11.1 The world of wind instruments

We have said a little about wind instruments in earlier chapters, but now it is time to go into more detail. It is useful to start by comparing the wind instruments with the bowed-string instruments. Both types of instrument are capable of producing sustained tones, and we learned back in section 8.1 that this automatically means that they must all involve nonlinearity in some essential way. In the case of a bowed string, the main source of nonlinearity was from the frictional interaction of between bow-hair and string. In the wind instruments, the nonlinearity is associated with things going on near the mouthpiece end, which we will discuss in detail shortly.

An obvious difference between stringed instruments and wind instruments is that wind instruments usually have no equivalent of the stringed instrument body, which influences the sound of the instrument by imposing an extra stage of filtering between the string motion and the radiated sound. In most wind instruments, the sound is radiated directly by air motion in the tube which is the analogue of the string: this tube provides a set of resonances, sometimes approximately harmonically related, which interact with the nonlinearity to produce the pitch and waveform of the played note.

There are two other, less obvious, differences between bowed strings and wind instruments. First, the resonances of air in a tube are usually much more highly damped than the resonances of a string. A violin string might have Q factors of the order of thousands, whereas tube resonances are more likely to have Q factors of the order of hundreds or less. The second thing is that the nonlinearity associated with friction is (as we saw back in Chapter 9) a very severe one, whereas wind instrument nonlinearities tend to be less vigorous.

If you recall the distinction between “smooth” and “non-smooth” nonlinearities from section 8.2, this suggests that rather different mathematical approaches might be called for in the two cases. The big disparity in Q factors also pushes us in a similar direction. The result is that the modelling of wind instruments places a lot of emphasis on periodic regimes of oscillation, thresholds, and bifurcations between regimes. With the bowed string, we saw that there was no useful concept of a threshold of vibration, where the motion is “almost sinusoidal”: a bowed string goes into strongly-nonlinear stick-slip vibration straight away, and theoretical treatment tends to rely more heavily on “brute force” numerical simulation.

The next step is to sort the enormous diversity of wind instruments into a few general categories. If you look at the wind instruments in a western symphony orchestra or wind band (see Fig. 1 for an example), a three-way classification seems self-evident. There are the reed instruments (clarinet, oboe, bassoon), the flute-like instruments (flute, piccolo) and the brass instruments (trumpet, trombone, French horn). But if we cast our net a little wider than the symphony orchestra, we soon find out that things are more complicated and less clear.

Figure 1. A wind orchestra, showing many different orchestral wind instruments. Image: Julien Bertrand, Public domain, via Wikimedia Commons

For a start, by focussing on orchestral instruments we are missing one category of instrument entirely: the “free reed” instruments, like the harmonium, accordion or harmonica. This gives a four-way classification, which will turn out to allow us to pigeon-hole virtually all wind instruments (including the human voice). However, the categories are not always as distinct as you might think. It is important to be clear what the basis of the classification is. One thing that definitely does not influence the classification of instruments is the material they are made of. Despite the names, some “woodwind” instruments are made of metal while some “brass” instruments are made of wood.

Rather, the key observation is one we have already mentioned: all wind instruments are capable of producing a sustained tone, so they must all involve nonlinearity in some essential way. The four-way classification separates instruments based mainly on the details of the nonlinearity that allows them to work. Three of the four are related, to the extent that they blur into each other at the boundaries. These are the reeds, the brass and the free reeds. The fourth member really is different, though. Flutes and their relatives are driven by an air jet interacting with a sharp edge. This excitation mechanism involves no mechanical moving parts, unlike the reeds or vibrating lips of the other three categories of wind instrument.

I will give a brief description of the characteristic features of the four categories of instruments here, but more detail will be given in separate sections on each type: sections 11.3—11.6. Figure 2 shows a sketch of the mouthpiece end of a clarinet-like reed instrument. The player’s lips seal round the mouthpiece and reed, and then the player applies a suitable blowing pressure so that air flows through the narrow gap. As we already discussed briefly in section 8.5, a crucial feature of this kind of backward-facing reed is that the player’s blowing pressure tends to close the reed. If the pressure difference between the mouth and the inside of the mouthpiece gets too big, the reed closes completely against the rigid part of the mouthpiece (the “lay”) — or at least, almost completely. In reality there will be a bit of leakage because the reed and the lay do not meet perfectly.

Figure 2. Sketch of the mouthpiece end of a reed instrument like a clarinet.

Of course, the reed has a resonance frequency of its own. Indeed, it has many resonances if we look sufficiently high in frequency, but the lowest resonance is the most important for the normal musical functioning of a clarinet. Even this lowest resonance is usually placed higher than the fundamental frequency of any played note on the instrument, so reed resonance is a relatively minor contribution to the behaviour of the instrument. Much more important are the resonance frequencies of the tube, which the player manipulates by opening and closing tone-holes. The playing frequency of any particular note is predominantly governed by these tube resonances.

All this is in sharp contrast to our second category of instrument, which we will continue to call “brass” instruments despite the comment about materials made above. Figure 3 shows a sketch of the mouthpiece end of a typical brass instrument. The tube is terminated by a cup-shaped mouthpiece. The player presses their mouth firmly against this cup, and they “buzz” their lips. This makes the lips open and close at the frequency of the note being played, and high-speed video recordings of brass players [1] show that, when air is flowing out of the mouth into the instrument, the lips open outwards as sketched in the figure. So the lips act in a somewhat similar way to a clarinet reed insofar as they provide a kind of nonlinear valve controlling air flow into the instrument, but we will see in section 11.4 that the behaviour is crucially different because of the outward-opening motion: higher pressure in the mouth tends to make the gap get bigger, not smaller as in a clarinet.

Figure 3. Sketch of the mouthpiece of a brass instrument, with the player’s lips pressed against it and slightly opened during a “buzzing” cycle.

The tube of the instrument still has resonances, of course, but they do not dominate the playing pitch to the same extent as in a reed instrument. The brass player has to adjust their lip tension (“embouchure”) in order to achieve the correct buzzing frequency for the desired note. The tube resonances help a lot: players describe the sensation of the pitch being “slotted”. But a sufficiently skilful and forceful brass player can coax many pitches out of an instrument, especially when the resonances do not provide much support — we will explain more in section 11.4. This leads to some brass players’ party tricks: playing tunes on unlikely “instruments” such as vacuum cleaner hoses, or (particularly striking) holding a steady, unchanging note on a trombone while moving the slide in and out!

Things are different again for the third category of instruments, the free reeds. Figure 4 shows a sketch of a reed from a harmonium or accordion. A thin cantilever beam, usually of brass, is fixed to a heavier brass plate, over a slot which is just a little bigger than the reed. As the reed vibrates, it can move through this slot. If the amplitude is large, it will move away from the plate on both sides at different stages in the cycle. As in the instruments we have already looked at, the player creates pressure on one side of the reed plate, and the moving reed provides a nonlinear valve between the two sides of the plate. But this time, that valve opens wider on both sides. In both the other cases, the valve tends to open when air flows one way, and to close when it flows the other way: the difference between the two cases lies in which way round these two things happen.

Figure 4. Sketch of a single free reed, riveted to its reed plate above a slot.

The other big difference between a harmonium or accordion and any of the brass or reed instruments is that it has no resonating tube. The playing frequency is usually very close to the resonance frequency of the reed. But now we can see an example of blurring between these categories of instrument: many Asian “mouth organs”, like the ones seen in Fig. 5, do have resonating tubes. These are still formally classified as free reed instruments, but the playing pitch (and the sound spectrum) is influenced by these tubes. A related effect may be more familiar to western audiences: a harmonica player is able to “bend” notes, a particularly vital ingredient of performance technique for blues players. As will be explored in section 11.5, this is achieved by manipulating a different source of resonance: the player’s mouth cavity and vocal tract.

Figure 5. Hmong musicians perform on traditional free-reed instruments in Upper Lang De in Guizhou, China. Image: Michael Mooney from Chicago, USA, CC BY 2.0 https://creativecommons.org/licenses/by/2.0, via Wikimedia Commons

Thinking of the vocal tract reminds us that there is another familiar “wind instrument”: the human voice. Where does this fit into the scheme we have been outlining? When we speak or sing, the origin of the sound is that air flow from our lungs is modulated by vibration of the vocal cords located down in the larynx at the top of the trachea. (See this Wikipedia page for more anatomical detail.) Figure 6 shows an animation of how the vocal cords vibrate. The motion, and the fact that the vocal cords are made of throat tissue not very different from the lips, reminds us of the brass player’s lip vibration sketched in Fig. 3.

Figure 6. Animation of human vocal cords in action, singing a steady note. Image: Reinhard, Public domain, via Wikimedia Commons, from https://en.wikipedia.org/wiki/Vocal_cords

So should the human voice be classified as a brass instrument? Perhaps, but some of the other key ingredients of a brass instrument are lacking. Although the sound from the vocal cords does connect to a tube with resonances (the vocal tract), this does not serve to provide “slotted” pitches. The pitch is governed directly by the “reed resonance”, determined by how tightly your throat muscles stretch the vocal cords. In that respect, the singing voice is more like a free reed instrument. The vocal tract resonances are important, but not to modify the pitch. Instead, they provide spectral colouration of the sound by the time it emerges from your mouth, by amplifying some harmonics relative to others. These patterns of amplification around vocal tract resonances are the “formants” which determine your perception of vowel sounds, as mentioned back in section 5.3. This makes the voice a relatively rare example of a wind instrument involving a functional component somewhat similar to the body of a stringed instrument.

Our final category describes wind instruments that depend on an edge tone of some kind. Figure 7 shows a schematic sketch of part of the mouthpiece of a recorder, or a flue organ pipe, or a referee’s whistle. Some examples of flue organ pipes are shown in Fig. 8: wooden pipes in the centre, flanked by metal ones. Air is blown through a slot, and emerges as a jet. A short distance away from the slot is a sharp edge, and the air jet interacts with this edge in some complicated, fluid-dynamical way.

Figure 7. Sketch of an air jet from a slot, interacting with a sharp edge
Figure 8. Wooden and metal flue organ pipes in the Evangelische Christuskirche Stuttgart. Image: Andreas Praefcke, CC BY 3.0 https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons

In the presence of acoustic resonances (tube resonances in the case of the recorder or the organ pipe, a Helmholtz resonance for the whistle) this interaction may settle into a regular, periodic pattern and thus produce a musical note. Even without an acoustic resonance, an air jet can sometimes interact with an edge to produce a periodic variation in the flow pattern. Another possibility involves resonant behaviour not of an acoustic tube, but of the structure providing the edge: this effect is used deliberately in an Aeolian harp, in which stretched strings are excited by the wind, and it happens accidentally when power cables “sing” or “gallop” in the wind.

These air-jet instruments are probably the most complicated of the four categories in terms of modelling the underlying physics, but we need to understand a bit about fluid flow to make progress with all four types of wind instrument. This is the task of the next section: it will give a qualitative introduction to some key concepts and phenomena of fluid dynamics. After that, the remaining sections of this chapter will look at the four categories of instrument in turn in a bit more detail.

Before that, in the remainder of this section, we will deal briefly with some features held in common by several categories. First, a reminder of what we learned in section 4.2 about acoustic resonators, and especially about resonances in tubes with various bore profiles and boundary conditions. Figures 9–12 reproduce some plots from that section.

Figure 9 shows the case of a straight pipe, open at both ends. This could be an idealised model for a flute, recorder, or flue organ pipe (as in Fig. 8). The fundamental mode has a half-wavelength between the ends, so its frequency corresponds to a sound wave with wavelength twice the length of the pipe. The higher modes have resonance frequencies in ratios 2, 3, 4… to this fundamental frequency, so they fill a complete harmonic series. In reality, the frequencies will all be a little lower because the pipe will “feel” a little longer than its physical length: as explained in section 4.2.1, there is an “end correction” to be added to the physical length, whose value depends on the detailed geometry around the open ends of the pipe.

Figure 9. A straight pipe, open at both ends (in red), and the first few pressure mode shapes. These resonance frequencies are in the ratio 1:2:3:4

Figure 10 shows the corresponding plot for a straight pipe that is open at one end but closed at the other. This could be an idealised model for a clarinet, with the closed end at the mouthpiece, or for a “bourdon” organ pipe, which is open at the mouth but closed at the top end. This time, the fundamental mode has a quarter-wavelength between the ends, so its frequency corresponds to a sound wave with wavelength four times the length of the pipe. The higher modes have resonance frequencies in ratios 3, 5, 7… to this fundamental frequency, odd-numbered frequencies of a harmonic series. Again, for accurate frequencies it would be necessary to allow for end corrections.

Figure 10. A straight pipe, open at at one end and closed at the other (in red), and the first few pressure mode shapes. These resonance frequencies are in the ratio 1:3:5:7, but the fundamental frequency is an octave lower than the fundamental of an open-open tube of the same length, so relative to the frequencies noted in Fig. 9 these frequencies would be 0.5, 1.5, 2.5 and 3.5.

Figure 11 shows the corresponding plot for a conical tube, which could be an idealised model for an oboe or saxophone. The pressure mode shapes look superficially similar to the closed-open modes of Fig. 10, because the pressure waveforms all have a horizontal tangent at the left-hand end. This will happen for any closed tube, whatever the detailed shape: once the distance from the end is much smaller than the wavelength of sound, the pressure is bound to be approximately uniform in space, just as we observed when discussing the Helmholtz resonator back in section 4.2.1. Despite this resemblance to the closed-open case of a straight tube, the resonance frequencies are in fact exactly the same as for a straight open-open tube like the one seen in Fig. 9. One familiar consequence is that a clarinet can produce a lowest note that is an octave lower than an oboe with the same tube length.

Figure 11. A conical pipe (in red), and the first few pressure mode shapes. These resonance frequencies are in the ratio 1:2:3:4, and in the idealised case would be identical to the four frequencies indicated in Fig. 9, for a tube of the same length.

Figure 12 shows the corresponding plot for an approximate model of a brass instrument like a trumpet. There are no simple mathematical solutions for such shapes: these mode shapes are numerically computed, as described back in section 4.2. The black vertical lines mark the positions of the “effective end point” for each mode: the position where the wave in the flaring horn switches from oscillatory to evanescent behaviour. The behaviour revealed by these lines is that the mode shapes look a bit like the ones in Fig. 10 for a closed-open straight tube, except that the flaring bore shape has “squashed” the lower modes into a shorter portion of the tube, while the higher modes reach progressively closer to the open end of the bell. The frequency ratios are approximately 0.7:2:3:4. This is approximately a harmonic series, except that the fundamental frequency is way out of tune. This accounts for the fact that an instrument like this offers “slotted”pitches that fill a complete harmonic series rather than just the odd terms of the series — except that the fundamental of the series is missing.

Figure 12. A simple model of a brass instrument with a flaring bell, and the first four pressure modes shapes computed as described in section 4.2. If the frequency of the second mode is scaled to the value 2, the frequencies of these five modes are 0.75, 2, 2.97, 4.07, 5.07.

This last comment gives a good opportunity to mention an important fact about the overtone frequencies of any wind instrument tube: they play two quite distinct roles in the musical behaviour. On the one hand, they provide resonances that can interact with the nonlinear excitation mechanism to determine the pitch of a played note. On the other hand, they influence the sound spectrum and, more subtly, the “playability” of the note. Imagine you are playing the lowest note on an oboe, with a frequency matching the lowest resonance in Fig. 11. The nonlinear excitation mechanism (from the reed in this case) also generates exact harmonics of the played note. If the higher resonances of the tube are close in frequency to these harmonics, as they would be for the idealised case in Fig. 11, they will all be resonantly excited by the nonlinear harmonics. The result will be a sound richer in those harmonic components.

But also, it seems a plausible guess that the note will be made a bit easier to play. All these resonances are, as it were, in agreement about what exact pitch should be played. On the other hand, if the tube resonances were a bit out of tune, so that each was shifted a bit away from exact harmonic relations with the fundamental, surely these different modes would have slightly different views about what the played pitch “should” be. That might lead to a more sluggish transient as a compromise was “negotiated” between the different resonances. This idea, involving “cooperative regimes of oscillation”, goes back to an early study of the mechanics of wind instruments by Bouasse [2], and more recently it was persuasively argued by Benade [3]. Benade constructed a demonstration instrument he called the “tacit horn”: the resonances were deliberately mis-aligned, with the result that it was virtually impossible to sound any note on the instrument!

These two roles of tube resonances operate differently in our different categories of instrument. In the reed and air-jet instruments, players normally use oscillation regimes based on the first or second mode of the tube, not the higher modes. In that case, the main role of the remaining resonances is to do with tone quality and “playability”. But in most brass instruments, players will use many of the resonances of the tube as the basis for varying the pitch. Indeed, in instruments like the bugle or natural horn there is no way to change the effective length of the tube, and these regimes are the only resource a player can draw on. Instruments with valves or slides are more versatile because the tube length can be changed, but players still make use of many more tube modes than a woodwind player will do.

There are a few more topics for us to discuss here, because they are important for wind instruments in more than one category. One thing we have already mentioned is that the standard measurement used to characterise the linear acoustics of a wind instrument tube is the input impedance. It is worth looking at a simple example of input impedance, corresponding to the ideal cylindrical pipes seen in Figs. 9 and 10. The input impedance of a cylindrical pipe, open at the far end, is easy to calculate. The details are given in the next link, and Fig. 13 shows an example of the result. This particular case shows a tube 1 metre long, with internal diameter 20 mm. Realistic damping has been included, using formulae from the literature: again, the details are in the side link.

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Figure 13. The calculated input impedance of a cylindrical tube with length 1 m and internal diameter 20 mm. The impedance has been normalised by the value for an infinite tube, and plotted on a dB scale.

This gives us a chance to make an important observation about the interpretation of input impedance for instruments in different categories. Where do we find the resonances of the tube, by looking at a plot like this? The answer depends on the boundary condition at the end of the tube where the impedance has been measured. If the tube is closed at this end, as in Fig. 10, there can be no air flow in and out of the stopped end, but the pressure variation can be large at a resonance. This means that pressure divided by the volume flow rate must be very large — so the resonances correspond to peaks in the input impedance. Sure enough, the peaks in Fig. 13 show the odd-harmonic pattern that we expect from Fig. 10.

But if the tube is open at the end, as in Fig. 9, we have the opposite situation. Air can flow in and out of the end of the tube, but the pressure variation must be more or less zero because the open end is exposed to the outside world, with a fixed ambient pressure. So this time, at a resonance we expect very small values of the impedance, or equivalently we expect very large values of its inverse, called the input admittance. This admittance is plotted in Fig. 14: because of the decibel scale used here, the image is simply the same as Fig. 13, plotted upside down. Now the peaks fall half-way between the peaks in Fig. 13, and they are equally spaced in a complete harmonic series. This is exactly what we expect from Fig. 9.

Figure 14. The input admittance of the tube from Fig. 13: on this decibel scale, the plot is simply the same one, upside down. The admittance has been normalised by the value for an infinite tube, and plotted on a dB scale.

Finally, we look at the role played by tone-holes in the wall of the instrument tube. Most reed and air-jet wind instruments have tone-holes. Less familiarly, some “brass” instrument are played by covering tone-holes with the fingers: examples are found in older instruments such as the cornett (or cornetto) and the serpent. The purpose of tone-holes seems self-evident: making a hole in the wall of the tube is rather like cutting the tube short at that point, so that the frequencies of all the resonances go up. However, as with so many things in musical acoustics, it is a bit more complicated than that.

As a first step we can imagine a single tone-hole cut into an infinitely long cylindrical tube, as sketched in Fig. 15. We can send a sinusoidal sound wave at a chosen frequency down the tube. When this wave reaches the hole, some of its energy is reflected back, while some of it is transmitted past the hole and continues to propagate in the direction it was already travelling. This problem is analysed in the next link, by thinking about the air flow and the pressure in the small volume near the hole, shown as a dotted box. Simple expressions are found for the reflection coefficient $R$ and the transmission coefficient $T$. These depend on the geometrical dimensions of the hole and the tube bore, and they also vary with frequency.

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Figure 15. Sketch of an infinite cylindrical tube with a single tone-hole cut in it. A sound wave is sent in from the left, then the hole generates a reflected and a transmitted wave travelling off in the two directions.

Figure 16 shows three examples of how the energy reflection coefficient $|R|^2$ behaves when the area of the hole is varied from zero up to the cross-sectional area of the tube, at three different frequencies. For the lowest frequency (red curve), when the hole is as big as the cross-sectional area of the tube virtually all the energy is reflected. This is the effect we were expecting: the sound wave is more or less confined to the left-hand side of the hole, just as if the tube had been cut off at that position. As the size of the hole is reduced, the reflection coefficient drops gradually, reaching zero as the hole area vanishes. Well, that is no surprise: if the hole is no longer there, of course no energy is reflected from it.

Figure 16. The fraction of energy reflected by the single tone-hole of Fig. 15, as the area of the hole is varied from zero up to the cross-sectional area of the tube. The three curves correspond to different frequencies of sine wave: 500 Hz (red), 1000 Hz (blue), 1500 Hz (black).

At the two higher frequencies shown in Fig. 16 the reflection coefficient is systematically smaller. Even with a very large hole, a significant fraction of the energy is not reflected, but continues along the tube. As we will explore shortly, this means that what happens further down the tube from the tone-hole can have a significant impact on the behaviour at these higher frequencies. To get a different view of the frequency dependence, Fig. 17 shows the reflection coefficient plotted against frequency. Curves are shown for flutes of three different kinds, using bore size and tone-hole dimensions measured by Wolfe and Smith [2]. All three curves show a similar trend: perfect reflection at very low frequency, falling to rather low values at 2 kHz. Notice also that the three types of flute are systematically different, and they are arranged in order of age. There has obviously been a systematic evolution of flute design: we’ll come back to that in a moment.

Figure 17. The reflection coefficient, as in Fig. 16, shown here as a function of frequency. Bore and tone-hole dimensions for flutes of three different kinds have been used, taken from Wolfe and Smith [2]: baroque flute (red), classical flute (blue), modern flute (black).

When our infinite tube is replaced with a finite tube, we would like to know what happens to the resonance frequencies of this tube as the single hole is gradually closed. The answer turns out to be what you might guess. With a large hole, the resonances are those of a shorter tube, cut off near the hole. When the hole is reduced to a pinhole and then vanishes altogether, the resonances are those of the full length of tube. Well, in between the resonance frequencies change smoothly between these two limits. With this rather primitive wind instrument with just a single tone-hole, the player would in principle be able to produce a pitch that can be varied over this range by using a finger to part-close (or “shade”) the hole progressively.

The physics behind this gradual shift of resonance frequencies can be visualised in terms of an “end correction”. With a single hole of any size, the reflection is never perfect. This means that the effective length of the tube is always longer than the distance to the centre of the tone-hole. With a large hole and a correspondingly large reflection coefficient, this extra bit of effective length is quite small, but as the reflection coefficient reduces with a reduction of hole size, the sound wave penetrates further past the hole and the end correction determining the effective length (and thus the frequency of resonance) grows. Eventually, as the hole becomes vanishingly small, this “end correction” reaches all the way to the end of the tube, where virtually all of the sound energy is reflected back into the tube. Some details of the calculation lying behind this description are given in the next link.

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This variation of the end correction with hole size is something that instrument makers have taken advantage of, especially in earlier instruments that rely on fingertips to cover holes, rather than using key mechanisms. There are limits to how human fingers can be spread along the tube of a wind instrument. This sometimes means that a tone-hole in the “logical” place would be too far away from the others to be reached by the player’s finger, especially in larger instruments like the dulcian (or curtal), a predecessor of the bassoon: see Fig. 18. The solution was to use a smaller hole with a correspondingly bigger end correction, so that it could be placed in a more convenient position. The resulting odd-looking distribution of holes can be seen very clearly in the image.

Figure 18. A dulcian of 1700 (Museu de la Música de Barcelona), a predecessor of the bassoon. Note the spacing and sizing of the finger holes, to allow the holes to be reached by a player’s fingers. Image: Sguastevi, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

Of course, real wind instruments have more than one tone-hole. Figure 19 shows a schematic sketch of a “flute”, with a typical fingering: there are several tone-holes, the first few being covered while the remaining ones are open. This introduces several complications. The first open tone-hole is no longer in splendid isolation: not very far away are other open holes, before we reach the end of the tube. Also, the closed tone-holes affect the profile of the tube. The fingers or key-pads do not produce the same effect as not having the hole in the first place, because they add a small local volume to the tube, which can perturb the resonance frequencies a bit.

Figure 19. Schematic sketch of a flute, fingered with a combination of open and closed tone-holes.

To see what happens to the frequency response of a real instrument tube with various combinations of open and closed tone-holes we can look at some measured input impedances, taken from Joe Wolfe’s comprehensive web site on flute acoustics here. We can look first at some results for a baroque flute, which is a relatively simple instrument not too far removed from the sketch in Fig. 19. It has 6 finger-holes, plus a 7th hole which is operated by a key: you can see a picture in Fig. 22 below.

Figure 20 shows the input impedance for four different fingerings. Because we are looking at a flute, which is open at the mouthpiece end, we must keep in mind that the resonances of the tube are indicated in these plots by the minima, not the peaks. The first plot shows the result with all the holes closed, in the fingering for the lowest note of the instrument ($\mathrm{D}_4$, 294 Hz at the modern pitch standard, but a semitone lower on this flute at 277 Hz because it is adjusted for a pitch standard with $\mathrm{A}_4$ at 415 Hz). The plot shows an orderly sequence of peaks and minima, very reminiscent of the idealised version seen in Fig. 13.

The top right plot in Fig. 20 shows the impedance for a fingering rather like the sketch in Fig. 19. It has three closed holes, then three open holes, but the very last hole is closed by the key. The pattern is much less regular. The explanation is that a baroque flute like this has relatively small tone-holes (for a reason we will come to in a moment), and so a sound wave coming from the mouthpiece end would only be partially reflected at the first open tone-hole. A significant fraction of the energy will continue down the tube and interact with the other open tone-holes as well as with the open end of the tube. This complicated interaction leads to disruption of the pattern of resonances.

The lower left plot shows the fingering for $\mathrm{A}\flat_4$. This one is a “fork fingering” or “cross fingering”, with closed holes coming after the first open hole. The possibility of using such fingerings to achieve intermediate chromatic notes is probably the main reason that flutes like this have rather small tone-holes. If the holes are to be closed by the player’s fingers, unaided by complicated key mechanisms, there can only be as many holes as the player has available fingers. This is not enough to offer a full chromatic scale by simply opening one more hole for each semitone step. So instruments like this one are tuned to give a diatonic scale in a particular key, and the player has to use techniques like fork fingering to sound the missing notes. Fork fingerings can only work if the reflection coefficient at the first open tone-hole is fairly small: the hole’s end correction is then large enough that the influence of later closed holes can be felt.

The final plot in Fig. 20 shows the result with only one tone-hole closed. This one highlights a feature of the pattern of resonances which started to show up in the previous two plots, but less clearly. There are only two strong resonances (deep dips in the plot, remember), and for all frequencies above about 1 kHz the resonances are significantly attenuated. This feature is called the “tone-hole cutoff”, and it arises from the fact that there is a fairly regular series of open tone-holes following the last closed one. An approximate formula for this cutoff frequency in terms of the tone-hole geometry was first given by Benade [3], on the assumption that the “lattice” of open tone-holes was infinitely long, and uniformly spaced. A neat derivation of the result can be found in the Appendix of reference [2].

It is interesting to contrast this behaviour with some corresponding plots for a modern flute, taken from the same web site and shown in Fig. 21. Photographs of typical baroque and modern flutes are shown in Fig. 22. The sketches accompanying each impedance plot show that a flute like this has many more holes than the player has fingers. A complicated and ingenious system of keys is used to open and close these holes — it was invented by Theobald Boehm in the mid-19th century. A key system like this frees the designer from some of the constraints of human fingers. As well as allowing more holes, the individual holes can be larger because the key-pad can be shaped to close a hole of virtually any shape. This is the explanation for the contrast in reflection coefficient seen back in Fig. 17: the modern flute (black line) showed much stronger reflection at all frequencies than the baroque flute (red line).

The results of Boehm’s ingenuity can be seen in two features of the plots in Fig. 21. The first plot, with all the holes closed, looks very similar to the corresponding plot in Fig. 20. But after that, the behaviour is very different. The modern flute shows an orderly sequence of resonances for all the cases shown — there is very little of the disruption we saw in Fig. 20. The second thing we can see is that the tone-hole cutoff for this flute is significantly higher in frequency, up around 2 kHz. Both these features are, qualitatively at least, what we might have anticipated from the contrast seen in Fig. 17. Except for some advanced playing techniques such as “multiphonics” (we’ll come to those in section ?), there is (almost) no need for fork fingerings on this flute. Each semitone has its own key and hole combination.

Now we have dealt in some detail with open tone-holes, we should have a brief look at the perturbing effect of closed tone-holes. As we noted earlier, these create a small extra volume, which will have a similar effect to a slight bulge in the bore profile. Any perturbation to the bore will influence frequencies slightly. This might interfere with carefully-planned alignment of resonances, to improve the sound quality and playability. An instrument maker might try to counter such effects by making deliberate modifications to the bore profile.

So for one reason and another, it is of interest to know the effect on resonance frequencies of a small change in bore profile. As explained in the next link, Rayleigh’s principle (see section 3.2.5, and the application to a marimba bar in section 3.3.1) gives us an easy way to answer this question. We can describe the result qualitatively. For any given mode of a cylindrical tube, like the ones shown in Figs. 9 and 10, the effect of a local enlargement in the bore is biggest near a nodal point, or near an antinode (a position of maximum pressure). An enlargement near a node (including near an open end of the tube) has the effect of increasing the frequency, whereas an enlargement near an antinode has the effect of reducing the frequency. Since different modes have their nodes and antinodes in different places, it follows immediately that the effect of a single enlargement, such as a closed tone-hole, will be different for each mode, and so it will indeed tend to interfere with the harmonic relations between frequencies. But the same theoretical expression can be used to guide an instrument maker wishing to adjust the bore to bring resonance frequencies to desired values.

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Our final topic for this section is to note that not all holes in the tube wall of a wind instrument function as tone-holes, to control the pitch of the note. Most instruments also have register holes, to aid the player in switching between a playing pitch based on the lowest mode of the tube, and one based on the second mode. Players describe these alternatives as different registers. Sometimes, a player can switch between registers by adjusting the details of how they blow. For example, blowing harder into a recorder or tin whistle may cause the pitch to jump to the next register, often called “overblowing”. But life is easier for a player if they have a controlled way to switch registers, and this is where register holes come in.

Consider the example of a cylindrical tube, open at both ends, like a flute or recorder. As Fig. 9 reminds us, the first register is based on a pressure mode shape with one half-wavelength in the length of the tube, whereas the second register is based on a mode shape with a nodal point at the centre, and two half-wavelengths in the length of the tube. If a small hole is drilled in the tube wall near the centre, this will not affect the second mode but it will perturb the first mode (which has a pressure maximum at this position). It may shift the frequency of the first mode, and it may add some dissipation and thus reduce the height of the resonance peak. Both effects will tend to make it more difficult to play a note based on this mode, and thus to make it more likely that the player will get a note in the second register.

This is the principle of a register hole. Why does the hole need to be small? The answer to that is that we don’t want to affect just one note, we would like to encourage the register switch in other nearby notes, based on different effective tube lengths because tone-holes have been opened or closed. Each of these notes will have the second-register nodal point in slightly different positions, so the hole can’t be perfectly positioned to suit them all. But if the hole is fairly small, this doesn’t matter very much. If carefully designed, it can have the desired register-changing effect on several notes.

If we are thinking about a clarinet rather than a recorder or flute, we can make a similar argument based on the mode shapes in Fig. 10. The same argument applies, except that the place we need to drill our register hole is approximately 1/3 of the way down the tube from the mouthpiece, rather than in the centre — because that is where the nodal point of the second pressure mode shape occurs.

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[1] Murray Campbell, Joël Gilbert and Arnold Myers, “The science of brass instruments”, ASA Press/Springer (2021). See section 3.1.3 for examples of video recordings of a brass-player’s lips.

[2] Joe Wolfe and John Smith, “Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes”, Journal of the Acoustical Society of America 114, 2263—2272 (2003)

[3] Arthur H. Benade, “On the mathematical theory of woodwind finger holes”, Journal of the Acoustical Society of America 32, 1591—1608 (1960)