Natural Harmony

The August 1997 Scientific American has a one page geophysics news note on
page 18. Folks have been doing more and more analysis of sounds in the
ocean. Somebody found a tone without any harmonic overtones floating around
out there. The article points out that there's a more common class of seismic
signals called harmonic tremors with a nice overtone series.

Strikes me a tuning FAQ ought to address the question, are integer overtone
series natural or artificial, where do they come from?

Here are the three ways I see integer overtones arising:

The sound generated by a periodic motion will have integer overtones, by some
classical theorem of Fourier or whoever. Furthermore, a system with one
degree of freedom is pretty much constrained to move in a periodic fashion,
at least for bounded trajectories.

Ideal one-dimensional systems of coupled linear oscillators, like vibrating
strings or columns of air, will have an integer overtone series.

Many types of distortion generate integer multiples of the signals that are
getting distorted. These distortion mechanisms also generate the sum
difference frequencies when excited by several incoming frequencies all at
once. This is the basis of heterodyne circuits in radios, the Theremin
musical instrument, etc.

The next FAQ question could be, do systems with non-integer overtones also
arise?

Answer, they're everywhere. A real stringed instrument, because the strings
are not quite uniform and the endpoints are not quite rigid, will not
generate exactly integer overtones. Two dimensional systems like drumheads
produce powerful overtones that are nowhere close to integer multiples of the
fundamental. And anyway all real systems are actually three dimensional.

I wonder just how close to integer multiples are the overtones of those
harmonic seismic tremors?!

Other basic FAQ material - unique prime factorization, Fourier analysis,
logarithms.

BTW Arthur Benade's book FUNDAMENTALS OF MUSICAL ACOUSTICS (which Dover
republished in 1990) is a wonderful intro to the physics of overtones.
See also his HORNS, STRINGS & HARMONY.

A pretty wild trip can be induced by perusing DYNAMICS: THE GEOMETRY OF
BEHAVIOR by Ralph Abraham and Christopher Shaw. Lots of pictures, not very
many math formulas, but still very sophisticated ideas. This kind of
geometric dynamics is where my "one-dimension implies periodic" argument is
trying to come from.

Jim



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