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Helmholtz and combination tones.
A recent post from Randy Randhawa says, "Consider that even Helmholtz
had to appeal to non-linear processes (never really described) in the
auditory system to account for the missing fundamental and combination
Because this comment raises questions about what Helmholtz did and did
not describe, I would draw attention to Appendix XII in "On the
Sensation of Tone." There Helmholtz begins with the simple harmonic
oscillator dynamical equation and adds a quadratic term to the restoring
force, clearly conceived as just the second term in an expansion in the
displacement. He solves this to first and second order in small
quantities and finds that the second order term leads to combination
tones, which could include a missing fundamental.
An interesting feature of his solution is that summation tones are much
weaker than difference tones, which agrees with observation.
Specifically, for two frequencies f1 and f2, the summation tone
amplitude goes as 1/[(f2+f1)^2-fo^2] and the difference tone amplitude
goes as 1/[(f2-f1)^2-fo^2], where fo is the natural frequency of the
PS Singularities in the amplitudes occur because there is no damping in
the dynamical equation and resonances are unbounded.