Klarinet Archive - Posting 000786.txt from 1998/12

From: Roger Shilcock <roger.shilcock@-----.uk>
Subj: Re: [kl] Clarinets, saxes, octaves, and the trumpinet....
Date: Mon, 21 Dec 1998 12:21:28 -0500

Let's not forget that this is about so-called "standing waves" (which,
nonetheless, can move) and (more or less) equilibrium cnditions.
Roger Shilcock

On Mon, 21 Dec 1998, Tony Pay wrote:

> Date: Mon, 21 Dec 1998 14:09:51 GMT
> From: Tony Pay <Tony@-----.uk>
> Reply-To: klarinet@-----.org
> To: klarinet@-----.org
> Subject: [kl] Clarinets, saxes, octaves, and the trumpinet....
>
> 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
> | |ay Oxford OX2 6RE GMN family artist: www.gmn.com
> tel/fax 01865 553339
>
> "'Tis with our judgments as our watches, none
> Go just alike, yet each believes his own."
>
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