Of phlutes and fysics

How does a flute work? A flute is a fascinating physical system where chaotic turbulent oscillations of a blow of air get selectively amplified by a resonant column of air in an open tube. The pitch of the flute is not so much determined by the turbulence of the blowing of the flute, but by the geometry of the resonant column of air making up the body of the flute or whistle. (Note: a referee whistle is different: the pitch is determined for a large part by the blowing speed –it is a different mechanism.)

Design drawing by Fred Morgan after a Stanesby recorder. Thomas Stanesby and his son produced recorders in England in the early 1700s. The notes on this drawing seem to suggest that this particular Stanesby recorder was owned by Frans Brüggen, a famous dutch recorder player and conductor.

So how do we calculate the pitch of a flute of a certain length? Let us for simplicity assume that all the tone holes of the flute are closed, so the flute is essentially a tube with two open ends (one opening at the foot end of the flute, and one at the embouchure hole).

Because the pressure perturbations at those open ends must vanish (any pressure anomaly gets “dissipated” by sound radiation away from the ends), we find that the tube supports standing sound waves with pressure nodes at the ends. This then immediately suggests that the wavelength L of the resonant sound waves must satisfy L = 2l/n, where l is the length of the flute/tube, and n is any integer number.

The lowest pitch, corresponding to n=1, has a standing wave with half the wavelength fitting in the tube, l = L/2. This is the basic tuning of a flute.

I made a traverso flute myself from a PVC pipe and tuned its length, somewhat arbitrarily, so that the flute is pitched in D, corresponding to a frequency of around f=294–298Hz (depending a bit on how the flute is blown.) A digital tuner gives accurate determination of the pitch, and hence the frequency. (Anyone who plays a traditional flute such as an Irish flute or a Chinese dizi, would realize that this flute is effectively unplayable, because at this low pitch, the tone holes are too far apart to play without keys.)

Homemade PVC traverso flute, with stickers from the DIY store as “decoration” and white electricity tape covering the tone holes. On the table is also the Korg CA-1 chromatic tuner I used for frequency measurements.

With a soundspeed of c = 344m/s (this is accurate for room temperatures, allowing for a variation of perhaps 2m/s over different temperatures and humidities), the wavelength of my PVC flute sound is L = c / f = 1151–1173mm, notionally corresponding to a flute length of l = L/2 = 576–587mm, the acoustic length of the flute.

Now for the mystery: the measured flute length (embouchure hole – flute foot) is l=535–540mm (the embouchure hole is 5mm wide; the location of the plug in flute head is not crucial for the basic pitch of the flute). Taking this length at face value and applying our naive theory, the flute should sound a pitch somewhere between an E4flat and an E4, not a D4.

In other words, the acoustic length of the flute is more than 8% longer than the actual physical length of the flute; the flute sounds as if it was 8.5% longer than it really is.

Why is that? Some websurfing suggests “end effects”, essentially having the pressure node a bit further out of the tube. These end effects suggest a difference of around half the diameter of the tube, which clearly is nowhere near the observed length difference of about 40mm.

In fact, I spent a lot of time solving rather hard equations for standing waves in tubes of varying diameter (imagining that the flute end can be modeled as a tube with rapidly increasing diameter), and this fancy theory does indeed predict a difference in acoustic length and physical length of the tube, but the measured difference remains surprisingly large. It appears to me that the end effects cannot be the whole story.

The effect of tone holes to produce other pitches on a flute is even more enigmatic. The hole locations are not at all related to obvious locations of pressure nodes of the various pitches. Instead they seem to coax the flute into one resonance or another, rather than identify the locations of the pressure nodes.

As far as I am aware, the location of the tone holes in a recorder, or any woodwind instrument for that matter, have been determined empirically and modern recorder makers copy old tried-and-tested designs. It is something of a miracle that the 8 holes in a good quality baroque recorder can produce all the chromatic notes over about two octaves with incredible accuracy.

So we end up with a woefully inadequate/inaccurate naive theory (the one that is plastered all over the internet) and a much better fancy theory (it is too technical to include here), but still apparently insufficient to predict even the basic pitch of a flute.

A more successful conclusion is that, following my research in flute acoustics, I bought myself a treble recorder (having never played flute or recorder in my life) and am now practicing some Corelli sonatas. It is somewhat unnerving to play an instrument of which I do not understand the physics.