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Re: AUDITORY Digest - 31 Jan 2004 to 1 Feb 2004 (#2004-31)

Hi Neil and Jim,

My own discussion was sort of hooked on the phrase "precisely" in the
SciAm article that Jim cited.

In many practical situations we can do a whole lot quite well using
approximate methods. The tuning problem in musical acoustics is a great
example as Neil's wonderful bell-tuning or Orduna-Bustamante's (J. Acoust.
Soc. Am. 90, 1991, 2935-2941) and Bork's (Appl. Acoust. 46, 1995, 103-127)
numerical tuning of undercut marimba bars work shows.

To relate the geometric tuning problem to the inverse problem for geometry
only: Tuning means: give me a shape that will sound with these
frequencies. Inverse means: what shape or shapes (if any) will sound with
these frequencies. The tuning problem is satisfied if one shape is found.
The inverse problem is satisfied if all possible shapes (if any) are
found. The tuning problem hence doesn't have to be exhaustive. The tuning
problem is the inverse problem if the inverse is unique, but that's hard
to show.

In practice we may not care, because we have a bunch of considerations
that do help confine the search space to find one "suitable" solution. We
may only require the lowest couple of partials to be well tuned, which
further eases the problem, the instrument builder may worry about
minimum thickness, looks, etc to provide additional constraints that
reduce the search space.

 Finally let me quote another sentence from the SciAm article that started the
discussion. It contains also the sentence:

"Overtones are what distinguish a Stradivarius from an ordinary violin;
they add richness to the sound."

I find this wording confusing, because it suggests that an ordinary violin
lacks overtones and that this is the defining quality parameter. Of course
we know that to be nonsensical and I don't think that this is what they
mean to say. Maybe the danger of too much simplification in

- Georg

On Mon, 2 Feb 2004, Automatic digest processor wrote:

> ------------------------------
> Date:    Sun, 1 Feb 2004 20:23:13 -0600
> From:    beauchamp james w <jwbeauch@UX1.CSO.UIUC.EDU>
> Subject: Re: reverse engineering of acoustic sources
> Hi Neil and all,
> I think the upshot of the inversion problem is that if you have a
> reasonably good acoustic model to begin with, you can fill in the
> "blanks" (the parameter values) which will yield sounds that match
> the corresponding original sounds. (I've heard Neil's bells and they
> really do sound good.) But the trick is to find a plausible  model
> for an arbitrary sound, and I don't think an algorithm exists that
> will pull that out of the hat.
> A hypothetical question is: Can different musical instruments mimic
> one another?
> Jim Beauchamp
> Neil McLachlan wrote:
> >Hi all,
> >
> >I have tackled this inversion as a 3D design problem to generate the
> >geometry for bells with harmonic overtones and other tunings.
> >
> >We used gradient projection shape optimization methods to iteratively
> >alter a finite element model toward the desired overtone frequencies. As
> >you would expect from the earlier conversations there were multiple
> >solutions to the geometry of many of the bell tunings.
> >
> >The optimisation package 'Reshape' we were using is limited to linear
> >FEM problems, although I believe that the optimisation algorithms in
> >Ansys could be used for non-linear FEM problems. However optimisation in
> >Ansys lacks the spatial resolution possible in Reshape.
> >
> >You can see and hear the bells at www.ausbell.com and see my article in
> >JASA 114(1), 505-511
> >
> >-Neil McLachlan