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 will find that a simple model can be made with only 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 reasonable first approximation, most reliable at lower frequencies) 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. [1].
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 the physics suggests that brass players probably 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. (Of course, that isn’t how a player would describe the feeling of playing: they cannot be directly aware of the lip resonance frequency, only of the “tightness” of embouchure.) This, at least, is our working hypothesis. We will want to put it to the test 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 may be connected to inaccuracies in the impedance measurement at very low frequency, and also 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: as is shown in the next link it gives a good approximation to the correct damping factors, but of course there are small deviations in the details.
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 increased 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. 23.) 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 (apart from the influence of the player’s lips and vocal tract, which we are ignoring in this preliminary discussion).
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. The horizontal blue lines mark the impedance peak frequencies — but note that for these, the left-hand scale should be read as actual frequency rather than lip resonance frequency.
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. Once again, blue lines mark the peak frequencies of the impedance (blue curve in Fig. 9). 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. Indeed, the blue line that would have marked that resonance frequency cannot be seen because it is off the bottom of the plot. 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.
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. [2], 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. 22. 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 [4] 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. 23, 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. 24. 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 25 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 26 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 27 shows the frequency deviation from the nominal value, similar to Fig. 17 for the trombone. Figure 28, similar to Fig. 18, shows the normalised playing frequency along the right-hand column of Fig. 26, 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. The blue lines in these two plots show that the playing frequencies fall rather close to impedance peaks (Fig. 28), while the lip resonance frequency generally needs to be rather well below those impedance peak frequencies (Fig. 27).
Figure 29 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. 28 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 30 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. 25, corresponding to the lower of the two green circles in Fig. 26. 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. 31, as a reminder.
Finally, Fig. 32 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.
E Back to the trombone
In the last two subsections we have seen some simulation results for a trombone and a cornetto, using a very simple model of the action of the player’s lips. This model gave qualitatively plausible behaviour, but if we want to get somewhere close to quantitative agreement with the behaviour of real instruments, we need to look a bit harder at this model. We will explore in the context of the trombone rather than the cornetto, because the trombone has been far more extensively studied.
For a first step, we can look at the influence of a parameter we haven’t said much about: the assumed damping of the lip resonance. The trombone results we have shown so far have all assumed a value 15 for the Q-factor of the lip resonance. Figure 33 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 totally implausible for a resonance of squashy flesh in the lips — surely a vibrating lip could not “ring” for 30 cycles or more? Even the value 15 seems too high 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 [5] estimating lip parameters for trumpet playing which suggests Q-factors of the same order we been using — but they do it by comparing measured thresholds from a test rig using artificial lips with simulations using essentially the same model used here. Their results suggest that as players increase lip tension to raise the resonance frequency in order to play higher notes, the Q-factor also increases. This trend seems quite plausible, but there is as yet very little in the way of direct physiological measurements to test whether the actual values of Q-factor apply to real lips.
What evidence there is (see for example Figure 3.30 of Campbell, Gilbert and Myers [4]) suggests that human lips have a significantly lower Q factor than the artificial lips (which consist of sausage-shaped balloons filled with water). So we have a challenge: can we tweak the simulation model to be capable of predicting that notes could be played at realistic threshold pressures using lips with a Q-factor as low as 1?
Before we start exploring enhancements to the model, it is useful to review the (rather sparse) set of known facts about trombones played by human lips. These provide the “ground truth” against which simulation results must be tested. The first source of information comes from the experience of trombonists. For a given slide position, a trombonist expects to be able to elicit notes forming (more or less) a harmonic series. They will have to adjust their lips and blowing pressure to achieve these different notes. So far, so good: we have already captured that aspect of behaviour, qualitatively at least, in our pressure-lip resonance diagrams.
But there is an important aspect of a trombonist’s experience that we have not reproduced. Figure 18 showed a plot of the predicted playing frequency as a function of the lip resonance frequency, for a particular set of simulations. The playing frequency in every case was near to the corresponding tube resonance, or above that frequency. In trombonist’s language, the simulated trombone notes could be “lipped up” to a higher pitch by adjusting embouchure, but they could not be “lipped down” to a lower pitch. But in reality, trombonists report that notes can be lipped down just as easily as they can be lipped up.
It would obviously be useful to have some quantitative information about played trombone notes, but published information is very sparse. Our main resource will be a set of measurements by Boutin, Smith and Wolfe [6]. A professional trombonist played three versions of a single note: one normal, one lipped down and one lipped up. Using a combination of sensors and high-speed video recordings, the authors measured a variety of waveforms for each of these notes, including the pressure inside the mouthpiece, the volume flow rate into the tube, the trajectories of the tips of upper and lower lips, and the area of the lip opening. For all three of their measured notes, the player’s blowing pressure was around 1 kPa.
Conveniently, the note chosen for those measurements is one already included in our simulation results: the second natural note with the trombone slide in first position, $\mathrm{B}\flat_2$ (nominally 116.5 Hz). To see how our simulation model is doing before we start enhancing it, we can make a first comparison of their normally-played note with the simulation marked by a green circle in Fig. 33. This is the correct note, played with the correct pressure which puts it just above the threshold for the particular model used for Fig. 33.
A comparison of waveforms is shown in Fig. 34. The level of agreement is encouraging, but perhaps rather surprising given the crudeness of the simulation model being used here. This case used a Q-factor of 7, surely too high for human lips. But Fig. 33 suggests an immediate problem if we were to reduce the Q-factor in the simulation model. It shows that threshold pressures go up as the Q-factor is reduced, and we are already near the threshold at this blowing pressure with $Q=7$. If we are to find playable notes at a blowing pressure near 1 kPa with a far lower value of the Q-factor, something will have to change in the model to reduce the threshold pressure for this note.
We can draw inspiration from Boutin et al. [6]: they highlighted some effects not included in the current simulation model, and suggested that these might have a big influence on the energy balance, and hence on thresholds. First, as also observed in earlier work by Copley and Strong [7], they found that the horizontal and vertical components of the lip motion are not in phase: the tips of the upper and lower lip each trace out a loop in space. Figure 35 shows these loops, for the same note shown in Fig. 34. They also pointed out effects associated with an additional component of the flow into the tube caused by the forwards-and-backwards motion of the lips.
In order to investigate these suggestions we need to incorporate them into an extended simulation model, described in the next link. We will do this in the simplest possible way, but inevitably the resulting model becomes more complicated because it has more parameters to explore. We will show some results in pictorial form here: the full details behind the pictures are given in the side link, together with more plots of the predicted behaviour.
To allow phase differences somewhat like the results seen in Fig. 35, we can use a trick. Recall that the simulation model works by stepping forwards in time in small steps. In order to create a phase lag in the vertical lip motion, relative to the horizontal motion, we can simply calculate that vertical displacement using a delayed value of the horizontal displacement, stored from a few time-steps previously. The value of this time delay will be a key parameter in the model. An approximate physical interpretation of this model is that we still represent the lip motion by a single resonance, but now the mode shape corresponding to this resonance has a phase difference between the different components of motion. The tip of the “lip reed”, which is representing a combination of the two lips, moves round an elliptical path in space rather than along a straight line as it did in our model for a clarinet reed in earlier sections. (In mathematical terms, we are using a “complex mode shape”.)
Cutting a long story short, with carefully-chosen parameter values this extended model does indeed allow notes to be “played” on our simulated trombone with a Q-factor of the lips as low as 1: far lower than we were able to use with the previous model. Figure 37 shows a sequence of three pressure–lip resonance diagrams, in the same format as previous ones, all using this value of Q-factor. As you move down the sequence in the left-hand column, the time delay is increasing in steps of 0.5 ms. With no time delay, the entire diagram would be black, but the other three (with delays 0.5 ms, 1 ms and 1.5 ms) all show a good spread of coloured pixels. Note that all these delays are quite short compared to the period of the notes being played: the nominal fundamental frequency for this configuration of the trombone is 58.3 Hz, corresponding to a period 17.2 ms.
Each pressure–lip resonance diagram is accompanied by a plot in the same format as Figs. 18 and 28, showing the normalised playing frequency for any non-black pixels in the column corresponding to blowing pressure 1 kPa, which is the pressure used in the measurements by Boutin, Smith and Wolfe. In the top row there are no playable notes at this blowing pressure, although the pressure–lip resonance diagram shows that it would only need a slightly higher pressure for the third natural note to reach its threshold and start to play. In the second row, we see that the second, third and fourth notes can be played with this value of delay, but that the red stars all lie above the blue lines so that the notes can only be “lipped up”, not “lipped down”.
However, in the third row some stars fall below the blue lines, indicating that lipping down has become possible. The horizontal dotted lines mark semitones, so the plot shows that the second natural note can be lipped down by about a semitone in this case. The suggestion, based on the results of Boutin et al., is that a proficient trombone player can exercise some control over the effective time delay by (somehow) manipulating the details of their embouchure. For normal playing or lipping-up, they might use a modest delay like the middle row in Fig. 36. But when they want to lip a note down, they change embouchure so as to increase the delay to something like the bottom row in the figure.
It is worth looking rather carefully at the behaviour revealed by Fig. 36. The first conclusion is an obvious one: the time delay is a critical parameter in this model. With no delay, no notes can be played within this range of blowing pressure and lip resonance frequency. But with even a small delay, notes up to the sixth (indicated by the yellow colour) become possible. Increasing the delay further moves the pattern around: there are many subtle differences between the three pressure—lip resonance diagrams which have implications for the player. (Or, at least, they would have these implications if this very crude model is really capturing enough of the underlying physics to give a good guide.)
Increasing the delay seems to help with producing the lower notes, but makes it harder to produce the higher notes. If you look carefully at the bottom of the three diagrams, you can see that the first note (indicated by a rather dark red colour) shows as a bigger region in the lower images, and also the threshold blowing pressure for this note gets progressively lower. But at the same time the yellow colour at the top of the images progressively retreats: according to this model it would be easiest to play that sixth note with the shortest delay.
However, this is by no means the end of the story. The time delay is not the only parameter which has a sensitive effect on the simulation results. I will illustrate the influence of four other parameters with pictures. I will take the middle row of Fig. 36 as my reference case, and show how the pressure—lip resonance diagram changes when a significant change is made to one parameter at a time. In each case, the reference plot will be repeated on the left-hand side, to be compared with the modified version on the right.
The first comparison illustrates something we already know. Figure 37 shows the effect of increasing the Q-factor of the lip resonator to 2 — in other words, halving the damping. The region of coloured pixels grows, as we should expect. Every note is easier to play with the increased Q-factor.
The second comparison brings in a parameter we haven’t previously mentioned. It describes the initial position of the lips, before the player starts to blow. In Figure 36, it was assumed that the lips were just touching: no gap, but also not actively pressed together. In the right-hand plot of Fig. 38, we see the result of pressing the lips together a little. This is represented in the model by pushing the lips 0.5 mm closer together than the previous case. This amount of lip-pressing makes every note a bit harder to play: the coloured region has retreated to the right everywhere. However, more detailed investigation reveals that the original case is not quite the “best” choice for easy playing: a slightly bigger region of coloured pixels appears with 0.1 mm of pressing-together of the lips. At least qualitatively, this is consistent with what trombonists report: the lips should be lightly pressed together for best embouchure.
The next case is shown in Fig. 39. This time, a change has been made in the scale factor that relates the vertical lip motion to the horizontal motion. In the reference case, the maximum vertical opening of the lips is 4 times the maximum horizontal motion, whereas in the modified case it is only 2 times that horizontal motion. The plot shows that this change makes the coloured region retreat to the right, so all notes are harder to play.
The final comparison is done in order to investigate one of the effects pointed out by Boutin et al. [6]. If the effective area of the vibrating lips varies during the oscillation, this may increase the supply of energy available to drive the lip motion, and thus reduce thresholds. The reference case includes this effect of area modulation, and Fig. 40 shows the result of switching the effect off. At first glance you might think the two plots are very similar, but if you concentrate on the bottom part of each plot you can see that the two lowest notes are significantly affected by the change. In the right-hand plot, the lowest note doesn’t appear at all: the plot is all black, with no dark red as can be seen in left-hand plot. The region of coloured pixels for the second note is also significantly bigger on the left than on the right. So this effect can indeed influence the ease of playing of these two lowest notes on the simulated trombone.
F The limitations of simple models
The results shown here, together with the extra detail provided in the previous link, tell us some encouraging things about the trombone model. The simulations seem to be able to match all the main trends of qualitative behaviour, and to come at least within striking range of matching some quantitative observations. However, I want to close this section by emphasising some weaknesses and limitations. There are two types of concern about the extended lip model, one based on physics and the other on the relation to what players actually do. These concerns do not just apply to the particular model discussed here, but to other models that have been proposed for more realistic lip motion, such as the pioneering study by Adachi and Sato [8].
The physics-based concern arises directly from the very high damping of the lips. Within the broad area of musical acoustics, it is very unusual to have an oscillator of any kind that forms a crucial part of understanding how an instrument works, and which has a Q-factor anywhere near as low as 1. For all the other problems we have looked at throughout this e-book, we have been able to get away with approximate descriptions of damping that are based on an assumption that damping is small. That is certainly not the case for the behaviour of brass-players’ lips.
Throughout the wider subject of structural vibration, it is well accepted (but not often talked about explicitly) that physical models of highly-damped systems are intrinsically questionable. The reason goes all the way back to the underlying general theory that was outlined in Chapter 2 and its side links. There are good reasons why we can expect the standard model of linearised vibration of undamped systems to work pretty well, but damping was only ever included in an ad hoc manner, without a corresponding deep justification. There simply is no general theory of damped systems.
The ad hoc approach is usually good enough when damping is small, but there is no convincing reason to expect any of the usual approximations to work with any level of accuracy for highly damped structures like lips. A full discussion of this issue would take us too far from the things we really want to be talking about, but as an example the options to model damping in standard computer packages for such things as Finite Element Analysis generally have no solid physical justification, and might lead to quite misleading answers for a highly-damped system.
In the context of the particular lip model used here, it is worth noting that we are on fairly safe ground with the aspect of that model that might seem most odd, the phase difference we introduced between the horizontal and vertical displacement of the lips. For any linear vibrating system with damping that is not small, we can confidently expect that if the concept of vibration modes still has any meaning, those modes will exhibit phase differences rather like this. In the mathematical jargon of the subject, mode shapes will invariably be complex.
The second concern to be aired about the lip model is quite different. When we built a model for the action of a clarinet reed, all the parameters that entered were directly related to physical quantities. When we used simulations based on that model to generate pressure—gap diagrams in section 11.3, the variables on the two axes had direct physical interpretations that made sense to a clarinettist: blowing pressure and initial reed gap (corresponding directly to bite strength). In order to compare simulations with measurements, we could determine all the relevant parameter values from the tested instrument, and thus hope to get a direct and quantitative comparison of waveforms. The only doubt, perhaps, would have concerned the exact nature of the initial transient used to start the note, but we would hope that the final steady portion of a note should be reliably reproduced.
Our trombone lip model is different. There is a disconnect between the parameters of the model and what the player actually does, or thinks about, when controlling their embouchure to produce a particular sound. They do not think about their lip resonance frequency or the phase lag between horizontal and vertical motions of their lips. Instead, in the process of learning to play proficiently they have used feedback from the sound they produce to learn to do particular things involving the muscles around their mouth, their blowing pressure, and so on. This means that the trombone model is not truly predictive in the same way that the clarinet model was: to achieve that, we would need some kind of neuro-muscular model of embouchure described by parameters that are directly related to player actions.
We know from observations like those of Copley and Strong [7] that the player’s muscular actions in fact result in complicated lip motion including a phase difference between the horizontal and vertical components. We have included this phase difference in our model and found that it makes a big difference to the behaviour, but we really know rather little about how players control that phase difference, or what range of phase difference is accessible to a player, or whether the phase difference is linked to other aspects like lip tension. Indeed, we saw in Figs. 37—40 that phase lag and lip resonance frequency are by no means the only parameters that have a sensitive effect on the pattern of simulated response. The lip model used here, in common with other proposed lip models, contains many parameters (some 8 or 9 in our case). All these parameters may in reality be interlinked in some complicated way by what a player is actually able to do.
But in the computer model, all the parameters can be specified independently. This is both good and bad. The good aspect is that we were able to use the model to explore the effect of each parameter in turn, and thus isolate the sensitivity to different aspects of the underlying physics. The bad thing is that it probably means that many of the parametric combinations included in the simulation results are not in fact possible for a human player. The result is a model that sheds useful light on the physics of the trombone, but that may not say anything very useful to a trombonist.
When it comes to assessing the accuracy of the model, we noted earlier that the major source of quantitative comparison data is the set of waveforms measured by Boutin et al. [6]. We already saw one comparison with a simulated case (in Fig. 34), and the previous link shows comparisons with all three of the cases they measured. But there is snag with trying to use this data to discriminate between variations of the simulation model, and it is illustrated in Figs. 41 and 42. Here we see the same set of measured waveforms, compared with two different (carefully chosen) cases from the simulation model. Both show a match that is about as close as we could reasonably hope for — but the two simulated cases have very different parameter values. In fact, apart from the blowing pressure and the value of the time delay in the lip model, every other parameter has a significantly different value in the two cases. For example, Fig. 41 uses a lip Q-factor of 5, while Fig. 42 uses the value 1. The other parameter values are listed in the figure captions.
This story is repeated with the other two measured cases of Boutin et al. The simulation model is capable of moderately good matches to each of these measurements, but this can be achieved with many different combinations of parameter values. I have deliberately avoided any kind of numerical search for the “best fit” to these waveforms within the complicated space of all the model parameters. Such a search would be misleading because matching to these particular waveforms does not provide clear discrimination between the different possibilities, and in any case we noted earlier that in reality the player can probably only access certain combinations of parameter values.
[1] 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)
[2] 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).
[3] A. López-Carromero, D. M. Campbell, J. Kemp and P.L. Rendon, “Validation of brass wind instrument radiation models in relation to their physical accuracy using an optical schlieren imaging setup”, Proceedings of Meetings in Acoustics, 28, 035003 (2016).
[4] Murray Campbell, Joël Gilbert and Arnold Myers, “The science of brass instruments”, ASA Press/Springer (2021)
[5] 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).
[6] Henri Boutin, John Smith and Joe Wolfe, “Trombone lip mechanics with inertive and compliant loads (‘lipping up and down’)”, Journal of the Acoustical Society of America 147, 4133—4144 (2020)
[7] D. C. Copley and W. J. Strong, “A stroboscopic study of lip vibrations in a trombone”, Journal of the Acoustical Society of America 99, 1219–1226, (1996).
[8] Seiji Adachi and Masa-aki Sato, “Trumpet sound simulation using a two-dimensional lip vibration model”, Journal of the Acoustical Society of America 99, 1200—1209 (1996).