We now turn our attention to brass instruments (bearing in mind that they are by no means always made of brass — the name is simply a convenient label for this family of instruments). You might imagine that we will need to develop new modelling to describe the physics of these instruments: they seem to be quite different from the reed instruments. But there is a surprise. When we come to computer simulations of brass instruments, we only have to make a tiny change to the program we have already used for reed instruments: just a single minus sign has to be introduced.
A. What do brass players do with their lips?
But this minus sign makes a crucial difference, and we should begin our discussion of brass instruments with this. We need to understand what a brass player does with their lips, in order to make the instrument sound. Figure 1 shows a repeat of an earlier sketch, of the cup mouthpiece of a brass instrument with the player’s lips pressed against it. Figure 2 is another reminder, of the mouthpiece of a reed instrument like the clarinet, discussed extensively in earlier sections.
For the clarinet, we were never in any doubt about the fact that, because the reed faces into the player’s mouth, the reed tends to blow shut when the player blows hard. What is the corresponding description of the lips of a brass player? The mechanical behaviour of lips, made of squashy flesh, seems less clear-cut than the behaviour of a small cantilever made of cane. Indeed, the scientific literature on brass instruments contains considerable discussion on this matter. But the consensus is that (for a good first approximation) lips behave rather as indicated in Fig. 1: when they vibrate as the player “buzzes” them, they open into the mouthpiece, away from the player’s mouth, and if the player blows harder, the lips tend to open further rather than closing like a clarinet reed. This is where our crucial minus sign comes in.
Having made this decision, the simplest model for how the lips behave is exactly the same as the clarinet model set out in section 11.3.1, apart from this change of sign. Figure 3 shows the result. The dashed red curve is the now-familiar nonlinear valve characteristic of a reed mouthpiece, and the blue curve is what we need for a model of brass playing. It is exactly the same curve, flipped over. I have shown the two curves as completely identical, for didactic purposes, but of course the parameter values relevant to lips (mass, stiffness and so on) will be somewhat different from those of a reed mouthpiece. So the functional form of the curve will be the same as for the reed, but the details will a little different. The specific lip parameters used here are taken from Table II of Velut et al. .
But now, if you can remember as far back as section 8.5 when we first explored the clarinet, you should spot a major snag. Figure 4 reminds you of three diagrams from that section. If a note is to get started on the clarinet from a gentle initial transient, it must grow from small amplitude. To visualise that, we thought about the tangent to the curve. As the player increases blowing pressure, the “operating point” on the valve curve moves to the left. All the time the tangent at this operating point has a negative slope, as in the first two diagrams in Fig. 4, the reed curve simply contributes extra energy dissipation.
But when the operating point moves over the hump of the curve, as in the third diagram, the tangent line then slopes the other way, and it behaves like a source of energy rather than dissipating energy. When that energy source is strong enough to overcome the physical energy dissipation inside the clarinet tube, the note will start — this is the threshold condition, forming the lower edge of the wedge-shaped regions in most of the pressure-gap diagrams from sections 11.3 and 11.4. (The exception was where we noted evidence of an “inverse bifurcation”, allowing a note to be sustained in a region where it would not be possible to start it from a gentle transient.)
Returning to the blue curve in Fig. 3, we can see that this reversal of slope will never happen! The tangent to the curve always slopes the same way, so it will always be a source of energy dissipation. So how does a brass player ever get a note to start? Well, I cheated in the description I just gave of the clarinet, because I failed to mention an important assumption. The argument relied on the fact that the reed would displace in the same direction as the force acting on it from the pressure difference on the two sides. That seems intuitively obvious, but in fact it is only true at frequencies below the resonance frequency of the reed or lips.
We need to recall something we saw way back in section 2.2, when we looked at the very simplest vibrating system: a mass on a spring, as sketched in Fig. 5. If we apply a sinusoidal force to the mass, indicated by the arrow, then if the frequency of the force is below the resonance frequency, the effect of the spring dominates over the effect of the mass, and the mass moves in phase with the force. But the resonance frequency is determined by the condition that the effects of the spring and the mass are exactly equal. For a forcing frequency higher than that, the effect of the mass now dominates. The mass then moves in the opposite direction to the force: the phase has reversed.
This phase reversal is the key to understanding what brass players do with their lips. We have seen that the slope of the tangent to the nonlinear valve characteristic is always negative. But if the player can arrange that their lip resonance is a little lower than the note they are trying to play, the negative sign of the tangent slope is “cancelled out” by the phase reversal. The negative slope then behaves like a source of energy, and the note can grow. So brass players must constantly adjust the tension in their lip muscles, to change the resonance frequency of the lips to lie just below each note they are trying to play. That, at least, is my claim. We will want to be convinced of all this by seeing some simulated examples.
B. Input impedance and the role of the mouthpiece
But before we start to look at simulation results, there are some important aspects of the acoustics of a typical brass instrument tube that we should look at. We have already said a bit about the mode shapes and resonance frequencies of such tubes, back in section 4.2. The shape of a typical brass instrument is sufficiently complicated that there is no equivalent of the super-idealised models we used when we started to look at the clarinet and the saxophone. We need to resort to computation, and Fig. 15 of section 4.2 showed the result of a numerical solution of the “Webster horn equation”, with a bore profile chosen to resemble a brass instrument like a trumpet or trombone, with a cylindrical section followed by a gently flaring section, and finally an abruptly flaring bell.
Those computed modes can be plugged directly into the formula derived in section 11.4.1, to estimate the input impedance. Our first step should be to check whether the result gives a reasonable match to a direct measurement of input impedance. We will look at the example of a trombone (with the slide in first position). The red curve in Fig. 6 shows the result of applying the procedure just described, and the blue curve shows a measurement on a real trombone (without its mouthpiece – we come to that in a moment). The blue curve has been displaced downwards by 40 dB for clarity, but Fig. 7 shows a comparison of the same two impedances zoomed to the low-frequency range, superimposed without a shift.
Figure 6 shows that both the real trombone and the simulated version have many resonances: the peaks are still going strong at the limit of the measured frequency range, 4 kHz. Figure 7 shows that the simulated version does a pretty good job of matching the measurement: the general level is accurately matched, the peaks line up reasonably well, and the peak-to-valley excursions are matched well, apart from the first few modes. For these low modes, the simulated version shows higher peaks and deeper valleys, compared to the measurement.
Part of the reason for this discrepancy is connected to the assumed damping in the simulated version. The model from section 4.2 gave undamped mode shapes and frequencies. In order to convert to a realistic impedance, the damping model described in section 11.1.1 has been used. This model represents energy losses associated with viscous and thermal losses along the walls of the pipe. The specific formula is widely quoted in the literature, but the measurements shown in Fig. 6 shed some interesting light on it: as explained in the next link, the formula gives an estimate of the modal Q-factors which always lies within a factor of 2 of the measured values, but the measurements clearly show a different trend with frequency, suggesting that this formula should probably be revisited.
Now we need to add a mouthpiece to the trombone. The shape of a typical brass mouthpiece is sketched in Fig. 8, and we can see in Fig. 9 that such a mouthpiece has a profound influence on the input impedance. Concentrate for the moment on the blue curve, a direct measurement of the same trombone as the blue curve in Fig. 6, now fitted with a mouthpiece. Instead of peaks extending all the way to high frequencies, we now see only a dozen or so strong peaks, after which the curve smooths out to a rather featureless shape.
To see why this has happened, we turn to our simulated trombone and add a mouthpiece model to that. We can’t use the Webster equation directly for the mouthpiece, by simply changing the bore profile near the end of the tube. The reason is that the Webster equation is based on an approximation which assumes that the bore only varies slowly along the length of the instrument. But the mouthpiece bore profile, from Fig. 8, changes dramatically over just a few centimetres.
So we use a different approach. The mouthpiece sketched in Fig. 8, once it has been closed at the left-hand side by the player’s lips, should remind you of something we saw earlier: the Helmholtz resonator, from section 4.2. The cup traps a volume of air, and the backbore makes a narrow entrance to that volume. The result will be a resonance frequency governed by the balance of the stiffness of the enclosed air, and the mass of the air in the backbore. For low frequencies, while the wavelength of sound is very long compared to the dimensions of the mouthpiece, the acoustical behaviour should be well approximated by a simple model based on this effective mass and stiffness. The next link described how such a model can be coupled to the input impedance of the tube without mouthpiece. The result of that calculation is the red curve in Fig. 9, and it can be seen that it looks reassuringly similar to the measured blue curve.
The Helmholtz resonance frequency in this case was set at 460 Hz, and it is clear from the plot that it is above this frequency that the resonance peaks in the admittance begin to fade away: the mouthpiece imposes a “cutoff frequency”. What is happening is that above the Helmholtz resonance frequency the mass associated with air in the backbore starts to dominate the behaviour. That mass is less and less willing to move as frequency rises above the resonance. The result is that sound waves in the tube are reflected by this immobile mass, so that the pressure variation cannot reach the player’s lips on the other side of the cup volume. This automatically reduces the heights of the impedance peaks to create the cutoff effect.
The simple Helmholtz resonator model tells us how the resonance peaks should be affected by adding the mouthpiece. The model predicts that below the mouthpiece resonance frequency the frequencies are all reduced, while their Q-factors are slightly increased. Above the resonance frequency, the Q-factors are decreased. The frequencies continue to decrease, but the rate of reduction slows down and each frequency tends towards the frequency of the next-lowest of the original frequencies. Something of this pattern can be seen directly in Fig. 10, which shows the measured impedance with and without the mouthpiece, over the low-frequency range. Each peak in the blue curve lies lower in frequency than the corresponding peak of the black curve, and the first few peaks are a bit taller and narrower because of the reduced Q-factor.
We can see this behaviour in action by processing the various impedance functions to extract modal parameters. Figure 11 shows one aspect of the behaviour that is revealed. This plot shows the “effective fundamental frequency” of each mode, calculated by dividing each frequency by the mode number. The two lines at the top, in black and red, show the results without the mouthpiece, from the impedances in Fig. 6. The measured results (in black) are, predictably, a little less regular than the idealised synthetic values (in red). But the general pattern of both curves is the same: all frequencies except the lowest lie close to a horizontal line, demonstrating that the frequencies are very close to the pattern of an ideal harmonic series. For reasons explained earlier (see section 4.2) the fundamental frequency does not follow this pattern: it is always far too low to fit into the harmonic pattern.
The other lines in Fig. 11, with points marked by circles rather than stars, show results with the mouthpiece. The magenta line shows the pattern of frequencies from the measured impedance in Fig. 9, and the green line shows the results for the computed model with the mouthpiece, the red curve in Fig. 9. The blue line shows a halfway house between these: it shows the results of applying the Helmholtz resonator model to the measured admittance without mouthpiece. The green and blue lines lie very close together, both showing frequencies reduced relative to the no-mouthpiece results. The magenta line shows a similar pattern but with a larger reduction of frequency.
Figure 12 shows the frequency plotted against the Q-factor, for the synthesised impedances with and without mouthpiece. The colours and symbols match Fig. 11. Looking carefully at this plot, you can see the pattern described above. The two curves cross near the mouthpiece resonance frequency. Below that crossing, each green circle is a little higher and to the left of the corresponding red star: the frequency has been reduced, while the Q-factor has been increased. Above the crossing point, the Q-factors on the green curve fall away to significantly lower levels. The frequencies always fall in the gaps between the frequencies marked by the red stars: one green circle per gap.
Figure 13 shows the same comparison involving measured results — again with colours and symbols matching Fig. 11. The black and blue curves show a very similar pattern to the curves in Fig. 12. The magenta curve, showing the measured results with mouthpiece, shows a similar pattern to the blue curve but the drop-off of Q-factors above the mouthpiece resonance is even more marked. The deviation between the blue and magenta curves here is probably a result of the Helmholtz resonator approximation becoming inadequate as frequency rises, and wavelength reduces.
Before we move on to look at simulation results for the trombone, it is interesting to compare the results of Fig. 11 with corresponding results for other instruments. Figure 14 shows a plot in the same format as Fig. 11, but now including measured frequencies from the clarinet and the saxophone from section 11.4, and for a cornetto that we will come to later in this section. (The cornetto is an early instrument which is played with a brass-type mouthpiece but which is made of wood and has finger-holes. You can see a picture of one a bit later, in Fig. ?.) For the clarinet, the “effective fundamental frequency” has been calculated by dividing the frequencies by the numbers 1,3,5,… rather than by 1,2,3,… based on the behaviour of an ideal cylindrical tube.
The trombone, with or without its mouthpiece, is the only one of these instruments to produce an essentially horizontal line of points in Fig. 14, showing that it has the most “harmonic” set of tube resonances. This seems slightly paradoxical. All the other instruments are based on cylindrical or conical pipes, which in their idealised “textbook” versions give harmonic overtones naturally. In contrast, the flaring trombone only achieves the effect by dint of careful acoustical engineering. But in the world of real instruments, the effect seems to be the other way round.
In fact we should not be too surprised. The trombone is the only one of these instruments that makes direct, musical use of at least 7 or 8 of the tube resonances, to play notes which the player hopes will be in tune. All the others have finger-holes, so that players have more options for changing notes, by opening and closing holes. When they use a higher mode of the tube as the basis of a note, by playing in a higher register, they normally use a register hole, which will slightly change the tuning of the resonances. So when an instrument maker is designing and voicing such an instrument so that it plays in tune, they can adjust the position, size and detailed shape of these finger-holes and register holes. But a brass instrument designer must put all their effort into shaping the bore profile, because that is the only thing that determines the relative frequency tuning of the resonances.
C. Simulation results for the trombone
We now have all the ingredients in place to simulate some notes on the trombone. First, we can look at periodic waveforms predicted by the model, once the transient has run its course: a selection is shown in Fig. 15. In each case, the pressure inside the mouthpiece is shown by the top curve, in red; the volume flow rate through the lips is shown in the middle, in blue; and the motion of the “reed”, in other words of the player’s lips, is at the bottom, in black. The figure shows examples of a note in the first, second, third and fourth registers.
The plots reveal an interesting parallel with the “Helmholtz motion” of the clarinet, discussed back in section 11.3. The simulated clarinet waveforms were all recognisably related to an ingenious argument by Raman, which suggested that the pressure waveform should always be, approximately, a symmetrical square wave. Once the amplitude is high enough that the reed closes during part of the cycle, this square wave involves an alternation between two states. Either the reed is shut, so there is no flow, or else the pressure inside the mouthpiece is rather close to the mouth pressure, so that again there is rather little flow.
There is no argument as simple as Raman’s that applies directly to the trombone, but nevertheless Fig. 15 shows that the periodic motion adjusts itself into a rather similar state. Each of the four cases plotted in that figure shows a similar pattern involving the lips closing completely, once per cycle of the oscillation. Almost all the time, either the lips are shut (indicated by horizontal lines in the lower two waveforms), or else the pressure is almost constant at a value close to the mouth pressure (indicated by a near-horizontal line in the upper plot, lying near the mouth pressure indicated by the dashed line).
The result is that the flow rate through the lips is always rather small — small enough that you can see a bit of “digital noise” in the blue curve of the top left set, arising from the finite resolution of the simulation model. There is another indication of how small that air flow must be: during the intervals when the lips are open, you can hardly discern the deviation of pressure from the dashed line in the top plot for each note — even though it is this difference of pressure that causes the volume flow shown in the middle plots.
We are ready for a more systematic use of simulation. For the reed instruments, we selected mouth pressure and reed gap as two key parameters for a player, and we used simulation to populate the pressure-gap diagram to give an indication of “playability”. For the trombone, mouth pressure is still an important parameter, but rather than reed gap we will choose lip resonance frequency as the most natural second parameter. So we will generate a “pressure-lip resonance” diagram. Figure 16 shows an example, based on modal simulations using the measured input impedance shown in Fig. 9.
The nominal pitch of the tenor trombone with the slide in first position is $B\flat_1$ (58.3 Hz), so the colour shading in Fig. 16 as based on this nominal pitch. For cases that produced a note rather than silence, the colour indicates the playing frequency normalised by 58.3 Hz. It is immediately clear that there are horizontal bands of colour corresponding approximately to the values 1,2,3 up to 8. Each band requires a certain threshold mouth pressure in order to start, and this threshold gets progressively higher for the higher registers. The green circles mark the cases shown in Fig. 15, lying in the middle of the first four bands, at the highest mouth pressure considered here.
Figure 17 sheds a little more light on the playing frequency: it shows the same set of simulations, this time colour-shaded to indicate the frequency deviation in cents away from the relevant harmonic multiple of the nominal pitch. The lowest band has turned largely black in this plot: those notes play at least a semitone flat. But the higher bands all have a stripe of dark red in their lower portion, connoting something close to the desired frequency.
A different view of these variations of playing frequency is given by Fig. 18, which shows in graphical form the right-most column of Fig. 16. The vertical axis shows frequency on a logarithmic scale, and the horizontal dotted lines mark equal-tempered semitones. Examining this graph closely, you can see that the second band produces a frequency very close to the value 2, running along the corresponding semitone marker line until it turns abruptly upwards. The next band repeats the pattern, starting by following the semitone marker line at the value 3. All the higher bands have a similar pattern: with a carefully chosen mouth pressure, this simulated trombone seems to be capable of playing in-tune notes for harmonics 2—8 of the nominal frequency.
However, the lowest band is quite different. The playing frequency varies from 2 semitones below the nominal pitch to some 5 semitones above it, before the symbols jump up to the second register. This is a consequence of the fact that the lowest resonance of the trombone tube lies nowhere near the nominal fundamental frequency, as Fig. 11 showed. It is perfectly possible to sound a note down in this low register, but the tube provides very little help in setting the correct pitch: the player can “lip” this note up or down over a wide range. As an aside, this regime is the basis for a trombone-player’s party trick: it is possible for a skilled player to hold a fixed note in this register, while sliding the slide in and out with apparently no effect!
In Sound 1 you can hear the four notes shown in Fig. 15 and marked by circles in Fig. 16. The first note is indeed conspicuously flat compared to the harmonic series of the other three sounds.
Figure 19 shows a different view of these playing frequencies. This time, they are normalised by the lip resonance frequency, and the plot confirms something we anticipated: the playing frequency always lies above the lip resonance frequency, so that the values in the plot are all bigger than 1. Comparing this plot with Fig. 18, we can understand the pattern. Concentrate first on the second band of frequencies. In Fig. 18 we saw a horizontal portion close to the desired frequency, followed by a sharp rise. Figure 19 shows the converse pattern: the playing frequency gets nearer and nearer to the lip resonance frequency, but when it gets very close the curve flattens out because the playing frequency has to remain above the lip frequency. So the rising portion in Fig. 18 shows the playing frequency tracking upwards with the lip frequency.
Next, we can look at the transient behaviour of these simulations. Figure 20 shows a version of the same pressure—lip resonance diagram but now colour-shaded to indicate transient length. For the value of lip resonance frequency marked by the horizontal green line, Fig. 21 shows four examples of waveforms. The first case is for the first coloured pixel on the line, and shows a very long transient. The remaining cases correspond to alternate pixels along that row. The transient gets progressively quicker, as Fig 20 indicates. But for all these cases of “cold start” transients, the model is very slow to speak. Trombonists need to do articulatory tricks in order to create more abrupt starts to notes.
Next, we look at the influence of a parameter we haven’t mentioned: the assumed damping of the lip resonance. The results we have shown so far have all assumed a value 15 for the Q-factor of the lip resonance. Figure 22 shows what happens if that value is halved to 7 or doubled to 30. With the lower value, the threshold mouth pressures for all notes tend to rise, and the 7th and 8th notes do not play at all within the range of mouth pressure explored here. When the Q-factor is increased, the opposite trends are seen. The threshold pressures all reduce, and at the very top of the plot you can see a hint that the 9th note starts to sound.
Now, a Q-factor as high as 30 sounds rather implausible for a resonance of squashy flesh in the lips. Even the value 15 seems on the high side for plausibility. It is not at all easy to obtain direct measurements of this Q-factor by measuring real lips. There is a published study by Doc, Vergez and Hannebicq  estimating lip parameters for trumpet playing — but they do it by comparing measured thresholds with simulations using essentially the same model used here, so it is not surprising that they obtain similar values. They suggest that as players increase lip tension to raise the resonance frequency in order to play higher notes, the Q-factor also increases. This seems quite plausible, but there are as yet no direct physiological measurements to test the predictions. We must, of course, keep in mind that the simulation model includes various idealisations, especially in the way the lips are modelled, and it is possible that a more realistic model would change the detailed conclusions.
There is one more important comment to be made about the trombone. The simulations in Sound 1 capture a vaguely brass-like sound, but you may have thought that they do not entirely convey the impression of a trombone played loudly — even though the simulations used a very high mouth pressure. There is a characteristic sound of a trumpet or trombone playing a crescendo: the sound is initially fairly mellow, but as the level grows there is a gradual transition to what is often described as “brassy” sound. The sound gets brighter, indicating a marked increase in high-frequency content.
The explanation of this effect does not lie in the pressure waveform inside the mouthpiece, which is what we have been concentrating on so far. As first demonstrated by Hirschberg et al. , the form of pressure wave changes during its rather long journey along the tube, before it emerges from the bell as audible sound. There is a nonlinear process which happens with large-amplitude waves of various kinds, called “steepening”.
The most familiar example is what happens to surface waves on water as they approach a shelving beach. There is a viscous drag force near the ground, which has the effect of slowing down the troughs of the travelling wave. But the wave crests are further away from the ground and do not feel this drag. The result is that the wave crests travel a bit faster than the troughs. The wave might have been initially almost sinusoidal, but the front face steepens as the crest catches up with the trough. This is the effect that surfers take advantage of. Eventually the crest overtakes the trough, and the wave breaks.
Something similar happens when a high-intensity sound wave travels down a straight tube, like the section of a trombone up to the end of the slide. The speed of sound is sensitive to temperature. The sound wave is approximately adiabatic, so the air is warmer near the points of maximum compression, and cooler near the troughs of the pressure waveform. The louder the sound, the bigger these temperature changes. So, like the water wave, the crests travel a bit faster than the troughs and the wave steepens. If the tube is long enough, it reaches a critical point. But instead of the wave breaking, it forms a shock front: an abrupt jump in pressure. (A similar shock front is responsible for the “sonic boom” when a supersonic aircraft passes overhead.)
We have already seen the result, back in section 10.6: Fig. 12 from that section is reproduced here as Fig. 23. It shows schlieren flow visualisation of a shock wave emerging from the bell of a trumpet driven at high amplitude by a loudspeaker. A similar thing happens in response to normal playing at fortissimo level. The abrupt pressure jump across the shock front guarantees that the sound spectrum contains a lot of high-frequency content, and this is responsible for the “brassy” sound.
This effect is mainly confined to instruments like the trumpet and trombone. The reason is that the steepening effect needs a sufficiently long stretch of cylindrical tube in order to build up enough to form a shock. If the tube is flaring, spreading of the wave-front reduces the amplitude, and the nonlinear steepening effect is reduced. So instruments like the euphonium have too much flare, while instruments like the clarinet have a tube that is too short. This idea can be captured by a formula: Campbell, Gilbert and Myers  in section 6.1.4 define a “brassiness potential parameter”, and in Fig. 6.11 of that section they plot values of this parameter for brass instruments of different kinds. This plot indeed shows that trumpets and trombones are more likely to show “brassiness” than instruments like euphoniums or saxhorns.
D. Simulation results for the cornetto
For a contrasting style of “brass” instrument, we will show a few results for the cornetto. The cornetto, in various forms, was used throughout the medieval, renaissance and baroque eras and nowadays it is usually encountered in period ensembles. The typical instrument, like the one shown in Fig. 24, has a slightly curved tube with a conical bore. It is usually made of wood covered with thin leather, and it has a small brass-style cup mouthpiece and is played using finger holes.
The measured input impedance of a cornetto with all finger-holes closed is shown in Fig. 25. Comparing with Fig. 9, we can guess that the Helmholtz resonance frequency for the mouthpiece must be a bit above 1 kHz: beyond that frequency we see the same kind of high-frequency cutoff of peak heights that we saw for the trombone. We have already seen the frequencies resulting from a modal fit to this impedance, in the green curve in Fig. 14.
In order to run simulations based on this impedance, we need parameter values for the lip model. In the case of the trombone, we were able to use values from the literature. However, there is virtually no acoustical literature about the cornetto, so we are reduced to a bit of guesswork. Because the mouthpiece is so small, it seems reasonable that the width of the opening and the effective mass of the lips might both be significantly smaller than for the trombone case. For the purpose of the simulations to be shown here, a width 6 mm has been used. It then turned out that the simulation would not play notes with reasonable mouth pressures until the effective mass per unit area of the lips was reduced substantially below the trombone value (9 kg/m$^2$). The results to be shown here use the value 0.6 kg/m$^2$.
Figure 26 shows some typical periodic waveforms given by the resulting model: one in the first register, the other in the second register. The cornetto is not normally played in higher registers than that, because most note changes are effected using the finger-holes. Indeed, as we will see in a moment, it may not be possible for human lips to achieve a resonance frequency high enough to excite the third register — although the computer has no such difficulty, of course. Comparing these waveforms with the ones in Fig. 15, we see strong similarities. Again, the lips close completely for part of every cycle, although perhaps for a smaller proportion of the cycle than was the case with the trombone. Again, the volume flow rate through the lips is either zero (because the lips are closed) or rather small (because the pressure inside the mouthpiece is close to the mouth pressure).
Figure 27 shows the pressure—lip resonance diagram. It shows regions corresponding to the first three registers, but there is much more black in this picture than in Fig. 16 for the trombone. If this model is to be believed, the cornetto player has to place their lip resonance frequency with some precision in order to get a note to sound, especially in the first register where the band of colour is quite narrow.
Figure 28 shows the frequency deviation from the nominal value, similar to Fig. 17 for the trombone. Figure 29, similar to Fig. 18, shows the normalised playing frequency along the right-hand column of Fig. 27, with horizontal lines marking equal-tempered semitones. Between them, these two plots show that in the first register, the cornetto is much more well behaved than the trombone. The playing frequency only varies a little either side of the nominal (220 Hz), so that the player should have relatively little difficulty in playing the note in tune. The second register (and the third) tends to play sharp, by a fraction of a semitone. But don’t forget that this instrument has finger-holes, and we are only looking at a single note here. The intonation of the instrument as a whole will be determined mainly by the skill of the instrument maker in placing and shaping the finger-holes.
Figure 30 shows the playing frequency normalised by the lip frequency. As with the trombone, the plotted values are always greater than 1 — but here they don’t even come very close to 1. That is why we didn’t see the lines in Fig. 29 turning upwards as they did for the trombone. As the lip resonance frequency is increased, the note ceases to sound rather than entering a regime where the playing frequency is “carried upwards” by the lip resonance.
Figure 31 shows a typical transient waveform from a simulated note in the first register: in fact, it is exactly the same note that was shown in the left-hand plot of Fig. 26, corresponding to the lower of the two green circles in Fig. 27. The behaviour of the volume flow rate (blue curve) should be noted. Early in the transient, the flow rate through the lips is quite high, but once the point is reached where the lips begin to close at some point in the oscillation cycle the flow rate reduces conspicuously. The pattern is strikingly reminiscent of the clarinet transients shown in Figs. 23 and 25 of section 11.3: the Raman model prediction from Fig. 25 in that section is reproduced here as Fig. 32, as a reminder.
Finally, Fig. 33 shows the influence of two key parameters of the lip model. In the left-hand plot, the effective mass per unit area has been increased to 1 kg/m$^2$. The notes still play in all registers, but the threshold blowing pressures have increased. A similar effect, but significantly stronger, is seen in the right-hand plot. This shows the influence of halving the Q-factor of the lip resonance to 7.
 Lionel Velut, Christophe Vergez, Joël Gilbert and Mithra Djahanbani, “How well can linear stability analysis predict the behaviour of an outward-striking valve brass instrument model?”, Acta Acustica united with Acustica 103, 132–148 (2017)
 J.-B. Doc, C. Vergez and J. Hannebicq, “Inverse problem to estimate lips parameters values of outward-striking trumpet model for successive playing registers”, Journal of the Acoustical Society of America 153, 168—178 (2023).
 A. Hirschberg, J. Gilbert, R. Msallam and A. P. J. Wijnands, “Shock waves in trombones”, Journal of the Acoustical Society of America 99, 1754–1758 (1996).
 Murray Campbell, Joël Gilbert and Arnold Myers, “The science of brass instruments”, ASA Press/Springer (2021)