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Re: Helmholtz and combination tones.
Title: Re: Helmholtz and combination
It has been done. You get a stronger pitch percept when the
phases give strong envelope modulation, and less pitch percept
otherwise, such as for random phases. I don't have a ref handy,
but I saw a figure in something I was looking at just the other
John Pierce's 1991 JASA paper is another example of the pitch
depending on the phasing of high harmonics.
At 7:07 PM -0700 8/4/11, James Johnston wrote:
I don't know if it's been done, but it
would be interesting to see what happens when you set up the various
harmonics for a missing-fundamental probe in different phases, and see
what happens when they do and do not all cross zero at the same
On Thu, Aug 4, 2011 at 6:16 PM, William
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
PS Singularities in the amplitudes occur because there is no damping
in the dynamical equation and resonances are unbounded.
James D. (jj) Johnston
Independent Audio and Electroacoustics