11.4 Reeds II: towards the real world

PREVIOUS SECTION

In the previous section, we investigated the lowest notes of one cylindrical reed instrument (the clarinet) and one conical one (the soprano saxophone). In the interests of learning something about the playing behaviour of such instruments, we used a reasonably realistic model of the reed mouthpiece in order to perform computer simulations. However, in both cases our model of the instrument tube was highly idealised: a perfect cylinder and a perfect truncated cone respectively. In this section, we move closer to real instruments: we will look at a new set of simulations using the same reed model, but making direct use of measured input impedance of a clarinet and a saxophone, fingered for the lowest note with all tone-holes closed.

A. Input impedance, modes and modal simulation

The first step is to extract information from the input impedance in a form we can use in a simulation model. There is more than one way in which this could be done. In section 11.3 we did simulations based on the “digital waveguide” method: the same approach we had used earlier, when we simulated a bowed string in Chapter 9. For this approach, the input impedance from an idealised tube model was converted into a “reflection function”, as described in detail in section 11.3.2.

We could do the same thing with a measured input impedance, but there are two reasons why I will in fact choose to do something different this time. First, the process of converting an impedance to a reflection function always involves approximations and compromises, so that the result may be less accurate than we would really like. This didn’t matter very much when we were dealing with idealised models, because we weren’t trying to achieve an accurate match to the behaviour of a particular tube. Instead, we were looking for qualitative insights into how systems generically similar to a clarinet or a saxophone might work. But the main point of using a measured impedance function is that it should represent, accurately, the particular instrument that was measured.

The second reason is more pragmatic. The methods generally used to measure input impedance can be relied upon to give a reasonably accurate value of the impedance magnitude, but in some cases they are less accurate when it comes to the phase of the complex frequency response function. (This is not in fact a very big problem for the impedances we are about to use, but it will become more of an issue in section 11.5, when we look at brass instruments.) If sufficient information to run a simulation can be extracted without relying on complete accuracy of the measured phase, this would circumvent some potential problems.

There is indeed a way to achieve this. An acoustic input impedance, in common with many other frequency response functions such as the bridge admittances of stringed instruments discussed extensively in Chapter 5, can be expressed in terms of modal properties: resonance frequencies, modal damping factors and mode shapes. The underlying theory for the case of mechanical vibration was outlined in section 2.2.5. The corresponding theory for the case of an acoustic input impedance is given in the next link.

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Now we can make use of something we have already seen. In section 10.5, we met the idea of experimental modal analysis. A measured frequency response function can be analysed to extract the modal parameters that make it up, provided we know the mathematical formula for the relevant modal expansion. A very simple approach has been adopted here, individually fitting each visible peak in the measured impedance using a variant of circle fitting (see section 10.5). The details of the procedure are given in the next link.

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The results are shown in Figs. 1 and 2, for input impedance measurements of a clarinet and a soprano saxophone provided by Joe Wolfe. In both cases the measurement is shown in red, and the reconstructed impedance based on the modal expansion is shown in blue. The agreement is excellent at low frequency. There is gradual divergence at higher frequencies, probably caused mainly by the effect of missing modal contributions — associated either with resonance frequencies outside the measurement range or with peaks too small to circle-fit. There is no doubt that a closer fit could be achieved by a more elaborate procedure, but given that we are mainly interested in the lowest playable note with this fingering on each instrument, these fits are probably entirely adequate.

Figure 1. Red curve: the measured input impedance of a clarinet, fingered for the lowest note, taken from the dataset on this web page. The blue curve shows the reconstructed function based on the modal fit described in the text.
Figure 2. Red curve: the measured input impedance of a soprano saxophone, fingered for the lowest note, taken from the dataset on this web page. The blue curve shows the reconstructed function based on the modal fit described in the text.

Rather than using these fitted impedances to calculate a reflection function, the information is used directly to perform simulations based on the impulse response, the pressure response to a sharp pulse of volume flow through the reed. Once the modal parameters are known, there is a simple formula for this impulse response. Each mode contributes a sinusoidal contribution at its resonance frequency, decaying at a rate given by its damping factor. The results corresponding to Figs. 1 and 2 are shown in Figs. 3 and 4.

Impulse responses like these can be used directly in a simulation algorithm, as described in the previous link. At each time step, the pressure inside the mouthpiece can be computed by convolution of the impulse response with the history of the volume flow rate through the reed. But, as we have noted earlier (see section 9.5.2) we don’t need to compute this convolution directly, which might be rather slow — instead, we can use a recursive digital filter (“IIR filter”) for each mode in the expansion, and add the results together. The result is a modal-based simulation which actually runs faster than the ones we used in section 11.3, based on the reflection function.

Figure 3. The impulse response corresponding to the clarinet impedance shown in Fig. 1, computed from the fitted modal parameters.
Figure 4. The impulse response corresponding to the saxophone impedance shown in Fig. 2, computed from the fitted modal parameters.

This modal approach works well in the context of wind instrument synthesis, but the same approach would not have been so useful for the bowed string studies we saw earlier. There are two main reasons. First, for these wind instruments we only need to take a relatively small number of modes into account: about 15 for the two fits shown in Figs. 1 and 2. For a typical violin string, with a violin body coupled to it, we would need to include a lot more modes, and the process of fitting the parameters of all these modes would be more challenging.

The second reason concerns the modal damping: the damping of these tube modes is far higher than the corresponding damping of a string. The effect is immediately visible in Figs. 3 and 4: the impulse responses die away in a small fraction of a second, whereas a string (following a pluck, for example) rings on for far longer. The longer “memory” of a string makes the response to bowing much more “twitchy”: the predictions of a simulation model are sensitive to errors in the modal parameter values, and to other small details in the model.

The modal parameters deduced from the fitting process are of some interest in their own right. Figure 5, 6 and 7 show the frequencies, Q-factors and amplitudes for the two instruments. Figure 5 is perhaps the most interesting. For the idealised models we used in the previous section, a (complete) conical tube had harmonically-spaced resonance frequencies filling a full harmonic series: frequency ratios 1,2,3,4… The cylindrical tube of an idealised clarinet also had harmonically-spaced resonances, but only occupying the odd-numbered terms of a harmonic series with frequency ratios 1,3,5,7… The actual frequencies deduced from the input impedances used in the idealised models from section 11.3 are included in Fig. 5: “clarinet” in a dashed black curve, “saxophone” in a dotted black curve.

Figure 5. The modal frequencies for the clarinet (red curve) and the soprano saxophone (blue curve) deduced from modal fitting, each normalised by the fundamental frequency. The “error bars” indicate the half-power bandwidths of each modal peak. The lower green line shows the behaviour expected if the frequencies were perfectly harmonic with ratios 1,2,3,4… while the upper green line shows the corresponding behaviour for frequency ratios 1,3,5,7… The dashed black curve shows the values used in the idealised clarinet model from section 11.3, and the dotted black curve shows the corresponding idealised saxophone model.

Figure 5 shows that the two real instruments show behaviour somewhat intermediate between the idealised patterns shown by the two green lines. Both sets of measurements lie between these two lines. The saxophone (blue line) stays close to the lower green line until about the 5th mode, while the clarinet (in red) is already departing significantly from the upper green line by the third mode. The idealised “saxophone” (black dotted curve) looks very similar to the actual measurements, deviating slightly above the lower green line. However, the idealised “clarinet” has perfectly harmonic frequencies, and looks quite different from the actual measurement.

Are these departures from harmonic behaviour significant for the behaviour of the real instruments? One preliminary way to assess that question is to take into account the fact that the resonance peaks in the impedance have a finite bandwidth, associated with their damping. The standard measure of this effect is the half-power bandwidth (see section 2.2.7). These bandwidths are indicated in Fig. 5 by the “error bar” symbols around each resonance frequency.

If the green lines do not pass through the vertical range indicated by these bandwidth markers, we can conclude that the inharmonicity is likely to be significant. For the clarinet, this happens very quickly. If the clarinet plays a note based on the fundamental resonance frequency of the tube, the nonlinear action of the reed will generate exact harmonics of that frequency. Because of the inharmonicity, these will not fall sufficiently close to the higher resonances of the tube to gain strong resonant reinforcement. This would be expected to impact on the frequency content of the played note: it will be dominated by the fundamental, much more than was the case for the idealised model with harmonic resonances, so the sound will be less bright.

Inharmonicity may also have an impact on the playing frequency via “Benade-style” cooperative interactions between the resonances. Figure 5 shows that the resonances of the clarinet fall progressively flat, compared to the ideal harmonic behaviour. Suppose the player changes what they are doing in such a way that the amplitudes of harmonics generated by the reed increase (perhaps as a result of blowing harder). We can predict that the playing frequency is likely to respond by shifting downwards as the influence of the higher tube modes becomes greater: the note will play progressively flat. But Figure 5 shows that the saxophone exhibits the opposite tendency, so we might expect our soprano saxophone to play sharp rather than flat under similar circumstances. We will soon see simulation results, which will allow these predictions to be tested.

Figure 6 shows the Q-factors deduced from the mode-fitting process. This time, there is no big difference between the two instruments. Both curves are rather featureless, and one might guess that the behaviour would not change very much if we simply took a constant value for the modal Q-factor around 30, for all modes of both instruments.

Figure 6. The Q-factors deduced from the modal fitting process, for the clarinet (red) and the soprano saxophone (blue). The dashed black curve shows corresponding Q-factors deduced from the idealised clarinet impedance, shown in Fig. 1 of section 11.3.2 and used for the clarinet simulations of section 11.3. The dotted black curve shows the corresponding idealised saxophone model from section 11.3.3.

The dashed black curve in this figure shows the Q-factors used in the idealised clarinet model from section 11.3, while the dotted black curve shows the corresponding idealised saxophone model. The tube damping in the clarinet model was based on formulae for wall losses (ignoring energy loss from sound radiation) originally derived by Rayleigh, and summarised in section 8.2 of Fletcher and Rossing [1]. We can see from the plot that these formulae give results that, apart from the first two resonances, look nothing like the measurements on the real clarinet (red curve)! The results for the idealised saxophone, on the other hand, are quite close to the values for the saxophone measurements. This is no coincidence: the damping behaviour in the idealised saxophone model was tweaked in an ad hoc manner to give an impedance curve resembling the measurement.

Figure 7 shows a similar comparison of modal amplitude factors, for the two measurements and also the two idealised models from section 11.3. The idealised model of a cylindrical tube predicts that the amplitude should be exactly the same for every mode. This is, of course, not exactly true for the measured clarinet, but the measured results are not very far away from this super-simple theoretical pattern. For the conical saxophone, the amplitude is predicted to vary with mode number in a pattern that rises and then falls. The measured amplitudes match the idealised prediction quite well for the first three modes, but then they fall much more sharply than predicted. The measurements rise again for modes 13, 14 and 15, but it is not clear how reliable these measured values are. Looking back at Fig. 2, these high modes, above 3 kHz, were only visible as small wiggles in the impedance curve, and the simple circle-fitting strategy is probably too crude.

Figure 7. The amplitude factors (squared modal amplitudes) deduced from the modal fitting process, for the clarinet (red) and the soprano saxophone (blue). The dashed black curve shows the values used in the idealised clarinet model from section 11.3, and the dotted black curve shows the corresponding idealised saxophone model.

B. Simulation results for the clarinet

Finally, we are ready to see some results of simulation using the new model based on measured impedance. We start with the clarinet. First, we need a “sanity check” to make sure that the new simulation model gives results that are recognisably related to the ones we saw in section 11.3 with the idealised model. Figure 8 shows a typical “cold start” transient which resulted in a note being played. The upper plot shows the mouthpiece pressure, and after a few period-lengths this settles into a periodic waveform looking quite like the square wave we have come to expect. The lower plot shows the corresponding waveform of volume flow rate into the mouthpiece. We see that the reed closes once per cycle, and we also see that for a brief interval during each cycle air is flowing in the reverse direction: back into the player’s mouth through the open reed.

Figure 8. Example transient of the clarinet model: the upper curve shows the mouthpiece pressure, the lower curve shows the volume flow rate into the mouthpiece. This is a “cold start” transient: it corresponds to the bottom right-hand pixel in Fig. 9.

But, as we saw before, it is hard to learn much from an individual simulation. So we move directly on to pressure-gap diagrams, to give an impression of the “playability” of this clarinet, fingered for its lowest note. Figure 9 shows a first example. This shows the behaviour following cold start transients, over the same ranges of reed gap and mouth pressure that we saw in section 11.3. The particular transient shown in Fig. 8 corresponds to the bottom right-hand pixel of this diagram, with mouth pressure 4 kPa and reed gap 0.6 mm. The colour shading here is done on the same basis as Figs. 43 and 46 from section 11.3: it shows the playing frequency of each pixel deduced from the autocorrelation of the final waveform, expressed as a multiple of the nominal frequency. In this case, the colour is uniformly red, connoting the value 1: every single case that produced a note turns out to play at a frequency close to the nominal. (We will have more to say more about the playing frequency shortly.)

Figure 9. Pressure-gap diagram for the clarinet over the same ranges of mouth pressure and reed gap as the examples shown in section 11.3. These simulations were all run assuming a “cold start”, with the mouth pressure suddenly applied at time $t=0$. The colour, uniformly red, shows that whenever there was a note rather than silence, it had a period matching (approximately) the fundamental frequency of the tube, i.e. the first register of the clarinet. The assumed Q-factor of the reed was 2.5, as in Fig. 32 of section 11.3. The various sloping lines indicate thresholds of various kinds, exactly as discussed in section 11.3.

The various lines included on this figure have exactly the same meanings as in corresponding plots in the previous section. They show various thresholds. The magenta line is the lower limit for possible excitation of a note. Any note must lie above this line: how close it is possible to get depends on the amount of energy dissipation. As we saw in Fig. 6, the new model has higher dissipation at every resonance than the old idealised model, so the coloured pixels in Fig. 9 cannot get as close to the magenta line as before. The green dashed line shows the “beating reed threshold”, above which (according to the approximate Raman model) the reed closes completely at some stage in the cycle. The cyan line shows the “inverse oscillation threshold” above which it is possible for the reed to be held permanently shut by the mouth pressure. The highest line, in red, is the “extinction threshold” beyond which (according to the Raman model) no oscillation at the fundamental frequency is possible. Since we are dealing with cold starts, we expect the playable region to be more or less confined between the magenta and cyan lines — which indeed it is.

You may wonder how these thresholds based on the Raman model are calculated for our new model. The answer is that, since we are expecting a “Helmholtz motion” dominated by the fundamental, we can simply use the measured Q-factor of the lowest mode and choose a Raman model that matches this value. As later results will confirm, this gives an acceptable approximation to the various thresholds.

Figure 9 gives an answer to one question, but this is not the only question of interest to a clarinettist. We would also like to know the extent of the largest region within which it is possible to sustain a note, once it has been started. This question would be the direct analogue of the calculation behind the Schelleng diagram for a bowed string — outside Schelleng’s wedge-shaped region it is simply not possible to sustain Helmholtz motion in a bowed string. But the calculation, and the diagram, does not tell you anything about what kind of transient you need in order to start Helmholtz motion in the first place — for that, we needed something like the analysis behind the Guettler diagram.

To address this question for the clarinet model, we can change how we start each simulation. We want to encourage oscillation approximately at the fundamental frequency. The new mode-based simulation gives us a very simple way to do this, better and more versatile than the approach we used in section 11.3. Recall that each mode is represented separately by a recursive digital filter. For a cold start, we initialise all these filters with the value zero. But if we want to encourage oscillation near the fundamental frequency, we can initialise that particular filter with a non-zero value. This would represent a situation where there was pre-existing excitation of the fundamental mode, while all the other modes were silent.

The result of doing a set of simulations with this kind of initialisation is the pressure-gap diagram shown in Fig. 10. Compare this with Fig. 9. The lower edge of the red wedge is in the same position as before, but the top edge has moved up to lie close to the (barely visible) red line. Provided you can get the note started, the clarinet is capable of playing at (or at least near) the nominal frequency anywhere within this larger wedge-shaped region.

Figure 10. Pressure-gap diagram as in Fig. 9, but initialised in a way that encouraged oscillation at the fundamental frequency. The resulting red wedge indicates the region in this plane in which it is possible to maintain a steady note in the first register, once it is started. The three horizontal green lines mark the sets of waveforms to be shown in Fig. 13.

There are many other choices for how to colour-shade diagrams like this, to bring out different aspects of the behaviour. Figure 11 shows the same set of simulations as in Fig. 10, coloured to show the behaviour of the playing frequency. The colour scale now shows the deviation, expressed in cents, of the playing frequency relative to the nominal frequency of this note on the clarinet. That note is written $E_3$, but because the $B\flat$ clarinet is a transposing instrument it plays at $D_3$, 146.8 Hz. The plot suggests that the note tends to play sharp, but there may be an overall shift of frequency because the measurement of impedance does not include the small end correction arising from the flexibility of the reed. The important thing that Fig. 11 shows is that blowing harder and moving diagonally down the spine of the wedge-shaped region results in the note getting some 30 cents flatter. This is the effect we anticipated, in the discussion of Fig. 5. This pattern of significant flattening is very much in line with the measurements by Almeida et al. [2].

Figure 11. Pressure-gap diagram for the same simulations as in Fig. 10, but now colour-shaded to show the pitch deviation in cents from the nominal pitch of this note ($D_3$, 146.8 Hz).

Another choice of colour shading is illustrated by Fig. 12. This is an attempt to show something about the frequency content of the pressure waveform. The plot is directly comparable with Fig. 17 from section 11.3, and later plots in that section. It shows the highest-numbered harmonic of the waveform that has an amplitude no more than 10 dB below that of an ideal square wave. Turn back to Fig. 17 of section 11.3, and look at the colours. Comparing with the new plot demonstrates something else we predicted earlier. Whereas the old plot has a lot of bright yellow and white pixels, the new one never goes beyond red. The idealised model used before had perfectly harmonic tube resonances, but the measured clarinet has significant inharmonicity. This results in a big reduction in high-frequency content in the pressure waveform.

Figure 12. Pressure-gap diagram for the same simulations as in Fig. 10, but now colour-shaded to indicate the frequency content of the final waveform by the value of the highest harmonic that remains close to the level predicted by the Raman square wave. This diagram is in the same format as Fig. 17 and later plots from section 11.3.

To give a direct impression of the waveforms that lie behind these various plots, Fig. 13 shows the full set of pressure waveforms along the three lines marked in green in Fig. 10. To help interpret this plot, notice that the lowest waveform in the right-hand column is the same as the one shown in Fig. 8. It is a recognisable version of the square wave, and all three columns of Fig. 13 show waveforms like this in the middle of their range. But near the edges of the wedge-shaped region, for example the waveforms at the top and bottom of the left-hand column, the shape looks more like a sine wave.

Figure 13. The final portion of the waveforms along lines marked in green in Fig. 10, for the cases that produced notes. The left-hand column corresponds to the upper line, the middle column to the middle line, the right-hand column to the lower line. Left-to-right rows in Fig. 10 become top-to-bottom stacks here, so that mouth pressure increases as you move down each column.

I will show one final group of plots relating to this clarinet model. If we make a version of Fig. 9 over an extended range of blowing pressure and reed gap, we get the result shown in Fig. 14. Notice that a few white pixels have appeared, indicating a frequency approximately three times the nominal. In this set of notes generated by cold-start transients, a few cases have spontaneously chosen a playing frequency based around the second resonance of the tube, rather than the fundamental resonance. In musician’s terms, our simulated clarinet is over-blowing to the second register.

Figure 14. Pressure-gap diagram matching Fig. 9, but taken over a wider range of mouth pressure and reed gap. A few white pixels indicate conditions that led to over-blowing to the second register (with a frequency close to the second mode of the pipe, approximately 3 times the fundamental frequency, a musical twelfth).

Figure 15 shows a version of Fig. 10 over this same extended range. When the simulation is “primed” with some oscillation at the fundamental frequency, it is perfectly capable of sustaining a note in the first register over the whole wedge-shaped region. Now compare this with Fig. 16, in which the simulations were primed with the second tube resonance rather than the first. Now we see a large number of white pixels: there is a big region of the diagram over which it is possible to sustain a second-register note, once started. This white region lies entirely within the red region of Fig. 15: in that range the clarinet is capable to sustaining two different regimes of oscillation (and maybe others too) — very much like what we saw with the bowed string.

Figure 15. Pressure-gap diagram over the same ranges as Fig. 14, but for transient conditions matching Fig. 10.
Figure 16. Pressure-gap diagram corresponding to Fig. 15, except that the oscillation was initialised to encourage the second register rather than the first register.

These three plots tell us something important about playing the clarinet. None of the initial transients used in these simulations will be an accurate representation of what a human player really does. They do not “sing” into the tube to prime the first or second mode, nor are they capable of switching on their blowing pressure in the instantaneous way that the computer simulations do it. Instead, skilled players learn to manipulate subtle details of their transient to shape the sound, and to land on the regime they are seeking with a musically-acceptable transient sound. The study of such articulatory gestures in wind instruments is in its infancy. Up to a point these issues can be explored by experiment and by the kind of simulation studies started here, but there remains a research challenge. No-one has yet carried out an analysis of wind instrument transients comparable to Knut Guettler’s bowed-string study that gave rise to the Guettler diagram.

C. Simulation results for the saxophone

We can now look at some similar plots for the soprano saxophone. As an introduction, Fig. 17 shows the pressure waveform of a successful cold-start transient, which develops into a periodic pulse-like waveform. This is the expected form of the “Helmholtz motion” for this instrument.

Figure 17. The waveform of pressure for a successful transient of the saxophone model.

Figure 18 shows a pressure-gap diagram for cold-start transients. Unlike the corresponding results for the clarinet in Fig. 9, the saxophone shows regions where a note in the first, second or third registers is produced — at least according to this correlation analysis. We will see shortly that the reality is a bit more complicated. The behaviour shown by Fig. 18 is similar to what we saw earlier with the idealised saxophone model, but the real instrument looks slightly better-behaved: the regimes appear in well-defined and contiguous regions, giving the player a fighting chance of achieving the one they are aiming for. But the comment we made about clarinet playing applies here too: real transients will not be as abrupt as the computer’s version of a cold start. A skilled player will learn to fine-tune their initial articulation so that the register they are aiming for is achieved reliably.

Figure 18. Pressure-gap diagram for the saxophone over the same ranges of mouth pressure and reed gap as the examples shown in section 11.3. These simulations were all run assuming a “cold start”, with the mouth pressure suddenly applied at time $t=0$. The colour indicates which register of the saxophone each note plays in. The assumed Q-factor of the reed was 2.5. The three horizontal green lines mark the sets of waveforms to be shown in Fig. 18, and the circles mark cases whose frequency spectra are shown in Fig. 20.

Figure 19 shows three sets of waveforms, in the same format as Fig. 13. They correspond to the three rows of Fig. 18 marked by horizontal lines. It is worth looking rather closely at these waveforms, because the pattern of behaviour is rather complicated. The right-hand column gives the most detail. The top four waveforms correspond to red pixels in Fig. 18, and they show the pulse waveform characteristic of Helmholtz motion. Indeed, the second of these is the case we have already seen in Fig. 17.

Figure 19. The final portion of the waveforms along lines marked in green in Fig. 18, for the cases that produced notes. The left-hand column corresponds to the upper line, the middle column to the middle line, the right-hand column to the lower line: the format is the same as Fig. 13. The second waveform from the top in the middle column is the case shown in Fig. 17.

Next, we come to four yellow pixels in Fig. 18, and Fig. 19 reveals that these all show two pulses per period, “double-slipping motion” in bowed-string terms. Of the next 6 pixels in Fig. 18, 5 are white. Fig. 19 reveals that these all show rather similar waveforms, with three pulses per cycle — very much what we would expect for the third register. The odd one out is an isolated red pixel, for which Fig. 19 shows a kind of double-pulse waveform, but the two pulses are not equally spaced so that the periodicity of this waveform does indeed correspond to the fundamental frequency, not the the second register.

Next, we come to three yellow pixels in Fig. 18, and Fig. 19 shows that the corresponding waveforms look very similar to the earlier “double slipping” waveforms. Finally, there are three red pixels. But these do not show what we might have been expecting from the discussion of the idealised saxophone in section 11.3. They do not show the “inverted Helmholtz” motion, with a single upward pulse in each cycle. Instead, the first two of them show a kind of asymmetric double pulse waveform rather like the isolated red pixel we passed earlier. The final waveform is different: it shows, somewhat unexpectedly, a kind of square wave, rather like what we saw from the clarinet in Fig. 13.

It is useful to look at the frequency spectra of a selection of these waveforms. Figure 20 shows these, for the 5 cases marked by circles in Fig. 18. The top one, in blue, shows strong peaks at all harmonics of the fundamental frequency. The second, in orange, is a bit more surprising. We have described this as a note in the second register, implying a frequency twice that of the fundamental tube resonance. But the frequency spectrum reveals that although the even-numbered peaks are much stronger than the odd-numbered ones, nevertheless the odd-numbered peaks are not entirely missing. So this note is dominated by frequency components in a harmonic series an octave higher than the first register, but it still contains traces of that fundamental frequency. This pattern is repeated in the third spectrum in Fig. 20, in yellow. This note has been described as being in the third register. Well, the third peak is certainly the strongest, but all the harmonics of the fundamental are present at some level. The remaining spectra match what has already been described: the fourth is rather like the second, and the last one (in green) has strong peaks at all harmonics, like the first one.

Figure 20. Frequency spectra of the steady waveforms of runs marked by circles in Fig. 18.

So, you should be wondering, what do these all sound like? Sound 1 gives you a chance to hear all the simulated notes along the bottom row of Fig. 18. Each simulated note is 0.4 s long, and these have simply been strung together to make this sound file. You should hear quite clearly the “bugling” effect of going up through the first three registers, and then down again. But if you pay attention to individual notes, you can hear changes in tone quality which are associated with the waveform and spectrum details we have just discussed.

Sound 1. The simulations from the bottom row of Fig. 18, or the right-hand column of Fig. 19, strung together so you can hear the regime transitions.

Figure 21 shows a version of the same pressure-gap diagram as Fig. 18, colour-shaded to show the frequency deviation in cents from the nominal frequency of the relevant register (i.e. twice the nominal frequency for the yellow pixels in Fig. 18, and three times the fundamental for the white pixels). We can see that the third-register notes tend to play a little flat, compared to the ideal harmonic series. Looking back at Fig. 20, the spectra confirm this: if you look closely at the peak positions, the yellow spectrum has all its peaks at slightly lower frequencies than the other four.

Figure 21. Pressure-gap diagram for the same simulations as in Fig. 18, but now colour-shaded to show the pitch deviation in cents from the nominal frequency, taking the different registers into account.

We will say a bit more about frequency deviation in a moment, in the context of our next set of plots. Just as we did with the clarinet, we now want to ask where in this pressure-gap plane it is possible to sustain a note in the first register, once it has been started. We can investigate this in exactly the same way as we did before, by computing a pressure-gap diagram with the filter representing the fundamental tube mode primed with a non-zero value. The result is shown in Fig. 22, and just as in the clarinet case it shows a large area, uniformly coloured red connoting the first register. One thing can be noticed immediately: the red pixels in this plot extend a little below the magenta threshold line, something we have not seen before. But in Fig. 18, with cold-start transients, the pixels were all above the line. Figure 22 is suggesting that once you get a note started on the saxophone, you may be able to sustain it with the pressure reduced below the line.

Figure 22. Pressure-gap diagram as in Fig. 18, but initialised in a way that encouraged oscillation at the fundamental frequency. The resulting red wedge indicates the region in this plane in which it is possible to maintain a steady note in the first register, once it is started. The three horizontal green lines mark the sets of waveforms to be shown in Fig. 25.

Figure 23 shows a version of the same pressure-gap diagram, shaded to show frequency deviation. Figure 24 shows another version, shaded to show something about the frequency content of each waveform — but this time that has been done in a different way, by plotting the spectral centroid of each final periodic waveform. (If you printed the spectrum on cardboard and cut it out, the spectral centroid would be the centre of gravity of the cardboard shape.)

Figure 23. Pressure-gap diagram for the same simulations as in Fig. 22, but now colour-shaded to show the deviation in cents from the nominal frequency.
Figure 24. Pressure-gap diagram for the same simulations as in Fig. 22, but now colour-shaded to show the spectral centroid of the final periodic waveform.

The two plots reveal some interesting features. First, Fig. 24 shows a pattern of variation that looks superficially similar to the corresponding plot for the clarinet, seen in Fig. 12. But think carefully — it actually shows the exact opposite behaviour! The “core” of the wedge-shaped region in Fig. 24 shows darker colours, so the spectral centroid is lower there. For the clarinet, the sound was predicted to be brighter in the core of the wedge, but for the saxophone it is predicted to be less bright.

To see what is going on to cause the wedge of orange colour in the middle of Fig. 24, we need to look at waveforms in detail. Figure 25 shows a plot in the same format as before. If you scan down the middle column of this plot, you see that at first the waveforms show the pulse-like Helmholtz motion that we have seen before. But the waveform shapes gradually morphs into something like a square wave, and then as you continue down the column it gradually turns into the inverted pulse waveform that we saw in section 11.3. The orange colours in Fig. 24 correspond, more or less, to the square wave cases. A symmetrical square wave has weakened even-number harmonics, and that is probably responsible for reducing the spectral centroid.

Figure 25. The final portion of the waveforms along lines marked in green in Fig. 22, for the cases that produced notes. The left-hand column corresponds to the upper line, the middle column to the middle line, the right-hand column to the lower line: the format is the same as Fig. 13.

We haven’t finished with Figs. 23 and 24 yet: there is another interesting thing to be noticed. Earlier in this section when we were discussing the resonance frequencies of the clarinet and saxophone, we suggested that the two instruments might show opposite trends of pitch deviation, when the blowing conditions are changed. We have seen one of our guesses confirmed: the clarinet tends to go flat when blown in a more vigorous way, causing more harmonic generation by the nonlinear reed behaviour. So what about the saxophone? The instrument behaves in a more complicated way than the clarinet, as we have just seen, but we can see something relevant to our guess if we look at the behaviour near the lower edge of the wedge-shaped region. Figure 23 shows reddish colours near this lower edge, getting progressively yellower as you move to the right. This means that the note plays sharper as the blowing pressure is increased, in this zone fairly close to the threshold. This is indeed where an increase in nonlinear effects could be anticipated, as the beating reed threshold is approached and passed.

As blowing pressure is increased further, the waveform morphs into the square wave as we just saw. At that stage, the frequency deviation shifts back towards more orange colours. Blowing harder still produces the complicated series of waveform changes we just described. These changes interact with the pitch deviation in a complicated manner, leading to a pattern of stripes in Fig. 23.

D. Multiphonics

So far, we have given an impression that when a reed instrument plays a note, the pressure signal will settle down to a periodic waveform of some kind. But this is by no means always the case. Sometimes, the result is something called a “quasi-periodic waveform”. These may be produced by mistake when a player is trying to produce a regular note, but sometimes they are used deliberately for musical effect — they are then usually called “multiphonics”. We will look at one example of a multiphonic fingering for a clarinet, taken yet again from the data set on Joe Wolfe’s web site. He labels this particular example “D4B5”.

The input impedance for this fingering is shown in Fig. 26. By using an unconventional fingering, the player has created a rather irregular distribution of resonance peaks. It is hard to guess what will happen now when they blow into the mouthpiece. Figure 27 shows a pressure-gap diagram from a set of simulations, coloured to show the spectral centroid. The portion of the plane with an orange colour corresponds to more-or-less steady notes, but in the strip along the lower edge of the wedge-shaped region, something else happens.

Figure 26. The red curve shows the measured input impedance of a clarinet fingered for a multiphonic labelled “D4B5” in the dataset on this web page. The blue curve shows the reconstructed version, from the modal fit.
Figure 27. Pressure-gap diagram computed with the multiphonic fingering giving the impedance in Fig. 26. The three circles mark the cases used in Figs. 28 and 29, and in Sound 2.

We will look at three examples, drawn from the bottom row of this diagram and marked by three circles. The first half-second of the simulated pressure waveforms of these three cases are plotted in Fig. 28. The top plot (in blue) corresponds to the orange pixel in the right-hand corner of Fig. 27. After quite a short transient, this settles into a periodic waveform. But the other two cases lead to non-periodic signals. The bottom plot, in yellow, has a very long transient before it settles into the quasi-periodic state.

Figure 28. The pressure waveforms from pixels 30, 28 and 26 in the bottom row if Fig. 27 (marked there by circles).

Figure 29 shows the frequency spectra of the same three waveforms, with the same layout and colour coding. The blue curve shows a series of harmonic peaks. But the other two show a complicated mixture of frequency components. In Sound 2, you can listen to the three waveforms in succession. The first is a fairly normal-sounding note, but the other two are not. You can perhaps hear more than one pitch in these sounds: this is, of course, the origin of the term “multiphonic”.

Figure 29. Frequency spectra of the three waveforms shown in Fig. 28, calculated from the next half-second of the signals after the portion shown in the plot.
Sound 2. Sound demonstration of the three pressure waveforms shown in Fig. 28.