Subject: AW: Helmholtz and combination tones. From: "reinifrosch@xxxxxxxx" <reinifrosch@xxxxxxxx> Date: Fri, 5 Aug 2011 11:47:38 +0000 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>------=_Part_1576_32819571.1312544858509 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Hello! For those interested in combination-tone calculations: Chapter 43 ("Quadrat= ic and Cubic Difference Tones") of my book "Introduction to Cochlear Waves"= is based on Appendix XII ("Theory of Combinational Tones") of Helmholtz' b= ook "On the Sensations of Tone". In my text, misprints in Helmholtz' equati= on for x_2(t) are corrected, and the corresponding equation for x_3(t), co= ntaining the LCDT (lower cubic difference tone) is given. Reinhart Frosch, CH-5200 Brugg. reinifrosch@xxxxxxxx . ----Urspr=C3=BCngliche Nachricht---- Von: hartmann@xxxxxxxx Datum: 05.08.2011 03:16 An: <AUDITORY@xxxxxxxx> Betreff: Helmholtz and combination tones. Dear List, A recent post from Randy Randhawa says, "Consider that even Helmholtz=20 had to appeal to non-linear processes (never really described) in the=20 auditory system to account for the missing fundamental and combination=20 tones." Because this comment raises questions about what Helmholtz did and did=20 not describe, I would draw attention to Appendix XII in "On the=20 Sensation of Tone." There Helmholtz begins with the simple harmonic=20 oscillator dynamical equation and adds a quadratic term to the restoring=20 force, clearly conceived as just the second term in an expansion in the=20 displacement. He solves this to first and second order in small=20 quantities and finds that the second order term leads to combination=20 tones, which could include a missing fundamental. An interesting feature of his solution is that summation tones are much=20 weaker than difference tones, which agrees with observation.=20 Specifically, for two frequencies f1 and f2, the summation tone=20 amplitude goes as 1/[(f2+f1)^2-fo^2] and the difference tone amplitude=20 goes as 1/[(f2-f1)^2-fo^2], where fo is the natural frequency of the=20 oscillator. Bill Hartmann PS Singularities in the amplitudes occur because there is no damping in=20 the dynamical equation and resonances are unbounded. ------=_Part_1576_32819571.1312544858509 Content-Type: text/html;charset="UTF-8" Content-Transfer-Encoding: quoted-printable <html><head><style type=3D'text/css'> <!-- div.bwmail { background-color:#ffffff; font-family: Trebuchet MS,Arial,Helv= etica, sans-serif; font-size: small; margin:0; padding:0;} div.bwmail p { margin:0; padding:0; } div.bwmail table { font-family: Trebuchet MS,Arial,Helvetica, sans-serif; f= ont-size: small; } div.bwmail li { margin:0; padding:0; } --> </style> </head><body><div class=3D'bwmail'><P><FONT size=3D2>Hello!</FONT></P> <P><FONT size=3D2>For those interested in combination-tone calculations: Ch= apter 43 ("Quadratic and Cubic Difference Tones") of my book "Introduction = to Cochlear Waves" is based on Appendix XII ("Theory of Comb= inational Tones") of Helmholtz' book "On the Sensations of Tone". In my tex= t, misprints in Helmholtz' equation for x_2(t) are correcte= d, and the corresponding equation for x_3(t), containing the LCDT (lower cu= bic difference tone) is given.<BR>Reinhart Frosch,<BR>CH-5200 Brugg.<B= R>reinifrosch@xxxxxxxx .</FONT><BR><BR></P> <BLOCKQUOTE><FONT size=3D2>----Urspr=C3=BCngliche Nachricht----<BR>Von: har= tmann@xxxxxxxx<BR>Datum: 05.08.2011 03:16<BR>An: <AUDITORY@xxxxxxxx= L.CA><BR>Betreff: Helmholtz and combination tones.<BR><BR>Dear List,<BR>= <BR>A recent post from Randy Randhawa says, "Consider that even Helmholtz</= FONT> <BR><FONT size=3D2>had to appeal to non-linear processes (never reall= y described) in the <BR>auditory system to account for the missing fundamen= tal and combination <BR>tones."<BR><BR>Because this comment raises question= s about what Helmholtz did and did <BR>not describe, I would draw attention= to Appendix XII in "On the <BR>Sensation of Tone." There Helmholtz begins = with the simple harmonic <BR>oscillator dynamical equation and adds a quadr= atic term to the restoring <BR>force, clearly conceived as just the second = term in an expansion in the <BR>displacement. He solves this to first and s= econd order in small <BR>quantities and finds that the second order term le= ads to combination <BR>tones, which could include a missing fundamental.<BR= ></FONT><BR><FONT size=3D2>An interesting feature of his solution is that s= ummation tones are much <BR>weaker than difference tones, which agrees with= observation. <BR>Specifically, for two frequencies f1 and f2, the summatio= n tone <BR>amplitude goes as 1/[(f2+f1)^2-fo^2] and the difference tone amp= litude <BR>goes as 1/[(f2-f1)^2-fo^2], where fo is the natural frequency of= the <BR>oscillator.<BR><BR>Bill Hartmann<BR><BR>PS Singularities in the am= plitudes occur because there is no damping in <BR>the dynamical equation an= d resonances are unbounded.<BR></FONT><BR></BLOCKQUOTE><BR></div></body></h= tml> ------=_Part_1576_32819571.1312544858509--