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Helmholtz and combination tones.

Dear List,

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 tones."

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 oscillator.

Bill Hartmann

PS Singularities in the amplitudes occur because there is no damping in the dynamical equation and resonances are unbounded.