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Saxophones, oboes and generally generally most woodwind instruments I can think of behave like open pipes. That seems logical; when every key is closed, they actually are open pipes and, if I remember correctly, a pierced pipe behaves like a (slightly longer) open pipe.

The clarinet, however, behaves like a closed pipe. This has two major consequence I can think of: the partials of a clarinet songs consists of odd harmonics (at least in first approximation); a clarinet register key makes the instrument play a twelfth (thrice the fundamental) instead of an octave (twice the fundamental).

What is the physical explanation behind this phenomenon?

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I believe you have some incorrect assumptions here -- the closed/open pipe phenomenon is readily demonstrated with boomwhackers, but does not apply here since all of the mentioned reed instruments are closed at one end and open at the other. The difference in register key interval has only to do with the location of the vent created by the register key. –  NReilingh 17 hours ago
    
@NReilingh You are wrong. Here is a picture of a clarinet’s spectrograph, on which you can see there is no even harmonics (hence no octaves) in it’s signal. Also, a flute is opened at both ends. –  Édouard 15 hours ago
    
It's just very unclear from your question what you mean by a closed or open pipe, and what characteristics they are expected to have. All keyed reed instruments that I am familiar with (flute, clarinet, saxophone) operate by changing the length of the tube by way of closing and opening keys, and changing register by venting the tube near a desired harmonic node. There are differences as you have noted in the harmonic profile of the clarinet, but it's unclear why that is expected to be dependent upon the tube-ness of the instrument and not other aspects of its construction. –  NReilingh 14 hours ago
    
I did some additional reading on this and I think I understand--the difference in behavior you're asking about is all to do with harmonics, yes? It would be good to include a reference at the top of the question to the known differences and definitions between closed and open pipes--clarity like that always helps get better answers. The first paragraph is still a little confusing to me, particularly the "pierced pipe" bit. Actually, I'm more curious now as to why oboes and saxophones don't behave like the "closed pipe" clarinet, in contrast to the open-pipe flute. –  NReilingh 14 hours ago
    
The answer by @charlieparker, albeit short and not going into the maths of it, is completely right. I was mistaken when I said that a sax or a oboe is open: it’s closed. But it behaves like a conical closed pipe. As for pierced pipes, I mean pipes with opened holes on the side, which IIRC can be quite properly approximated by opened pipes. I’ll try to detail my question sooner rather than later. –  Édouard 10 hours ago

4 Answers 4

Saxophones and oboes are conical, and behave like closed conical pipes. They are closed at the reed, just like the clarinet.

Flutes are cylindrical, and behave like open cylindrical pipes. The sound is made by blowing across the opening at the head joint, and it is not closed like in other woodwinds.

Clarinets are cylindrical like the flute, but closed at the reed, so they behave like closed cylindrical pipes.

Each category has its own set of unique characteristics. You can find a detailed explanation here: http://www.phys.unsw.edu.au/music/ . It goes more in depth than most sources, and requires that you are comfortable with the math involved in a typical university physics course.

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+1, but an important note of clarity is that the main point asked about in the question is about the harmonics for each type--and that both the conical and open-cylindrical instruments are similar in that regard (with only the closed-cylindrical instruments, like clarinet, behaving very differently). –  NReilingh 14 hours ago

The clarinet has a cylindrical bore, which makes it behave as a closed tube, odd harmonics.

See http://en.wikipedia.org/wiki/Bore_(wind_instruments)

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I guessed that it’s related to it’s bore, but why would a cylindrical clarinet still behaves like an stopped cylinder pipe instead of an open one? Isn’t it an open pipe? If it behaves as closed on the mouthpiece end, why don’t other reed instrument do the same? –  Édouard Jan 27 '14 at 22:08
    
The flute also has a cylindrical bore and behaves like an open pipe - this does not help much (and the wikipedia article also offers no additional explanation). –  guidot Jan 28 '14 at 12:47

The state of a sound pulse at a particular point in a tube any particular time may be characterized by the direction in which the pulse is propagating, the direction of displacement of the air from its "neutral" position, and whether the pressure is higher or lower than the neutral pressure. If air moves so as to cause an increase in pressure, that means that a high-pressure pulse is coming from the source of the higher pressure, which means the wave will travel in the same direction of the air. If the air moves so as to decrease the pressure, that means that a low-pressure pulse is coming from the side where the air is being drawn, which means the pulse will travel in the opposite direction of the air.

When a pulse hits the open or closed end of a tube, it will be reflected. Because the direction of wave propagation will be reversed, either the displacement or pressure (but not both) will also be reversed. An open end will reverse pressure but keep displacement direction; a closed end will keep pressure but reverse the displacement direction.

The resonant frequency of a tube will be determined by how many round trips are necessary to end up with a wave whose displacement and pressure match the original. If both ends reverse displacement or both ends reverse pressure, then after one round trip the displacement and pressure will end up matching the original. If one end reverses displacement, however, while the other end reverses pressure, then after one round trip both displacement and pressure will be reversed; a second round trip will be necessary to restore both to their original directions.

In a flute, the mouthpiece is constructed so that a high-pressure pulse arriving from within the tube will increase the amount of air which is deflected away from the opening, thus generating a low-pressure pulse in the same way as would an open pipe. A reed assembly which is constructed so that a high-pressure pulse will decrease the amount of air blown into the pipe will behave similarly. A reed assembly may be constructed, however, so that a high-pressure pulse will increase the amount of air blown into the pipe, in which case it would generate a high-pressure pulse whose displacement was the reverse of the original. Using such a reed in a nearly-closed pipe would cause a pulse to regenerate after one round trip, but in an open pipe two round trips would be necessary.

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When I asked this question to a math professor, his answer was: just because the differential equations modelling the air column resemble those of a closed tube. All other explanations offered here do not help much (at least in the presented summaries), since the bassoon, also with a conical bore and a stopping reed at the end overblows into the octave, and therefore also behaves like an open tube. So I would summarize to "there is no easy explanation, dive into advanced math".

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I'm not sure what you're saying here; the bassoon "behaves like an open tube" because it has a conical bore. The basic rule is that open tubes and conical-bore instruments behave similarly and stopped cylindrical-bore instruments behave differently from them; understanding why that's true requires a lot of math, but applying it doesn't. –  Micah Jan 28 '14 at 19:14
    
What I‘m trying to say is: 1) cylindrical or conical does not decide alone - see flute (where everything but the head is cylindrical). 2) The construction of the mouth-piece - however closed it looks to the eye also does not cut it - saxophone and clarinet are quite similar in that respect. So simple recipes like taking that bore and that mouth-piece will give that tube behavior still seems ambitious to me, even if one can learn it for existing instruments easily. –  guidot Jan 28 '14 at 20:45
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What's relevant is the combination of the mouth-piece and the bore. A closed mouthpiece and cylindrical bore lead to "clarinet-like" behavior (overblowing at the twelfth, small even harmonics, lower fundamental); any other combination leads to "non-clarinet-like" behavior (overblowing at the octave, large even harmonics, higher fundamental). So a flute is non-clarinet-like because it has an open mouthpiece (despite being cylindrical) while a saxophone is non-clarinet-like because it has a conical bore (despite having a closed mouthpiece). –  Micah Jan 28 '14 at 21:50

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