11.3 Reed instruments

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Having covered some background material relevant to a wide range of musical wind instruments, it is time to turn to a more detailed look at the main families of instruments. We will begin with the reed instruments, starting with the clarinet because this is arguably the most-studied member of the family. We will find some surprisingly close analogies with the behaviour of a bowed string, which we studied in Chapter 9. Later in this section we will look at the soprano saxophone, as an example of an instrument with a conical bore rather than a cylindrical bore like the clarinet. This too will have an analogy with bowed string behaviour. If you have read Chapter 9 before reaching this point, these analogies will make immediate sense to you. But if you have dipped in at this point without knowing much about bowed strings, it would be prudent to break off here and take a look at the next link before continuing this section. It gives a short, non-technical summary of some key points of bowed-string behaviour.

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We have already looked at some aspects of the linear acoustics of wind instrument tubes, in section 11.1, so the main emphasis in this section will be on the excitation mechanism of reed instruments, and the consequences of that for sound and playing behaviour. For an immediate impression of how important the excitation mechanism is, look at this rather amusing video. In it, Joe Wolfe demonstrates what happens if you swap the mouthpieces of a flute and a clarinet. The clear conclusion is that any roughly cylindrical instrument played with a clarinet mouthpiece sounds very much like a clarinet, while one played with a flute mouthpiece sounds like a flute. For both the clarinet and the saxophone, we will use computer simulations based on an idealised model to produce diagrams somewhat analogous to the Schelleng diagram for a bowed string. These give an indication of how a player might perceive the variation of behaviour, depending on how they control the way they blow into the instrument.

A. Simplest model of the clarinet

We have already given a schematic description of how a clarinet mouthpiece works, back in section 8.5. That description was qualitatively correct, but now we can look at a version based on explicit modelling. Figure 1 shows a sketch. The reed is a tapered, flexible piece of natural cane or synthetic material, which is clamped against the rigid tube wall. The player seals their lips around the mouthpiece and reed, and they blow air in. As indicated in the sketch, there is a concentrated air jet in the narrow gap, then inside the mouthpiece it breaks up into turbulent eddies and dissipates the kinetic energy. For the purposes of this simple description, we will assume that the pressure inside the player’s mouth is simply constant. This is not always true, though, in real performance: expert players may manipulate the resonances of their vocal tract to make subtle adjustments.

Figure 1. Sketch of the air flow into a clarinet mouthpiece

We get a reasonable approximation to the physical effects if we assume that the simplest form of Bernoulli’s law applies to the jet in the region where it passes over the reed surface. The high speed of the jet produces a reduced pressure compared to the mouth pressure acting on the lower face of the reed. This pressure difference tends to draw the reed inwards, and eventually to close it against the “lay”, the rigid lip and side plates of the mouthpiece. As explained in the next link, this gives an explicit formula for the volume flow rate of air through the gap, as a function of the pressure difference between the player’s mouth and the inside of the mouthpiece.

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This function is plotted in Fig. 2, using parameter values appropriate for a clarinet reed. The exact shape depends on a small number of parameters: the width and height of the initial gap between reed and lay, and the stiffness of the reed. Of course, the reed has resonances — the previous link show how to include the lowest resonance in our modelling. We will come back to this a little later, at which time the resonance frequency and damping factor will need to be added to the list of parameters. But for the moment we will assume that the playing frequency is well below the reed’s first resonance, in which case the reed will behave like a simple spring.

Figure 2. The simplest model of a clarinet mouthpiece: the flexible reed provides a nonlinear valve, with a relation between pressure difference and volume flow rate like the red curve here. The sloping dashed lines indicate two possible positions of the line whose intersection with the red curve determines the new values of pressure and flow rate. Note that many authors plot this function with the horizontal axis running in the reverse direction: do not be confused if you are comparing this plot with textbooks like [1]!

Figure 2 also reminds us of another key ingredient of the model developed in section 8.5. In a simulation of the transient response, at each time step we first calculate the incoming pressure wave, returning down the clarinet tube after reflecting from the tone-holes and bell. This incoming pressure determines the position of a sloping straight line: two examples of possible positions are shown in Fig. 2. The pressure and volume flow rate at the next time step are determined by finding the intersection between the straight line and the nonlinear curve. The details governing the position and slope of the straight line were set out in section 8.5.3.

The plot in Fig. 2 shows the volume flow rate as a function of the pressure difference between the inside and the outside of the mouthpiece. However, for a model of the sound generation inside the instrument we want to express it is a function of the actual pressure $p(t)$ just inside the mouthpiece. To achieve this, we simply need to add the player’s mouth pressure. (Remember that for this introductory account we are regarding this mouth pressure as being constant: we are neglecting the effect of resonances inside the player’s mouth.) As the player increases their mouth pressure, the curve is shifted progressively to the right, as shown in Fig. 3.

Figure 3. The nonlinear valve characteristic from Fig. 2, plotted now as a function of pressure $p(t)$ inside the mouthpiece for four different values of the player’s mouth pressure: lowest for the red curve, highest for the black curve. The values of mouth pressure are 0.8 kPa, 2.8 kPa, 4.8 kPa and 6.8 kPa.The blue curve shows the case close to the threshold for the clarinet to start to sound.

We have already talked about the result of this shift, back in section 8.5. When the mouth pressure is low, as in the red curve in Fig. 3, the mouthpiece behaves like a mechanical resistance, tending to dissipate the energy associated with any pressure variation inside the tube. But once the pressure is high enough to shift as far as the blue curve, the tangent to the curve where it crosses the vertical axis starts to slope in the opposite sense, and the mouthpiece then behaves like a negative resistance, tending to amplify disturbances. This is the threshold condition for the clarinet to start to make sound. Blowing harder still can produce the green curve, and then the black curve. In both these cases the tangent is sloping steeply and the amplification effect is very strong.

Blowing harder and shifting a little further, the vertical axis would pass through the horizontal section of the curve where the reed is fully closed. The tangent is horizontal, and there is no amplification. Under those conditions, the instrument is likely to remain silent: it is “choked” by blowing too hard. However, we will see shortly that things are a little more complicated than that. With some kinds of initial articulation transient by the player, it is perfectly possible to produce sound from the instrument with a mouth pressure high enough to be in this region. This is a first hint of “playability” issues in our model clarinet, something we return to in more detail later.

Figure 4 illustrates possible effects on the nonlinear valve characteristic of varying two other key parameters in the model. The datum for comparison is the green curve, which is the same as the green curve in Fig. 3. The blue curve shows the effect of the player choosing a stiffer reed — reeds are sold in various grades of “hardness”. The stiffer reed, set to the same initial gap, requires a bigger pressure difference before it closes, so the curve is higher. The red curve shows the effect of setting the stiffer reed with its tip closer to the lay so that the closure pressure remains the same as in the green curve. The curve is lower, because the volume flow is reduced through the narrower gap.

Figure 4. Variations in the nonlinear valve characteristic from Fig. 2 when the reed properties are changed. The green curve is the same as the green curve in Fig. 3. The red curve shows the effect of reducing the initial reed opening from 0.6 mm to 0.4 mm while keeping the same closure pressure (so that a slightly stiffer reed is implied). The blue curve shows the effect of increasing the reed stiffness without changing the gap, so that the closing pressure is increased.

Referring back to Fig. 1, we can see that the player might have some influence on these two parameters during performance. The lower lip is in contact with the reed, and the player can choose the precise contact position, and also how hard to “bite”. Biting harder will reduce the gap, and the lip contact may also influence the effective stiffness of the reed. It will also change the reed resonance frequency, and its damping factor.

B. Why is a clarinet like a violin?

Now we can start to draw interesting parallels with a bowed violin string. The simplest model of a bowed string, discussed in section 9.2, involves a very similar set of ingredients to the clarinet model just outlined. There is a nonlinear function, the “friction curve”, and in a transient simulation the values of force and string velocity at the next time step are determined by the intersection of this curve with a straight line. The position of the straight line is determined by incoming reflected waves on the string.

Furthermore, a violinist can influence the friction curve through two of their main control variables: the effects are illustrated in Fig. 5. Look first at the red curve, which shows the typical shape of a friction curve. The player can change this to the blue curve by increasing their bow speed, or alternatively they can change it to the green curve by reducing the bow force. Comparing this figure with Figs. 3 and 4, we see obvious parallels. Bow speed behaves like mouth pressure, shifting the nonlinear curve sideways. Bow force behaves a bit like the two effects explored in Fig. 4: the nonlinear function stays a similar shape, but the peak value can be increased or decreased.

Figure 5. Variations on the friction curve, for the simplest bowed-string model. The red curve has normal force 10 N and bow speed 0.05 m/s; the blue curve has the same normal force but increased bow speed 0.1 m/s; the green curve has the same bow speed as the red curve, but reduced normal force 4 N.

So two of the violinist’s control variables have clear analogues in the clarinet. However, this is not true of the third major variable, the bow position on the string. Schelleng’s diagram (see section 9.3) showed us that the bow position has a very strong influence on how a violin string behaves. But the mouthpiece of a clarinet is not plugged into the tube at an intermediate point, analogous to the position of a violin bow. Instead, it drives the tube at one end.

There is an analogue in the bowed string, but it is a very extreme and unusual case: a clarinet behaves rather like a string bowed at its exact centre. The comparison is illustrated in the sketches in Fig. 6. The bowed string will respond with symmetrical motion in the two halves. If the wave speeds are the same on the string and in the clarinet tube, it is easy to see that the fundamental pitches agree. The clarinet will have a quarter-wave in the length, while the string will have a half-wave in a length which is twice as long.

Figure 6. The analogy between a clarinet and a string bowed at its mid-point

This comparison tells us something very interesting. When we first started looking at the behaviour of bowed strings, we described the work of Helmholtz and Raman. Well, the descriptions we used can be carried over directly to the clarinet problem. First, we have Helmholtz’s observation of how a bowed string moves. For the special case of bowing at the mid-point, the Helmholtz motion will have a pair of “Helmholtz corners” travelling in opposite directions on the string, to give the required symmetrical motion in the two halves. At the bowed point there will, as usual, be an alternation between sticking and slipping, once per cycle. The resulting waveform of string velocity at the bowed point will be a symmetrical square wave, which is the relevant special case of the usual rectangular pulse waveform.

Raman’s argument, explained in section 9.1.2, gives an explanation of this waveform for the idealised case with no dissipation. We will phrase the description in terms of the clarinet example, but it is exactly the same as described previously for the bowed string. If the tube has no damping, then the only way it is possible to have steady, periodic oscillation of the internal pressure is if the forcing, from the waveform of volume flow injected at the mouthpiece, is exactly constant. If it were not constant, it would contain Fourier components at the resonant frequencies of the tube, and in the absence of damping these would evoke a response which would grow without limit, contradicting the assumption of steady motion.

The result is that the volume flow can only switch between two values, positioned on a horizontal level on the nonlinear valve curve. For the clarinet, the pressure waveform (analogous to the string velocity waveform in the bowed string) will be a symmetrical square wave, so the pair of points must have equal and opposite pressure. Figure 7 illustrates what must happen: this shows the same four curves as Fig. 3, and three of them are annotated with circles in corresponding colours showing where the pair of points must lie. But for the curve with the lowest mouth pressure, the red curve, it not possible to find such a pair. We conclude that the clarinet version of ideal Helmholtz motion is possible with mouth pressures corresponding to the blue, green and black curves, but not for the red curve.

Figure 7. The four curves from Fig. 3, annotated to illustrate Raman’s argument. With no energy dissipation in the tube, the pressure waveform must be a square wave, alternating between two values on the same horizontal level. This is possible for the blue, green and black curves, as indicated by the circles, but it is not possible for the red curve so “Helmholtz motion” is not possible with this mouth pressure. Plot adapted from Fig. 9.10 of Chaigne and Kergomard [1].

As a reminder of the corresponding argument for the bowed string, Fig. 8 shows a copy of Fig. 1 from section 9.1.2. This illustration is not for a mid-point bow but for a more normal bow position close to one end of the string. The Helmholtz motion then involves a rectangular velocity waveform alternating between two speeds, but it is not a symmetrical square wave. Instead, the proportions of time spent sticking and slipping are in the same ratio as the position of the bowing point as a fraction of the string length. But it would be easy to find a different horizontal level on this plot, with a symmetrical pair of intersections on either side of the vertical axis. That line would define the velocities and friction force level for mid-point bowing of a string with no energy dissipation, directly analogous to the clarinet case shown in Fig. 7.

Figure 8. A copy of Fig. 1 from section 9.1.2, showing a schematic plot of friction force as a function of sliding speed, where $v_b$ is the bow speed. The black line and stars relate to Raman’s argument, explained in the text.

The analogy with the bowed string continues: we can go beyond the super-simplified Raman model by using an equivalent approach to the one described in section 9.2, in which the Helmholtz corner was allowed to become rounded, rather than being ideally sharp. In fact, we can do that more systematically for the clarinet than we were able to do for the violin string. There is a mathematical trick, described in the next link, that allows a measured or simulated input impedance to be processed to reveal the “reflection function” needed for an accurate simulation.

For the purposes of this section, we will use a sort of “one-note clarinet”: a cylindrical tube of the right length and bore diameter for a clarinet, but without finger holes (either open or closed). We can calculate the input impedance of such a tube, with plausible damping, in the way described in section 11.1.1. The details are given in the link, with some plots. The result of simulations using this reflection function should give a reasonably good representation of the lowest note of a clarinet — certainly good enough to bring out some interesting aspects of the physics.

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For a first example, we can look at (and listen to) three simulated notes based on the blue, green and black curves in Fig. 3, using three different mouth pressures while keeping everything else fixed. Each note was synthesised using a very gentle initial transient, in which the pressure and flow velocity history inside the tube were initialised to values close to the intersection of the curve with the vertical axis. This means that the note builds up very slowly when the mouth pressure is only just above the threshold, as in the blue curve of Fig. 3. This can be seen very clearly in Fig. 9, which shows the synthesised pressure inside the mouthpiece for the three notes: the first note has a very long initial transient, much longer than the other two notes. An artificial exponential decay to silence has been imposed on each note, in order to separate them. You can hear the sequence of three notes in Sound 1.

Figure 9. Three synthesised notes on the “one-note clarinet”, using three different values of mouth pressure. The three notes correspond to the blue, green and black curves of Fig. 3, in that order. The exponential decay at the end of each note has been artificially added in order to separate the notes clearly.
Sound 1. The synthesised notes shown in Fig. 9.

Waveform details cannot be seen at the resolution of Fig. 9, but the envelope shape of each note is very clear. Figure 10 shows a zoomed view of the periodic parts of the three waveforms, using the same colour code as the corresponding curves in Fig. 3. It is immediately clear that all three waveforms have a recognisable similarity to the square wave predicted by Raman’s argument. It is also clear that the amplitude increases as the mouth pressure increases, as suggested by the circle markers in Fig. 7. But the waveforms do not only differ in amplitude: the blue curve, corresponding to mouth pressure just above threshold, is much more rounded than the other two. This more rounded appearance is associated with a smaller amplitude of the higher harmonics, compared to the amplitude of the fundamental. The result is a reduction of brightness of the sound, as you should be able to hear clearly in the first note of Sound 1, compared to the other two.

Figure 10. Zoomed view of the steady portion of the waveforms of the three notes from Fig. 9, using colours corresponding to those in Figs. 3 and 7.

We can easily perform a corresponding synthesis using the Raman model in place of the realistic reflection function for the “one-note clarinet”. (The previous link gives details of how this was done.) The results, in the same format as we have just seen, are shown in Figs. 11 and 12, and Sound 2. Comparing Fig. 11 with Fig. 9, the amplitudes and envelope shapes look quite similar in the two cases. But the results certainly do not sound the same, and we can immediately see why by comparing Fig. 12 with Fig. 10. The only periodic waveform that can be produced by the Raman model is a perfectly sharp square wave. So the three notes only differ in amplitude and length of initial transient: the final periodic waveforms are all equally rich in higher harmonics, and they sound identical apart from a difference in loudness.

Figure 11. Synthesised notes using the Raman model, with everything else identical to the results in Fig. 9.
Sound 2. The sound of the three notes in Fig. 11.
Figure 12. Zoomed view of the steady portions of the three waveforms in Fig. 11, using thesame colour coding as in Fig. 10.

Next, we look at similar synthesised notes for the three variations of the nonlinear valve characteristic plotted in Fig. 4. Each group of three notes illustrates the green curve, the red curve, and the blue curve, in that order. The results are plotted in Figs. 13—16, in the same format as the ones we have just seen, using the realistic synthesis model and then the Raman model. Again, you can hear the corresponding sounds of the pressure waveforms inside the mouthpiece in Sounds 3 and 4. This time, the amplitudes of the three notes in each group are very similar because the mouth pressure is the same in all cases. For the realistic model, you can hear subtle difference of tone quality, while for the Raman model the sounds are virtually indistinguishable.

Figure 13. Three synthesised notes on the “one-note clarinet”, using three different variations of the nonlinear valve characteristic of the mouthpiece. The three notes correspond to the blue, red and blue curves of Fig. 4, in that order. The exponential decay at the end of each note has been artificially added in order to separate the notes clearly.
Sound 3. The synthesised notes shown in Fig. 13.
Figure 14. Zoomed view of the steady portions of the three waveforms in Fig. 13, using the same colour code as in Fig. 4.
Figure 15. Synthesised notes using the Raman model, with all other details the ssame as in Fig. 13.
Sound 4. The synthesised notes shown in Fig. 15.
Figure 16. Zoomed view of the steady portions of the three waveforms in Fig. 15, using the same colour code as in Figs. 4 and 14.

C. The pressure–gap diagram

We have seen in the preceding plots, and heard in Sounds 1 and 3, that the mouth pressure and the details of the valve characteristic both have audible effects on the sound of our synthesised clarinet. We learn some interesting things by taking a more systematic look at this. The mouth pressure is obviously a key control parameter for the player, and we noted a bit earlier that the “bite strength” might be another. The most obvious effect of a player biting harder on the reed is to decrease the gap between the reed and the lay. We can use these two physical parameters — mouth pressure and reed gap — to make a plot that plays a somewhat similar role to the Schelleng diagram for bowed strings. The first published version of a plot in this form, as far as I have been able to find, is in a paper by Almeida, George, Smith and Wolfe [2]. They used this format to present the results of measurements on a real clarinet, using a test rig that allowed controllable variation of pressure and bite force.

We can approach this in much the same spirit as our bowed-string studies back in Chapter 9, by doing what the nonlinear dynamics folk sometimes call “carpet bombing”. We can take a grid of points in the pressure—gap plane, perform a simulation for each grid point, then analyse the results and plot some interesting quantity in this plane, to see how it varies. Figure 17 shows an example. The horizontal axis shows mouth pressure, while the vertical axis shows the gap. This vertical axis is being regarded as a surrogate for the bite force, so the values of the gap have been plotted decreasing upwards, so that the bite force increases upwards as in the plots by Almeida et al. [2].

Figure 17. Pressure–gap diagram, colour-shaded to show the frequency content of the final waveform for cases where a note was produced, rather than silence. This example is based on the “one-note clarinet” model, with the kind of gentle initial transient used for Figs. 9, 11, 13 and 15.

Each simulation has first been analysed to test whether it produced “note” or “silence”: black pixels mark points where there was no significant amplitude of the pressure waveform at the end of the simulated time span. The coloured pixels show where a note was obtained, and they have been coloured to indicate something about the frequency content of the last few periods of the pressure waveform. The scale, indicated in the colour bar, shows how high up the series of harmonics you need to go before the level drops to 10 dB below that of the ideal square wave. So bright colours indicate bright sound, and dull reds indicate “mellow”or “muted” sound. (Ignore the various coloured lines for the moment — they will be explained shortly.)

The qualitative resemblance between Fig. 17 and the Schelleng diagram is quite striking. In a plane with axes showing two key control variables for the player, acceptable notes are only produced within a wedge-shaped region. Within that wedge the high-frequency content, and hence the brightness of the sound, varies significantly — albeit in a different pattern to the variation within the Schelleng diagram. The two diagrams encapsulate important aspects of violin and clarinet playing, which a beginner struggles to master and an expert learns to use with great subtlety.

Figure 18 shows a different plot based on the same set of simulations. This time, the colour shading indicates the frequency of the final waveform. The variations are not large in this case, but there is a definite tendency for the note to play progressively flat as conditions move down the middle of the wedge towards the bottom right-hand corner. The measurements of real clarinet behaviour by Almeida et al. [2] show a similar (but far stronger) tendency to flattening, in the same region of the diagram.

Figure 18. The same set of simulations as in Fig. 17, now colour-shaded to indicate the frequency of the final waveform.

The physical origin of this flattening phenomenon in our idealised example is far from clear. The two effects we already know about, which might have produced such an effect, are absent in this example. The resonances of our cylindrical tube are accurately harmonic, apparently ruling out “Benade-style” effects arising from collaborative regimes between inharmonic frequencies. However, such effects might well have been present in the measurements by Almeida et al. [2], potentially explaining why they saw a much bigger effect. The second possible effect takes us back to the bowed string. In section 9.2 we saw that when the straight line crosses the friction curve in more than one place, the result is a hysteresis loop which causes the pitch of the note to flatten (see especially Fig. 9 in that section). But Fig. 2 in the present section shows that the corresponding straight line is too steep to cross the nonlinear valve characteristic in more than one place, so the effect cannot arise.

Naturally, we want to know what determines the shape and position of the wedge-shaped region in these plots. We get a useful clue from Fig. 19, which shows another pair of plots like Figs. 17 and 18. The only difference is that this time each simulation was started with a more vigorous initial transient, giving a “kick” to get oscillation started. We see that this has expanded the region of possible notes. There is still a wedge-shaped region, but the top edge of the wedge has moved upwards. Roughly speaking, the wedge in Figs. 17 and 18 was confined between the lowest (magenta) line and the solid cyan line, but in Fig. 19 it extended upwards towards the top line (which is actually a solid red line and a dashed white line, very close together).

Those coloured lines were not drawn on the plots simply in order to follow the data. Instead, they represent thresholds between different types of behaviour of the simple clarinet model, which have been calculated by Dalmont, Gilbert, Kergomard and Ollivier on the basis of bifurcation studies [3]. We can describe them all in words. The magenta line is the one we already knew about, from the discussion in section 8.5. This is the threshold of oscillation, which occurs when the mouth pressure is big enough for the tangent slope on the vertical axis to change sign. The plotted line is simply the condition for the maximum of the valve characteristic to lie in that position: when there is energy dissipation, pixels that spontaneously generate a note must lie at least a little above this line.

The dashed green line represents the condition Dalmont et al. [3] called the “beating reed threshold”. Below this line, the reed remains open throughout the vibration cycle, but above the line the reed is closed for part of the time. The line is based on their analysis, calculated using the Raman model, but it should give a good guide to the behaviour in the more realistic model, as we will see shortly.

The solid cyan line represents the condition Dalmont et al. [3] called the “inverse oscillation threshold”: it shows when the mouth pressure, for a given gap, is just enough to close the reed completely against the lay. It is easy to see why this line gives an upper limit in the case of a very gentle initial transient, as in Figs. 17 and 18: beyond this line, the mouth pressure is high enough that the reed is closed initially, and the gentle transient means that the internal pressure is initialised with the same pressure — so the reed stays closed for ever after, and no note is produced.

But with a more vigorous transient, as in Fig. 19, the reed has a chance to open after the initial closure. For a pixel lying not too far above the cyan line, it is possible for a note to get going, and then to be sustained. However, for a pixel above the highest (red) line there is simply no possible solution (at least within the simplification of the Raman model, which Dalmont et al. [3] used to derive the condition). This line represents what they call the “extinction threshold”. The dashed white line, lying very close to the red line, is what they called the “saturation threshold”. This is the condition for the pressure amplitude to be a maximum, because (within the Raman model) the pressure and volume flow rate when the reed is open fall exactly at the peak of the nonlinear valve characteristic of Fig. 2.

In approximate summary, it may be possible to achieve a note anywhere between the magenta and red/white lines, but above the cyan line it is also possible to get silence — so the player has to use the right kind of articulatory gesture to achieve a note. The space between the cyan and red/white lines is dangerous territory where note production may feel unreliable, especially to a beginner. Indeed, the advice for a beginner would be similar to the corresponding advice to a violinist based on the Schelleng diagram: learn to stay near the middle of the wedge-shaped region in order to get reliable notes every time, and only explore towards the boundaries of the region when you have built up a bit of experience.

It is instructive to see comparable pressure—gap diagrams based on the Raman model. Figure 20 shows a pair of diagrams directly comparable to the ones in Fig. 19. The region within which a note is achieved is rather similar in the two cases, confined between the magenta and red/white lines as just explained. But the figures are much less pretty with the Raman model! If a note is achieved, it always has a square-wave pressure waveform, rich in harmonics, and it always has exactly the same frequency. So the colour shading is completely uniform in both plots.

Figure 21 shows Raman model simulations comparable to Figs. 17 and 18, with a very gentle initial transient. Again, the region in which notes are found is quite similar with both models, but now it stops at the cyan line for the reason explained above. Towards the bottom of the plots, both models show quite a few black pixels above the magenta line, more than in Figs. 19 and 20. The reason for this is simple: these pixels lie only just above the threshold of excitation, and the initial transient is so gentle that the growth is very slow. The pressure amplitude does not get sufficiently high by the end of the simulations to qualify as a “note”.

Finally, Figure 22 shows what happens with the lossless version of the Raman model. “Notes” now extend beyond the red/white line, all the way off the top of the plot. This behaviour is exactly what the theory predicts (see reference [2]). The red/white line here is entirely misleading: it has been plotted in the same position as the other plots, to guide the eye when comparing, but the two thresholds represented by these lines move “to infinity” when there is no energy dissipation. There is no upper threshold, exactly as the simulation results suggest. Notice, by the way, that the lower limit of “notes” is slightly different than in Fig. 20. Non-black pixels now extend right down to the magenta line, rather than stopping just above it. That is exactly what we expect from the original discussion of the threshold of oscillation: with no losses, the threshold lies at the peak of the nonlinear characteristic, rather than just beyond the peak.

D. Transients of the clarinet model

Up to now, as we have developed the clarinet model we have been finding surprisingly close analogies with the behaviour of a bowed string. But when we turn to examine transient behaviour, we will find a strong contrast. Figure 23 shows the transient waveforms of pressure and volume flow rate, for one of the cases used to make up Figs. 17 and 18. Specifically, this case has mouth pressure 3 kPa and reed gap 0.4 mm. Comparing with Fig. 17 reveals that this puts it in the middle of the wedge region, above the beating reed threshold (dashed green line). This is a case with a gentle initial transient, and it can be seen that the pressure variations start small, and grow quite slowly. But after a few period-lengths it settles to a periodic waveform looking recognisably like a square wave.

Figure 23. A clarinet transient from the set used for Figs. 17 and 18, with mouth pressure 3 kPa and reed gap 0.4 mm.

Figure 24 shows the same information, plotted (in red) on top of the nonlinear valve characteristic. This shows what part of that characteristic is used by this particular note. It can be seen that the red line extends only a rather short distance along the flat portion, where the reed is closed. This fits in with what we learned by looking at Fig. 17: we are above the beating reed threshold, but not very far above it.

Figure 24. The transient from Fig. 23, showing the region of the nonlinear valve characteristic that it occupies.

Now compare Fig. 23 with Figs. 25, which shows the corresponding transient computed with the Raman model. The pressure waveform is very square, as we expect from the Raman model, but it shows a very close resemblance to the pressure waveform in Fig. 23. The waveforms of volume flow rate look more different at first sight, but actually they are about as similar as they could be.

Figure 25. The same transient as in Fig. 23, simulated using the Raman model.

The Raman model jumps abruptly between different vales of pressure and volume flow rate, whereas in the more realistic model they both vary smoothly and continuously. The result is made particularly clear in Fig. 26, which is the equivalent of Fig. 24: the points on the nonlinear characteristic are sparse, rather than forming a solid red line. By the end of this portion of transient, the Raman model is simply alternating between two points, which are the leftmost and rightmost of the red stars in Fig. 26. This means that the waveform of volume flow in Fig. 25 is a square wave, whereas the corresponding waveform in Fig. 23 has to go up and over the hump of the curve each time the pressure switches from high to low. This produces a pair of “rabbit ears” in each cycle of the blue waveform in Fig. 23. If you ignore those, the waveform looks much more recognisably like the one in Fig. 25.

Figure 26. The Raman model transient from Fig. 25, superimposed on the nonlinear valve characteristic. The final periodic state corresponds to alternation between the rightmost and leftmost pair of points.

These examples demonstrate two things that appear to be typical: the transients of this model clarinet are always rather “simple”, and the Raman model usually matches the more realistic model quite well. Both these things are far from being true for bowed-string transients! Bowed-string transients are usually complicated, and the details are highly sensitive to everything you can think of — the player’s gesture, the details of the model of the linear system (string and instrument body), and also the details of the nonlinear model used to describe the friction force. In particular, switching to Raman’s model completely changes everything.

As a reminder of some of the discussion of this issue from sections 9.5—9.7, Fig. 27 shows a repeat of Fig. 6 from section 9.5. This shows three measured transients, differing only in the normal force between bow and string. The middle plot (in red) is the only one that echoes the simplicity of the clarinet transients. The upper plot (black) shows a “scratchy” transient featuring irregular string motion before eventually settling into a periodic waveform; the lower one (blue) shows a transient that leads to an entirely different periodic regime of string vibration (“double-slipping motion”). Neither of these phenomena has any obvious parallel within the clarinet model, and it seems that there is no need to look for an analogue of the Guettler diagram (see section 9.5).

Figure 27. Three measured transients of a bowed string, reproduced from Fig. 5 of section 9.5.

What makes the clarinet so different from a bowed string in terms of transient behaviour? There are three factors that probably contribute. First is the nature of the nonlinearity. The reed valve characteristic is probably somewhat more benign than the rather vigorously nonlinear friction curve. However, we saw in section 9.6 that the friction-curve model is probably not realistic. A different model is needed to give a satisfactory account of the frictional behaviour of violin rosin, and the thermal models that were discussed in that section behave in a more benign way than the original friction-curve model. So it is not clear how important this factor might be.

The second factor is a disparity of damping between a clarinet tube and a violin string. The Q-factor of the fundamental mode of our model clarinet tube is 26, whereas the Q factor of the fundamental of a typical finger-stopped violin string is of the order of 300. Damping has a stabilising influence on self-excited vibrations, and this factor of 10 difference may well help to make a bowed violin string more “twitchy” than the clarinet.

But the third factor is probably the most important. The clue to this lies in Raman’s argument. Raman originally formulated this argument in response to published measurements showing a range of possible vibration waveforms of a bowed string. In the simplest version of his argument, he imagined the bowed point dividing the string length in a ratio $1:n$, where $n$ is a whole number. Any possible periodic motion of the string, within his idealised model, will then have $n$ segments of constant velocity per period. The argument leading to Fig. 8 above means that the velocity within each of these segments can only take one of two possible values. Raman used this as the basis for a classification scheme for all possible motions of a (lossless) bowed string. He showed that there are indeed many possibilities.

But our clarinet model corresponds to bowing a string at its mid-point. In the lossless case, each cycle of a periodic motion at the fundamental frequency can then only have two segments of velocity. So there is only one possible way to have a two-velocity solution: it must be the kind of square wave we have been looking at. The lossless clarinet, or the mid-point bowed string, simply does not allow other periodic solutions (unless they have different periods, for example following a “period-doubling bifurcation” to give a note an octave lower). So one major aspect of the discussion of playability of bowed-string transients does not arise for the clarinet. We do not need to ask whether a given gesture leads to Helmholtz motion or to some other periodic regime like double-slipping motion, because there are no such alternative regimes!

E. Influence of the reed resonance

We haven’t finished with clarinet models yet. So far, everything has been based on the simplest mouthpiece model, in which the reed moved as if it was a simple spring. But of course the reed has its own resonance frequencies, and we already showed (back in section 11.3.1) how to incorporate the first reed resonance into a mouthpiece model. It is easy enough to incorporate this extra ingredient into a computer model, so we can look at some simulated results. For all the results to be shown here, the reed resonance frequency is chosen to be 2.8 kHz, based on a suggestion by Chaigne and Kergomard [1] (see section 9.2.2.2). The only other new parameter we need is the damping of the reed resonance. It is far from clear what value this damping should have, so we will look at results for several different values of the reed Q-factor. In practice, the reed damping is no doubt influenced by contact with the player’s lip, so it may vary with changes of embouchure.

Figure 28 gives a first example of a transient simulated with this new model. It is presented in the same format as Figs. 23 and 25, with the mouthpiece pressure in the top graph and volume flow rate in the lower graph. We will explain shortly which precise transient this is: for the moment, we just want to note some qualitative things about it. The pressure waveform looks quite similar to the one in Fig. 23, tending quite quickly to a slightly rounded-off square wave. But in the first few cycles you can see the effect of the reed resonance “ringing on” briefly after each jump in pressure.

Figure 28. A typical transient from the clarinet model, allowing for the reed resonance.

The lower graph in Fig. 28 looks a little different from its counterpart in Fig. 23, because the pairs of “rabbit ears” in the final periodic motion are no longer symmetrical: the second “ear” is much bigger than the first one. A different view of this phenomenon is given by Fig. 29, which shows the same data plotted in the pressure/flow-rate plane. The nonlinear valve characteristic that we used in the earlier models is also shown, in blue. Rather obviously, the new model shows a loop above and below this blue curve: it rarely follows the curve.

Figure 29. The final third of the transient from Fig. 28, plotted in red stars on top of the quasi-static nonlinear reed characteristic that we have seen before.

The explanation lies in the damping of the reed resonance. When the pressure jumps abruptly in the Helmholtz-like square wave, the reed no longer responds immediately, like the quasi-static spring model we used earlier. Instead, the damping imposes a slight lag. When the reed is open before the pressure jump, it stays open a little longer than before, so that the air flow through the gap is bigger. But if the reed is closed before the pressure jump, it stays closed a little longer and the air flow is reduced. The result is a hysteresis loop, followed in an anti-clockwise direction in this plot. This loop is reminiscent of the hysteresis loop we met back in section 9.2, for the frictional behaviour of a bowed string. In the bowed string, this effect was responsible for the “flattening effect”, so it should not come as a surprise that we will see shortly that it causes the pitch of our clarinet to fall: the note plays a little flat.

To complete the set of plots of this example transient, Fig. 30 shows the corresponding displacement of the reed. Positive values here correspond to the reed moving towards the lay, and when the value reaches 0.6 mm the reed closes — because that was the choice of initial gap used in this particular example. Throughout the plot you can see some evidence of transient ringing of the reed resonance, whenever the displacement changes abruptly in response to a pressure jump in the mouthpiece.

Figure 30. The reed displacement $y(t)$ from the transient of Fig. 28.

The natural next step is to use the new model to generate pressure-gap diagrams, to reveal the behaviour over a region of parameter space. Figures 31, 32 and 33 show three examples that are directly comparable with Fig. 19. They use three different Q-factors for the reed: 1.25, 2.5 and 5 respectively. All details of the plots are the same as Fig. 19, except that the range of behaviour encoded by the colours is different. The left-hand plot of each pair, representing the frequency content via the harmonic number that first falls 10 dB below the ideal square wave, has a scale that runs up to 25, compared to 50 in Fig. 19. All the waveforms from the new model tend to be less rich in higher harmonics: we will see an explicit spectrum plot in a moment.

The right-hand image of each pair is colour-shaded to show the frequency of the final periodic waveform. In this case, the scale represented by the colours is longer than in Fig. 19. The right-hand plot of Fig. 19 did show a slight tendency to flattening towards the bottom right-hand corner, but the total range of frequencies covered by that plot was 128.8—129.1 Hz, a total range of about 1/4 Hz. The corresponding range for the new plots (all three give a similar range) is 127.9—129.0 Hz, a range of 1.1 Hz.

To get an idea of what lies behind these plots, and how the new results relate to the earlier model, Fig. 34 shows the first few period-lengths of a particular transient for all the models. Specifically, it is the one in the bottom right-hand corner of the pressure-gap diagrams, marked by a blue circle in the right-hand plot of Fig. 33. The black curve is the result from the earlier model, where the reed was treated as a spring. The other three are drawn from the sets used to generate Figs. 31—33: the green curve has the reed’s $Q=1.25$, the blue curve has $Q=2.5$ and the red curve has $Q=5$. The changing damping of the reed resonance is immediately apparent in the extent of the “ringing” after each pressure jump. It is reassuring to see that the blue and red curves track quite close to the black curve — this is a useful check on the accuracy of coding of the new computer model. The blue curve here shows exactly the same transient used to generate Figs. 28, 29 and 30 earlier.

To get an idea of what lies behind these plots, and how the new results relate to the earlier model, Fig. 34 shows the first few period-lengths of a particular transient for all the models. Specifically, it is the one in the bottom right-hand corner of the pressure-gap diagrams, marked by a blue circle in the right plot of Fig. 33. The black curve is the result from the earlier model, where the reed was treated as a spring. The other three are drawn from the sets used to generate Figs. 31—33: the green curve has the reed’s $Q=1.25$, the blue curve has $Q=2.5$ and the red curve has $Q=5$. The changing damping of the reed resonance is immediately apparent in the extent of the “ringing” after each pressure jump. It is reassuring that the blue and red curves lie quite close to the black curve.

Figure 34. The first few period-lengths of the transient lying at the bottom right-hand corner of Figs. 19, 31, 32 and 33, corresponding to a reed gap of 0.6 mm and a mouth pressure 4 kPa. The black curve is for the old model without the reed resonance. The other three are from the new model, for the three cases shown in Figs. 31–33: green has $Q=1.25$, blue has $Q=2.5$ and red has $Q=5$.

Even on the short time-scale of this plot, you can see the flattening effect developing. All four pressure waveforms are synchronised at the start, but by the right-hand side the black curve (from the old model, with less flattening) has already got slightly ahead of the three coloured curves. The effect is easiest to see by comparing the near-vertical lines where the pressure jumps up or down.

All four of these transients led to a periodic waveform of pressure when the simulation was run for longer. Taking a chunk of this eventual waveform and applying the FFT, we can easily generate corresponding frequency spectra. The results are shown in Fig. 35, with the same colour code as in Fig. 34. The curves have been separated vertically for clarity, with the old model at the top, then the three cases of the new model in descending order of Q-factor.

Figure 35. Frequency spectra of the eventual periodic pressure waveforms following the four transients shown in Fig. 34. The colour code is the same as in Fig. 34. The curves have been separated by 60 dB for clarity. The blue curve (for $Q=2.5$) is marked with red stars to indicate the first few odd-numbered harmonics, and green stars for the corresponding even-number harmonics.

It is clear that the reed damping is having a significant effect on the frequency content of the note. With the old model (black curve) sharp peaks at the harmonics extend over the whole range of this plot. But with the reed resonance included in the model, the peak amplitudes die away with rising frequency — slowly with $Q=5$ (red curve), faster with $Q=2.5$ (blue curve), and very fast with $Q=1.25$ (green curve). This effect is quite audible, especially with the lowest Q-factor. Figure 36 shows a sequence of three notes from these simulations including the reed resonance, combined in the same way as in Figs. 9, 11, 13 and 15 (with an artificial exponential decay imposed to separate the notes). The sequence has $Q=1.25$ first, then $Q=2.5$, then $Q=5$. You can hear the resulting sound in Sound 5.

Figure 36. A sequence of three notes, combining the three simulations from Figs. 34 and 35 which include the reed resonance. The sequence is $Q=1.25$, then $Q=2.5$, then $Q=5$.
Sound 5. Audio demonstration of the three notes from Fig. 36.

Returning to Fig. 35, there is another feature we should mention. Our model clarinet tube is closed at one end and open at the other, so that the resonance frequencies match odd-numbered harmonics 1, 3, 5, 7… only. The tube has no resonances at the even-numbered harmonics. However, the nonlinear action of the reed valve can generate all harmonics, not just the odd-numbered ones. So the internal spectrum of the pressure, like the examples shown in Fig. 35, may contain even as well as odd harmonics. This is indeed what the plot shows. As a guide to the eye for identifying these harmonics, the blue curve has been annotated with red stars marking the first few odd-numbered harmonics, and green stars marking the corresponding even harmonics.

All four curves are dominated by odd harmonics at low frequency, but at higher frequencies the even harmonics become increasingly visible. But this plot shows the spectrum of the internal pressure, so the odd harmonics are supported and boosted by the tube resonances. The sound of the instrument, based on the external sound pressure radiated from the open end of the tube, may be significantly different. As Benade pointed out many years ago, the even-numbered harmonics correspond to antiresonances of the tube, not to resonances. At those frequencies the pressure is a maximum near the open end (or an open tone-hole), not a minimum as it is for the resonant frequencies.

The result is that any component of the internal pressure corresponding to an even harmonic may be radiated much more efficiently than the corresponding components from the odd harmonics. This effect goes some way to compensating for the even-odd pattern of harmonic amplitudes: the sound of a clarinet may contain significant levels of some even harmonics. This is indeed found to be the case when a clarinet sound spectrum is measured: Fig. 37 shows just such a measurement, of the lowest note of a clarinet with all holes closed, played forte. It is taken from Joe Wolfe’s web site. At the lowest frequencies, the odd-numbered harmonic peaks are higher than their neighbouring even-numbered peaks, but this even-odd pattern rapidly disappears as you move up in frequency.

Figure 37. Measured sound spectrum of the lowest note of a clarinet, with all tone-holes closed. Image taken from https://newt.phys.unsw.edu.au/music/clarinet/E3.html, reproduced by permission of Joe Wolfe.

Looking all the way back to Fig. 33, there is another feature that we should comment on. Along the lower edge of the wedge-shaped region of coloured pixels is a small wedge of white pixels. One of these is highlighted by a green circle. These white pixels represent what a clarinettist would call “squeaks”: instead of eliciting the expected tone based on the tube resonances, you sometimes get a much higher, and discordant, frequency. This has more to do with the resonance of the reed than with those of the tube. Indeed, you can make a squeak with a bare mouthpiece, detached from the tube.

A region in the pressure-gap diagram corresponding to squeaks rather than notes was found in the measurements by Almeida et al. [2]. It is rather encouraging that the simulation model produces a similar prediction, in roughly the same region of the diagram. If we take the transient marked by the green circle and assemble three notes in the same manner as Fig. 36 and Sound 5, we can hear the effect in action. The same sequence of reed Q-factors is used, and the assembled waveform is plotted in Fig. 38. You can hear the result in Sound 6. You will notice that the first note, with $Q=1.25$, shows a decaying transient rather than a sustained note — you just hear a kind of “thump”. The second note, with $Q=2.5$, gives a fairly normal clarinet-like tone, while the third one, with $Q=5$, gives the squeak.

Figure 38. A sequence of three notes in the same format as Fig. 36, for the three transients at the position marked by a green circle in Fig. 33. The sequence is $Q=1.25$, then $Q=2.5$, then $Q=5$. The first gives no note, just a transient decay. The second give a normal note, the third gives a squeak.
Sound 6. Audio demonstration of the three notes from Fig. 38.

F. Conical reed instruments

So far, all our effort has been devoted to the model clarinet, based on a cylindrical tube. But many reed instruments are based on tapered, conical tubes — the saxophone, the oboe and the bassoon, for example. We already know about one key difference between a conical tube and a cylindrical tube: we saw in section 4.2 that the resonance frequencies of a conical tube, tapering down to point, form a complete harmonic series. By contrast, a closed cylindrical tube like our clarinet model has resonance frequencies that are odd-numbered harmonics only.

We will study one idealised example of a conical reed instrument, to compare with the clarinet model we have already discussed. This will be based on a soprano saxophone, which has a conical tube somewhat longer than a clarinet. The lowest note, with all tone-holes closed, is a lot higher than the corresponding note of the clarinet, though, because the lowest mode has a half-wavelength in the length of the tube, whereas a clarinet has a quarter-wavelength.

Our model will use a similar level of simplification to the clarinet model: the aim in both cases is to bring out the essential physics of the two types of instrument with minimal complication, and draw some conclusions about the behaviour as a player might perceive it. Figure 39 shows a soprano saxophone, and immediately below it is the first stage of idealisation. With all holes closed, and ignoring the bell, we can approximate the internal bore shape by a straight-sided cone. But of course the real instrument does not taper all the way to a point. Instead, it is truncated and terminated by a mouthpiece: a small cavity carrying a flexible reed rather similar to the clarinet reed.

Figure 39. A soprano saxophone, with sketches of various stages of idealisation. Saxophone image copyright Joe Wolfe, reproduced by permission.

The third image in Fig. 39 shows the truncated cone, without the mouthpiece. It also shows the “completion” of the cone in a dotted red line, extending some way beyond the physical length of the instrument. The input impedance of a truncated cone like this can be calculated, as explained in the next link. The answer, slightly unexpectedly, involves the length of the “missing” portion of the cone as well as the physical length of the tube.

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The resulting formula for the impedance tells us several useful things. Its inverse, the input admittance, has resonance peaks at exactly the same frequencies as a cylinder of the same length. However, these are the resonance frequencies that would occur with both ends of the tube open. For our saxophone model the reed and mouthpiece will be like a closed end, just as in the clarinet model. The resonances are then given by the peaks of impedance, not of admittance. As we will see in some detail shortly, those peaks are not harmonically spaced — potentially bad news for our saxophone.

The final image in Fig. 39 shows a rather unexpected approximation to the acoustic behaviour, known as the “cylindrical saxophone” [1,4] and explained in the previous link. When the frequency is sufficiently low that the wavelength of sound is much bigger than the length of the missing piece of cone, the formula for the impedance becomes the same as the impedance of a cylindrical tube with the same length as the completed cone, open at both ends and excited at the position where the actual cone was truncated. So we can imagine a saxophone mouthpiece plugged in to the side-wall of the cylindrical tube, as sketched in figure. This cylindrical approximation is not accurate for the saxophone at higher frequencies, and we won’t rely on it when we come to run simulations. But it is illuminating because it tells us something qualitative about how a saxophone might behave. (Indeed, you can buy a real musical instrument based, rather loosely, on the idea of the cylindrical saxophone: the Yamaha Venova.)

The important thing is that the “cylindrical saxophone” frequency response applies equally well to a string, bowed at the same intermediate position. This allows us to extend the analogy between wind instruments and a bowed string, to incorporate the main ingredient that was missing when we talked about the clarinet. In the clarinet case, the equivalence was to the very special case of a string bowed at its mid-point. But the new analogy suggests that a saxophone (or an oboe or a bassoon) might behave a bit like a string bowed at a more normal position, relatively close to one end.

Immediately, we can make some guesses. Based on what we already know about bowed strings, the analogy suggests that the saxophone model might behave very differently from the clarinet model. First, the equivalent of the Helmholtz motion will no longer be a square wave of pressure. Instead, we expect something resembling the velocity waveform of a bowed string: a pulse wave, with the pulse occupying a proportion of each period determined by the position (as a fraction of the total tube length) of the “mouthpiece” in the bottom sketch of Fig. 39.

But something more fundamental will also change. For the clarinet, or the mid-point bowed string, we argued (following Raman) that the square wave was the only possible periodic waveform at the natural period of the tube or string. But now we have a “bowing point” that divides the length unequally — for the numerical cases we will see shortly, this division is roughly in the ratio 1:8. This opens the floodgates to a wealth of alternative waveforms, just as Raman found for a bowed string. So we might expect to see far more complicated behaviour of the saxophone model, compared to the clarinet model. We will shortly see some simulation results, and we will be on the lookout for behaviour that reminds us of the response of a bowed string, such as the appearance of periodic waveforms other than the Helmholtz motion.

There are a couple more details to mention, to complete a model of the actual (non-cylindrical) saxophone that we can use for simulation studies. One of these concerns the mouthpiece cavity, indicated by the sketch immediately below the photograph in Fig. 39. We have already mentioned that the impedance peaks for the truncated cone are not distributed according to an accurate harmonic series. But, at least for the first few modes, we can improve things a little by including the mouthpiece cavity in the model, and making a careful choice of its volume.

The approximation again works best when the wavelength of sound is very long compared to the missing portion of the cone. If the cone had been complete, then of course the air trapped in that final section would have been rigidly enclosed. We already know how such a confined volume of air behaves — we thought about it when looking at the Helmholtz resonator back in section 4.2. The trapped air behaves like a simple spring, and its stiffness depends only on the total volume, not on the shape. We also know that the complete cone would have harmonically-spaced resonances, so if we choose a mouthpiece cavity with the same volume as the “lost” section of cone we should find that at least the first few resonances still have approximately harmonic spacing. Sure enough, measurements of the volumes of preferred mouthpieces for a range of conical reed instruments (including some allowance for the extra “effective volume” arising from the reed’s flexibility) conform quite well to this predicted pattern, matching the volume lost by truncating the cone.

The final ingredient for a simulation model is the reed. But this one is easy — a soprano saxophone reed and mouthpiece are not very different from those of a clarinet, so we can use the same model with minor adjustments to parameter values. The details of the values used here are given in the previous link, but I have used some guesswork in choosing them: published data on saxophone reeds is in far shorter supply than on clarinet reeds. But this should not matter very much, since the aim here is to bring out qualitative behaviour, not to match quantitatively the behaviour of any particular saxophone — that is a task for future research.

So let us look at some simulation results. Figure 40 shows a successful transient, in the same format as Figs. 23 and 28: pressure inside the mouthpiece in the upper plot, volume flow rate through the reed in the lower one. The pressure waveform rapidly settles to a periodic state, and it is indeed a pulse wave as we anticipated from the “cylindrical saxophone” argument. The lower plot reveals that the reed closes once per cycle, for a rather short time that more or less matches the length of the pressure pulse.

Figure 40. A transient of the saxophone model, corresponding to the 6th pixel in the row of Fig. 43 marked with a green line. Upper plot: pressure inside the mouthpiece; lower plot: volume flow rate through the reed.

To see the other “cylindrical saxophone” prediction in action, we only have to change the mouth pressure a little. Figure 41 shows a transient with a slightly increased mouth pressure, and Fig. 42 shows the result after a further small increase. You can hear these three transients in Sound 7. The transient in Fig. 41 shows what in violin terms we would call “double slipping” or “surface sound”. The waveform settles to a periodic state involving two pulses per period, and two corresponding short episodes when the reed closes. But the sound is at the same pitch as the Fig. 40 note, not an octave higher. The reason is that the two pulses are not equally spaced in the cycle, so the periodicity of the note is unchanged. Pressure waveforms looking very much like this have been reported from real saxophones [5].

Figure 41. A saxophone model transient in the same format as Fig. 40, corresponding to the 8th pixel in the row of Fig. 43 marked with a green line.
Figure 42. A saxophone model transient in the same format as Figs. 40 and 41, corresponding to the 9th pixel in the row of Fig. 43 marked with a green line.
Sound 7. The three simulated saxophone notes from Figs. 40, 41 and 42, assembled in a similar format to the earlier sound demonstrations in this section.

But in Fig. 42 we see a different pattern. This time, the early part of the transient waveform shows groups of four pressure pulses (a “quadruple slip” motion in bowed-string terms). But by the end of the plot, two of these pulses have faded away and disappeared, and the remaining two are equally spaced. The result is that the final note sounds an octave higher, at least by the end. In the early part of this note, the third note in Sound 7, you can still hear a trace of the original pitch before it moves up to the octave.

The natural next step is to compute a pressure-gap diagram for the model saxophone: the result is shown in Fig. 43. I will explain the colour shading in a moment, but first we can note the wedge-shaped region of coloured pixels: the saxophone has a region of “note” rather than “silence” that looks broadly similar to what we already saw for the clarinet. The magenta and cyan lines have the same meanings as in previous pictures: they mark the threshold of vibration, and the threshold beyond which it could be possible for the reed to remain permanently closed. But the other thresholds shown in earlier pressure-gap diagrams are omitted because they have no direct counterpart here: they were derived from calculations with the Raman model, and a full set of equivalent conditions for the saxophone model has yet to be derived (but see section 9.4.8.3 of Chaigne and Kergomard [1] for some efforts in that direction). The horizontal green line in Fig. 43 indicates the row from which the transients of Figs. 40—42 were drawn. Counting from the left-hand side, these transients correspond to pixels numbers 6, 8 and 9. Pixel 6, corresponding to Fig. 40, is the first coloured pixel beyond the magenta threshold line.

Figure 43. Pressure-gap diagram for the saxophone model, with the same kind of vigorous initial transient used for Fig. 19 in the case of the clarinet. The green line marks the row containing the transients shown in Figs. 40, 41 and 42. The white lines mark cases illustrated in Fig. 44. The colour indicates the frequency of the final waveform as a multiple of the nominal frequency, based on analysis of autocorrelation.

To relate this to the waveforms we have seen, I need to explain the colour shading. First, we should look at a bigger set of waveforms. Figure 44 shows the full set of waveforms for the coloured pixels lying along the three white lines in Fig. 43. Three nominal period-lengths are plotted, from the end of each individual simulation. For each of the three sets, the top line shows a pulse waveform similar to the one in Fig. 40 (it just looks a little different because the vertical scale has been squashed). As you go down each stack of waveforms, the pulse waveform develops into shapes with more pulses. But before you reach the bottom of each stack you see a return to pulse waveforms — but upside down compared to the ones at the top of the stacks.

Figure 44. Some sample waveforms from the simulations lying behind Fig. 43. The three columns correspond to all the coloured pixels in the three rows marked with white lines. Left-to-right rows in Fig. 43 become top-to-bottom stacks here, so that mouth pressure increases as you move down each column. The last three period-lengths of each pressure waveform are plotted.

A more detailed example of one of these inverted pulse waveforms is shown in Fig. 45: this is from the right-most coloured pixel along the green line in Fig. 43. You can see what has happened: the reed opens and closes once per cycle as before, but now it is closed nearly all the time, opening only for a short pulse. This is the opposite behaviour to the case in Fig. 40.

Figure 45. A saxophone model transient in the same format as Figs. 40–42, corresponding to the 27th pixel in the row of Fig. 43 marked with a green line.

With these waveforms in mind, the colour shading of Fig. 43 can be explained. The last few period-lengths of each simulation were used to compute something called the autocorrelation function. The signal is multiplied, sample by sample, with another version of the same signal with a time lag, and the products are all added together. After normalising to allow for the absolute magnitude of the signal, the result is a number between $-1$ and $1$, and this number varies as the lag is changed. If the signal is periodic, then the autocorrelation reaches the value $1$ when the lag matches the period. So the period of the waveform can be deduced by looking for the smallest lag which yields a value of the autocorrelation above some chosen threshold. If the nominal period is divided by this lag, the result would be 1 for any signal at the nominal frequency, 2 for a signal an octave higher (i.e. at the second harmonic), 3 for a signal at the 3rd harmonic and so on.

This number is what has been used to colour the pixels in Fig. 43. If you compare with the scale in the colour bar, you can see that the chosen colours are almost invariably close to the values 1, 2, 3, 4 or 5. Now scan along the lowest of the three white lines, and compare pixel by pixel with the waveforms in the 3rd column of Fig. 44. The pulse waves at the start and at the end all give the dark red colour connoting the value 1: these cases all correspond to “normal” sound at the nominal frequency. The 4th coloured pixel along the line gives a bright red colour, connoting the value 2. Sure enough, the 4th waveform from the top shows a symmetrical “double slip”, sounding an octave higher. Continue this comparison, and the colour code should make sense.

So why have I chosen this way to colour the diagram? The answer to that lies in the behaviour we hope for in our saxophone. We might expect to find a region of the plane where it plays the “right” note. But then adjacent to that, we would not be surprised to find a region where the pitch jumps by an octave: this would correspond to “over-blowing to the second register”. A bit more, and we might over-blow to the third register, indicated by an orange colour connoting the value 3. We see exactly this pattern if we scan downwards from the top edge of the wedge region: a broad wedge with the value 1 (corresponding to “inverted” pulse waveforms like Fig. 45), then a thin line of red followed by a line of orange. After that it gets rather messy, because of the complicated waveforms revealed by Fig. 44. Near the bottom of the wedge we see another solid region of the value 1, from the pulse waveforms arranged as in Fig. 40.

So this first attempt at a simulation model for the soprano saxophone’s lowest note shows encouraging behaviour. It is indeed the case that low notes on a saxophone rather readily give “bugling” behaviour, where the note may jump progressively from register to register. The clarinet, by contrast does not show this behaviour: exactly what our plots such as Figs. 31–33 predict. When those plots show a note, it is always close to the expected pitch for the first register. To shift register on a clarinet, you need to open a register hole.

But possibly you think I am glossing over the regions of the saxophone diagram where the pattern becomes complicated? Possibly so — we simply do not have the data to know whether a real saxophone behaves in a rather messy way like these simulations, but we might suspect that the real instrument is a bit better-behaved: it is what we should expect, given that we have used a very crude model, lacking the subtlety of details that have been designed into real saxophones.

We can get an inkling of the sensitivity of the pattern to small details of the tube impedance by comparing with another case. We can’t make our model better without quite a bit of effort, but we can very easily make it a bit worse, and see what effect that has. We can omit the allowance for the mouthpiece volume, and simply use the acoustical behaviour of a truncated cone (as described in the previous link). This is quite a small change to the model, but as you can see from the results in Figs. 46 and 47 it has a significant effect. The wedge region is smaller, and a lot of the details have changed. However, Fig. 47 shows that the repertoire of waveforms is essentially the same: pulse waves both ways up, “double slipping”, and so on.

Figure 46. Pressure-gap diagram in the same format as Fig. 43, generated from the saxophone model without including the effect of the mouthpiece volume.
Figure 47. Some sample waveforms from the simulations lying behind Fig. 46, in the same format as Fig. 44. The three columns correspond to all the coloured pixels in the three rows marked with white lines. Left-to-right rows in Fig. 46 become top-to-bottom stacks here, so that mouth pressure increases as you move down each column. The last three period-lengths of each pressure waveform are plotted.

If we dig a little deeper into this comparison of results with and without allowing for the mouthpiece volume, we can see an interesting example of Benade’s idea of the playing frequency being determined by a collaboration and compromise between the resonance frequencies of the tube, when these are not exactly harmonically spaced. We will compare the spectra of notes produced by the two versions of the model, for a particular case: the chosen example has the same mouth pressure and reed gap as in Fig. 40, corresponding to a pixel just above the threshold of excitation. The two curves in Fig. 48 show the spectra deduced by taking an FFT of a long stretch of simulated periodic sound, for the two cases: the black curve includes the mouthpiece, the green one is without it. Both curves show sharp peaks at the harmonics of the note, and it is immediately apparent that the two models have “chosen” to play at different frequencies. To see why, we need to compare them with the corresponding input impedances.

Figure 48. Spectra of the final periodic waveforms from simulations corresponding to the same case as Fig. 40. The black curve shows the model including the effect of the mouthpiece, the green curve is without this effect. The sharp peaks mark the harmonics of the two notes.

We look first at the note simulated without allowing for the mouthpiece cavity. Figure 49 shows the green spectrum from Fig. 48, compared with the input impedance used in the simulation (blue curve) and also the “cylindrical saxophone” impedance (dashed red curve). This dashed red curve has resonances that are exactly harmonically spaced, so the blue/red comparison gives an immediate visualisation of the “imperfection” of the conical tube. Figure 49 shows that the played note has “chosen” a frequency to match the highest peaks in the impedance, with its 3rd and 4th harmonics. This has the result that the fundamental and the 2nd harmonic are sharp compared to the tube resonances, while the higher harmonics are flat.

Figure 49. The green curve is the same as in Fig. 48, showing the frequency spectrum of a played note from the saxophone model without including the mouthpiece effect. The blue curve is the input impedance used in the simulation, the dashed red curve is the corresponding input impedance of the “cylindrical saxophone” approximation.

Figure 50 shows the corresponding comparison for the case allowing for the mouthpiece. The black curve is the same one as in Fig. 48, the blue curve shows the input impedance that has been used in the simulation, and the dashed red curve again shows the input impedance of the “cylindrical saxophone” model. Compare this carefully with the blue curve. The first two peaks are virtually identical. The third peak shows a small separation, and then the higher frequencies show increasingly drastic divergence. Making the same comparison in Fig. 49, the blue and red curves are separated a little more clearly.

Figure 50. The black curve is the same as in Fig. 48, showing the frequency spectrum of a played note from the saxophone model including the mouthpiece effect. The blue curve is the input impedance used in the simulation, the dashed red curve is the corresponding input impedance of the “cylindrical saxophone” approximation.

This difference in impedance peaks with and without the mouthpiece seems like a small effect, but now compare with the black curve in Fig. 50. Our “saxophone” has chosen to play at a frequency that matches the first three peaks of the impedance curve: the apparently small effect of the mouthpiece has made these peaks sufficiently close to harmonic that their combined influence is able to “beat” the fact that the amplitude of the first peak is relatively low. (That low amplitude of the first peak is the reason that it is notoriously difficult to start low notes on a saxophone, oboe or bassoon very quietly.) The result is a simulation with a fundamental frequency very close to the first resonance of the tube. The higher harmonics of the played note diverge progressively from the peaks of the blue curve. They continue to coincide with the peaks of the dashed red curve, because those are accurately harmonic: as in Fig. 49, the blue/red comparison gives a direct visualisation of the “imperfection” of the conical tube, even with the mouthpiece compensation.

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[1] Antoine Chaigne and Jean Kergomard; “Acoustics of musical instruments”, Springer/ASA press (2013)

[2] Andre Almeida, David George, John Smith and Joe Wolfe, “The clarinet: How blowing pressure, lip force, lip position and reed ‘hardness’ affect pitch, sound level, and spectrum”, Journal of the Acoustical Society of America 134, 2247—2255 (2013)

[3] Jean-Pierre Dalmont, Joël Gilbert, Jean Kergomard and Sébastien Ollivier, “An analytical prediction of the oscillation and extinction thresholds of a clarinet”, Journal of the Acoustical Society of America 118, 3294—3305 (2005).

[4] Chaigne and Kergomard report that the “cylindrical saxophone” approximation was first proposed by A. Gokhshtein, “Self-vibration of finite amplitude in a tube with a reed”, Soviet Physics — Doklady 24, 739–741 (1979).

[5] Tom Colinot, Philippe Guillemain, Christophe Vergez, Jean-Baptiste Doc and Patrick Sanchez, “Multiple two-step oscillation regimes produced by the alto saxophone”, Journal of the Acoustical Society of America 147, 2406–2413 (2020)