATED DISCIPLINES - Physics, mathematics,
acoustics, architecture
ABSTRACT: Inharmonicity is a well-known property of stiff strings such
as those used in the modern piano. The inharmonicities
measured in a medium-sized grand piano are in excellent
correspondence with the predictions by formula from the
physical properties of the strings. Strings with higher
frequencies usually have higher inharmonicity than strings
with lower frequencies; this cancels out part of the effect
of inharmonicity on beat frequency. (Authors, abridged)

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