I've been trying to build a double reeded instrument out of plastic straws. I've run into a problem though, when I place fingering holes, the instrument doesn't seem to follow the $$f=/frac{nv}{4L}$$ formula. Is there a formula that would allow me to calculate where along the instrument I should place the holes?

  • I guess the answer is guess! :D It may also be trial and error. – xilpex Apr 3 '19 at 23:44
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    It seems that MathJax (the fraction code) isn't supported on Music SE. – Bladewood Apr 4 '19 at 10:36

For a pipe without any holes, the fundamental pitch of a pipe is determined by f = v/2L, where v is the speed of sound and L is the length of the pipe.

But when you start putting holes in, you get a mix of pipe lengths. It gets messy (mathematically) in a big hurry.

Placing a hole in the pipe shortens its length, but the new length - the effective length - isn't the distance to the hole, because that isn't the end of the pipe. The larger the hole, the more it will behave like the ideal. The smaller the hole, the longer the effective length will be.

Because the hole size is a variable, you're not going to find a formula that's going to fit every situation. That's because the hole size is a variable in relation to the diameter of the tube: a 1cm hole in a 10cm tube will have a different effective length than a 1cm hole in a 9cm tube.

Since no formula is going to work for all situations, you have do some trial and error. If the pitch is flat, you can enlarge the hole to shorten the effective length. If the pitch is sharp, you'll have to figure out a way to make that hole smaller (or make the whole tube longer - there's a reason woodwinds have multiple pieces!)

There are other variables, too... conical bores behave differently than cylindrical ones. But I'm assuming you're using a cylindrical tube.

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  • “In a perfect world” – that's a bit of a silly way to put it. You might also say in a perfect world the air flow is perfectly linear and laminar – however, in that case the mechanism by which flutes bring the air to vibrate in the first place also wouldn't work! And even assuming you get the air to vibrate in the perfect cylinder: nobody would be able to hear the sound, because all the energy is trapped! I wouldn't call that perfect. No, the world is exactly as good as is actually happens to be – what's imperfect are the models with which we describe certain phenomena in the world. – leftaroundabout Apr 4 '19 at 11:45
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    "In a perfect world" is a common idiom that means a world that conforms to the model in use. You're taking it too literally. – chepner Apr 4 '19 at 11:56
  • @leftaroundabout I changed the answer -- see if it is more readable now. – Carl Witthoft Apr 4 '19 at 13:04
  • @CarlWitthoft - I altered your edit. Helmholtz resonance is the frequency created by air "bouncing" at the mouth hole. It contributes to the overall timbre whether there are holes or not, and introducing holes does not change the Helmholtz resonance - that's fixed by the size of the mouth hole and the thickness of the tube material. Open holes don't have Helmholtz resonance frequencies, because the air pressure in the tube is always higher than that outside; the air simply escapes without additional vibration. – Tom Serb Apr 4 '19 at 13:47
  • @TomSerb thanks for doing that. I should have emphasized where Helmholtz was applicable (see "Ocarina" :-) ) – Carl Witthoft Apr 4 '19 at 14:34

The main problem is that it's an oversimplified assumption to consider an open hole as a perfect open boundary condition for the air column. In fact such a hole still has a significant impedance. On the other side, the mouthpiece is not a perfect closed (reeds) or open (flutes) boundary condition, and also a closed hole still affects the column somewhat. A pitch formula would need to take all those factors into account, which depend on hole size and bore. Doing this accurately would require a big CFD model.

In practice, probably almost every wind instrument maker has instead used empirical models, i.e. basically trial and error. It should certainly be possible to fit a formula to that which is more accurate than the overidealisation but still reasonably accurate, but whether one is available publicly I don't know.

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  • I suspect that Adolphe Sax used the same spacing ratios in (soprano, alto, tenor, bari) at least as a starting point, given the similarity in the bore shapes. – Carl Witthoft Apr 4 '19 at 13:06

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