Klarinet Archive - Posting 000956.txt from 2003/03

From: "Alan Woodcock" <alan.woodcock@-----.fr>
Subj: Re: [kl] Cylinders vs. Cones
Date: Mon, 24 Mar 2003 06:56:40 -0500

Why the clarinet overblows at the twelfth is pretty well explained on the
unsw site that Mark cites.
Why a "conical bore" overblows at the octave is less obvious.
In fact, if what I quote below is correct, most conical bores don't; some
non-conical bores might.

You may be interested in the explanation given by Ernest Ferron in his book
"Ma voix est un saxophone" (International Music Diffusion). M. Ferron is an
instrument repair and maintanance man, not a theoretical acoustician or
mathematician. What follows is (maybe mis-)translated and (perhaps wrongly)
abridged :

Take a cylinder closed at one end, with a fundamental at 50 Hz;
the third harmonic will be at 3*50 = 150 Hz (twelfth),
the fifth at 5*50 = 250 etc.
You have a pressure antinode *beyond the end of the tube* (end-effect).
Enlarge the open end. The antinode will approach the tube (intuitively you
have a less well-defined end, so less end-effect) and the frequencies will
rise, maybe a semitone; continue to enlarge the bore and the fundamental
tone will rise by a third, a forth... an octave ! The frequency of the
fundamental has doubled from 50 Hz to 100 Hz.

"Experiment shows that this gain of 50 Hz that we have obtained by making
our cylinder into a cone is a constant for all registers and we obtain :
50 + 50 = 100 Hz
150 + 50 = 200 Hz
250 + 50 = 300 Hz
We have a complete series of odd and even harmonics".

"But suppose we have done too well, and made too open a cone with a gain of
55 Hz, for instance; then the series obtained will be
50 + 55 = 105 Hz
150 + 55 = 205 Hz
250 + 55 = 305 Hz etc

"The instrument will be mistuned and difficult to play, as the registers are
no longer multiples of the 105 Hz fundamental...
"For any instrument, a bore having too open a cone will be too flat in the
(higher register).

Similarly if the cone is too closed...
"A bore having too closed a cone will be too sharp in the (higher register).

"This principle is more easily applied to brass instruments than to
instruments with a conical bore and lateral holes... (because of the mass of
the air in the holes) ... also the characteristics of the cone change every
time you open a key.

(and after more considerations on friction, heating and superposition of
vibratory systems we arrive at the conclusion that the cone will not have a
constant slope; in particular for the saxophone the bocal will not have the
same slope as the body of the instrument. In fact, the bocal is more of a
parabola than a cone).

So in summary:

Not all cones overblow at the octave.
It is not easy to make a conical woodwind which plays in tune.

Also note that the intuitive argument applies to enlargements of the bore of
any sort, not necessarily conical (for instance, oboe staples in cork
plugged into a clarinet body) so long as you don't expect to get a tuneful
scale out of the instrument. An oboe reed on a clarinet is not a
cylindrical instrument.

----- Original Message -----
From: "Mark Charette" <charette@-----.org>
Subject: RE: [kl] Cylinders vs. Cones

> > -----Original Message-----
> > From: Elgenubi@-----.com]
>
> > *I understand science, and am pretty good at math, but I have never
> > understood why a clarinet overblows the 12th. For all the
> > reasons discussed
> > already, there is no one line explanation.
>
> Visit http://www.phys.unsw.edu.au/music/ for some of the most lucid
reading
> on musical acoustics I've encountered.
>
>
>
> ---------------------------------------------------------------------
>
>

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