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A flute is an open cylinder air column instrument. This means that an idealised flute, the fundamental pitch of the flute should have a wavelength of twice the length of the flute.

This isn't exactly the case with a real flute though. For example, a standard D4 shakuhachi is around 54.5cm, but the wavelength of D4 is 117.48cm. Double the length is 109cm. This is about 7.8% shorter than ideal (so it should be ~1.5 semi tones high), but the flute still plays in tune. My shakuhachi is 53.3cm and it's still in tune - in fact, if anything it's slightly flat. If you account for the utaguchi the flute is about 5mm shorter again.

I'm not actually sure if Boehm transverse flutes have this difference - I just measured a C flute, and the full length of the sound chamber is about 67cm (slightly longer than the idealised 65.93cm), but the length from the mouth piece to the end is only 60cm, and I'm not entirely sure how the idealised tube length should be defined (I guess it's the length of the cavity, rather than the distance between openings). If it's from the mouthpiece, then the difference is even bigger, almost ~10%.

The shakuhachi has a slightly tapering bore, which a Boehm flue doesn't, so I'm wondering if it's related to that, but I don't understand how it would be. Is there some effect of the narrow bore on the speed of sound? Or is it because the sound actually bounces down the flute at a slight angle, thus lengthening it's path? Are there other parameters that affect the fundamental?

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2 Answers 2

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The vibrating air column in the bore extends past the openings of the tube. Any calculation of the pitch needs to take this into account and add end corrections for both ends of the flute. There's one summary of the calculations here, and a Google search for 'flute end correction' turns up plenty of others.

The open end correction (i.e. the length that has to be added to the physical length for the frequency calculation) for a typical flute of radius r is approximately 0.6r.
There will also be an end correction at the blowing end, but the shakuhachi embouchure complicates the calculation. According to this paper the blowing end correction for a shakuhachi is about 5mm.

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    Two links in previous comments mentioned end correction, but even that only increases the shakuhachi effective length by ~0.6cm, which makes it still 6% too short. I also just finished making a PVC flute in A(440), and it is 71.6cm with a cylindrical 24mm bore, and in good tune. Even with the end correction, it's only effectively 72.3cm, far shorter than the 78.41 I would expect
    – naught101
    Apr 2, 2022 at 13:15
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    @naught101 At least one researcher has confirmed the same effect: that the observed end correction much larger is than the calculated one: isjos.org/JoP/vol3iss1/Papers/JoPv3i1-2Recorder.pdf. This was for a recorder, but the slightly reverse-conical bore is similar to a shakuhachi's.
    – PiedPiper
    Apr 3, 2022 at 11:23
  • I've only seen images of shakuhachi but they seem to have a slight bell. This suggest The Bell Effect is probably somewhat significant. Apr 3, 2022 at 11:42
  • @ElementsinSpace There's no bell. The walls are much thicker at the end and the bore is actually narrower there.
    – PiedPiper
    Apr 3, 2022 at 14:03
  • @PiedPiper — oh okay, interesting. Apr 3, 2022 at 14:26
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I'm afraid, that the question boils down to Why are real instruments more complicated than extremely simplified theoretical models?

You need the simplifications to be able to calculate something at all, since the complications are staggering. Friction of the air at the instruments body, different speeds of air depending on location, huge impact by big diameter of tone holes respective to diameter of bore when it comes to wind instruments (since without holes it would not be an interesting instrument), thickness and mass of strings of the string instrument department - all these topics fill complete books. For wind instruments acoustic impedance is a good point to start research.

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    Yes, I understand that - that's why I expanded from the title. The last paragraph is the main question - essentially "what is a more complex model that can account for this difference?" Your last sentence is very useful though! From that I found this: newt.phys.unsw.edu.au/jw/AT/#end which is very cool. It says that a flute in air actually has an effective length of L+ 0.6r due to end effects. That doesn't account for much of the difference I'm seeing though.
    – naught101
    Apr 2, 2022 at 10:27

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