Re: Digest Number 200

Paul E. Get well! I hope you are feeling better already.

Re Dante's pages: Has anyone succeeded in reproducing Helmholtz's
computations? I've always found his description of how he computed the
dissonance curves opaque.

I've always thought that HP merely took Helmholtz's first octave curve,
smoothed it a bit, then reflected it about the horizontal axes and
rotated the result.

--John

>From: John Chalmers <jchalmers@ucsd.edu>
>
>Re Dante's pages: Has anyone succeeded in reproducing Helmholtz's
>computations? I've always found his description of how he computed the
>dissonance curves opaque.

Although in H's time they hadn't actually mapped the frequencly response of
the basilar membrane, as far as I can see his understanding of the
mechinism was basically right on.
His computations that produced the dissonance curve were based on assuming
that each fibre of the membrane behaves like a stretched string, and fibers
being stimulated in close proximity (small interval) interfere with each
other's motion, hence producing roughness. The math is in appendix XV.

There are also cochlear echos which can be picked up by a microphone after
an impulse is played into the ear, usually after a delay of 60ms or so.
After playing two tones x and y into an ear, an echo may be detected at 2x-y
and may be audible to the ear owner as a combination tone. I had always
thought that conbination tones were purely subjective, but apparantly not.
(This may be apropos to the subharmonic thread - If a person is singing and
his ear is acually producing various combination tone echos, then the
situation of what the person may be hearing in their own head may be alot
more complicated, even before considering it as a chaotic system).

>
>I've always thought that HP merely took Helmholtz's first octave curve,
>smoothed it a bit, then reflected it about the horizontal axes and
>rotated the result.
>
>--John

In some ways they look similar, but seem to be scaled differently.
Especially in the semitone closest to the unison, Partch has that area
looking not much more dissonant than the next semitone, whereas in Helmholtz
the roughness there is precipitous.

dante