Subject: Re: Helmholtz and combination tones. From: James Johnston <audioskeptic@xxxxxxxx> Date: Thu, 4 Aug 2011 19:07:20 -0700 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>--000e0cd15228ba4d0d04a9b88e41 Content-Type: text/plain; charset=ISO-8859-1 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 time. On Thu, Aug 4, 2011 at 6:16 PM, William Hartmann <hartmann@xxxxxxxx>wrote: > 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. > -- James D. (jj) Johnston Independent Audio and Electroacoustics Consultant --000e0cd15228ba4d0d04a9b88e41 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable 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-fundamenta= l probe in different phases, and see what happens when they do and do not a= ll cross zero at the same time.<br> <br><div class=3D"gmail_quote">On Thu, Aug 4, 2011 at 6:16 PM, William Hart= mann <span dir=3D"ltr"><<a href=3D"mailto:hartmann@xxxxxxxx">hartmann@xxxxxxxx= pa.msu.edu</a>></span> wrote:<br><blockquote style=3D"margin: 0px 0px 0p= x 0.8ex; padding-left: 1ex; border-left-color: rgb(204, 204, 204); border-l= eft-width: 1px; border-left-style: solid;" class=3D"gmail_quote"> Dear List,<br> <br> A recent post from Randy Randhawa says, "Consider that even Helmholtz = had to appeal to non-linear processes (never really described) in the audit= ory system to account for the missing fundamental and combination tones.&qu= ot;<br> <br> 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 d= ynamical 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 th= e second order term leads to combination tones, which could include a missi= ng fundamental.<br> <br> An interesting feature of his solution is that summation tones are much wea= ker 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], wher= e fo is the natural frequency of the oscillator.<br> <br> Bill Hartmann<br> <br> PS Singularities in the amplitudes occur because there is no damping in the= dynamical equation and resonances are unbounded.<br> </blockquote></div><br><br clear=3D"all"><br>-- <br><div>James D. (jj) John= ston</div><div>Independent Audio and Electroacoustics Consultant</div><br> --000e0cd15228ba4d0d04a9b88e41--