Klarinet Archive - Posting 000553.txt from 2001/02

From: Tony@-----.uk (Tony Pay)
Subj: Re: [kl] Difference Tones
Date: Thu, 15 Feb 2001 07:44:42 -0500

On Thu, 15 Feb 2001 06:44:16 -0500, bhausmann1@-----.com said:

> At 08:13 AM 2/15/2001 +0000, Tony Pay wrote:
>
> > On Wed, 14 Feb 2001 23:46:15 -0500, bhausmann1@-----.com said:
> >
> > > But all the evidence is that the walls, microphone diaphragms,
> > > etc., are vibrating AT 8 HZ! This would mean that SOMETHING is
> > > producing that frequency. If it is not the pipes, then what?
> > > Either these objects are responding to sound waves in the air
> > > produced by the organ pipes, or they are themselves creating the
> > > sound waves with their own vibrations. But if that is the case,
> > > what is causing them to vibrate AT THAT FREQUENCY?
> >
> > There's a third option other than your either/or.
> >
> > What causes the objects to vibrate at 8 Hz is the interaction
> > between one aspect of their properties (nonlinear response to
> > vibration), and the fact that they are being stimulated by two
> > frequencies that differ by 8 Hz.
> >
> > The thing to see is that something doesn't have to be *vibrated* at
> > a given frequency in order then to vibrate at that frequency.
>
> Is this not exactly what is happening to the AIR when the two pipes
> are played?

Perhaps my last sentence should have read:

> > The thing to see is that something doesn't *necessarily* have to be
> > vibrated at a given frequency in order then to vibrate at that
> > frequency.

The air, you see, is being vibrated at two frequencies, and the response
of the air is to vibrate at those frequencies, and at no other, *because
the air responds linearly*.

However, if that air vibration in turn stimulates a system that
*doesn't* respond linearly, other combination tones are generated in
that system.

The difficulty is that it's quite hard to have a simple feel for how
these combination tones are generated by the nonlinearity. (Indeed,
most of us probably don't have a very good mental picture of what
nonlinearity *is*.) The mathematics of Fourier transforms allows a
mathematician to talk about expansion of the response in a power series
that includes at least a term in x^2, perform self-convolutions, and,
bingo, there you are. But that's not very accessible to us. We just
have to take his word for it.

However, you can get a feel for the simpler phenomenon of the
generation of harmonics 2F, 3F etc in a nonlinear system excited by a
simple tone of frequency F. There is a simple mechanical model that
behaves nonlinearly outlined in Benade, beginning on p259, that does
this job. From there, the combination tones seem more plausible, even
if not fully explained; and in addition you see what sort of thing a
nonlinear system might be.

It's interesting that the model Benade chooses has a geometry that is
very like that of a reed vibrating on a clarinet mouthpiece. In this
geometry, the curvature of the mouthpiece facing means that the reed
moves against the facing in the 'closing' part of its cycle (so that the
point of contact is nearer the tip), and away from the facing in the
'opening' part of its cycle (so that the point of contact is farther
away from the tip). This arrangement means that the part of the
restoring force due to the springiness of the wood doesn't vary linearly
with the reed's deflection from its equilibrium position.

And indeed, this aspect of mouthpiece geometry is known to have a
significant effect on the existence and proportions of harmonics in the
sound of a clarinet. This is due to the precise nature of the
nonlinearity that it introduces into the system.

Tony
--
_________ Tony Pay
|ony:-) 79 Southmoor Rd Tony@-----.uk
| |ay Oxford OX2 6RE GMN artist: http://www.gmn.com
tel/fax 01865 553339

... Even a blind pig stumbles across an acorn now and again.

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