Helmholtz and combination tones.

[William Hartmann <hartmann@xxxxxxxxxx>]

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.