Klarinet Archive - Posting 000783.txt from 1998/12
From: Tony@-----.uk (Tony Pay)
Subj: [kl] Clarinets, saxes, octaves, and the trumpinet....
Date: Mon, 21 Dec 1998 12:21:25 -0500
On Mon, 21 Dec 1998 08:46:39 EST, SDSCHWAEG@-----.com said:
> ....what I want to know is why a clarinet behaves like a closed pipe
> when a soprano saxophone doesn't? I mean, they look kind of the same,
> so why shouldn't they be similar acoustically? Obviously, I'm not
> much of a scientist - try to explain in not-too-technical terms,
> please!
I used to use this example as the beginning of a discussion of what we
mean by the word, 'understanding', in various contexts. (OK, I know,
the bullshit meter quivers.)
But seriously, it's a very interesting example of how mathematics is
unavoidable if you want to understand the world, after a certain point.
The 'reason' why a clarinet overblows a twelfth, and thereafter the
other odd harmonics, is that it approximates a tube closed at one end,
and open at the other. Now, the wavelength is the distance after which
the wave is the 'same' -- the distance between two wave-crests, or two
wave-troughs at sea, for example. So because 'open' and 'closed' are
'different', the tubelength can only correspond to the distance between
a crest and a trough. (The distance between two crests, or two troughs,
would be the distance between two things that are 'the same'.)
The first such possibility is half a wavelength (the distance between
one crest and the neighbouring trough), the second one and a half
wavelengths, the third two and a half, and so on. This means that the
frequencies of the possible modes of vibration are in the ratios 1/2, 1
and 1/2, 2 and 1/2... or, multiplying throughout by two, 1, 3, 5, ....
A flute, though, is open at both ends, so the aircolumn corresponds to a
wavelength (crest to next crest, or trough to next trough). Then the
next one is two wavelengths, the next three, and so on. So we get the
ratios 1, 2, 3, ....
This is 'sort of' intuitive. In order to make it rigorous, you employ
the following strategy, S.
S: write down a differential equation, called the wave equation, that
describes how a tube of air vibrates; then, in order to solve that
equation, impose the conditions that the tube is open at one end and
closed at the other, or open at both ends, translated into the
mathematics of the variables appearing in the equation.
Then the solutions just come out as I've described above.
Now, how about the soprano sax?
The difference between a clarinet or flute, on the one hand, and a
soprano sax or oboe, on the other, is that the soprano sax and oboe
approximate not cylindrical tubes, but conical tubes.
Can you give an intuitive justification of why a conical tube overblows
an octave?
I don't think you can, directly: but what you can do is talk about the
strategy S, in a hand-wavy sort of way.
The thing is that the equation of how a cylindrical tube of air vibrates
is quite simple. You don't have to include the notion that the diameter
varies along its length, so you can use what's called a two-dimensional
approach, instead of a three-dimensional approach. A conical tube,
however, *does* have a varying diameter, so it looks as though the
business is much more complicated.
However, because *how the diameter varies* is simple, you can write the
wave equation in a different form, so that it *becomes* simple again.
And then, it just *happens to turn out* that applying the conditions
that it's open at one end, and closed at the other, to this new
wave equation, gives the ratios 1, 2, 3, .... just like the flute, but
*for a different reason* (that is, because the equation has a different
form).
So, if you ask mathematicians, do you understand why an oboe overblows
like a flute? they might answer, yes; but their understanding is
mediated by their familiarity with wave equations in Cartesian
coordinates, for the clarinet, and spherical polar coordinates, for the
oboe and sax.
But is that understanding intuitive?
For them, yes; for us, no.
Tony
--
_________ Tony Pay
|ony:-) 79 Southmoor Rd Tony@-----.uk
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"'Tis with our judgments as our watches, none
Go just alike, yet each believes his own."
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