To conclude this rather complicated chapter, I will suggest a few lines for future research into the playability of bowed-string instruments. Preliminary examples will be shown, but leaving a lot of scope for exploration. So far, we have concentrated on two things. First was the Schelleng diagram, based around the possibilities for steady motion of a bowed string and in particular mapping out where it is possible for a player to achieve Helmholtz motion. The second thing was the Guettler diagram, telling us something about the possibilities for short initial transients for a bowed note, leading rapidly to Helmholtz motion.
For both of these, we were able to use quantitative measurements from the Galluzzo test rig to compare with simulations. Those comparisons led to some enhancement and fine-tuning of the simulation model, with the end result that the simulation model now seems able to do a creditable job of predicting response to bowing — at least for an open D string on a cello, which was the setting for all the Galluzzo tests and for all the simulations performed to compare with those tests.
There is an important caveat, though: it seems clear both from the measurements and from the simulations that a bowed string exhibits “sensitive dependence”, suggesting that bowed strings are near the edge of chaotic behaviour. This means that repeat measurements under supposedly identical conditions are likely to give results that differ in details, and it also means that it would be futile to seek exact agreement between any particular measured waveform and a simulated version of it. Instead, we need to look for robust patterns in the string motion as parameters are varied, and try to match those patterns in simulation. “Parameters” here might mean quantities that a player controls, such as bow force, speed and tilt, but it might also include physical parameters relating to the behaviour of the string, instrument body, bow and rosin.
An obvious next step would involve using the simulation model to investigate what happens in response to different styles of bowing gesture. The Guettler diagram only scratches the surface of what a violinist or cellist does when they play. It is concerned with one particular family of gestures, in which the bow starts at rest in contact with the string and is then accelerated while keeping the bow force constant. But there are many other types of bow gesture: we will look briefly at four of them in this section.
Figure 1 summarises the cases we will consider. The solid lines in the top left-hand plot show an example of a Guettler transient: the bow force (plotted in blue) is constant throughout, while the bow speed (plotted in red) starts from zero, and rises linearly with time. But in reality a player does not keep increasing bow speed indefinitely: often they will settle to a steady speed before the note is finished. The dashed lines in the plot show what this might mean, for three different values of the final steady speed. In each case, the bow speed tends to a constant value with an exponential trend, and the time constant has been chosen so that the initial acceleration matches the solid line: the three time constants are marked by black ticks.
A natural companion to this kind of gesture is shown in the top right-hand plot. This is an idealisation of a string-crossing transient, in which the bow speed is constant, but the force rises from zero as the bow lands on the string. In this plot, the bow force has been allowed to level off to a constant value with an exponential trend. Two different values of the exponential time constant are illustrated: 10 ms (solid line) and 50 ms (dashed line).
The lower left-hand plot in Fig. 1 shows an idealised version of a sequence of spiccato notes, in which the bow goes back and forth with a sinusoidal velocity waveform while bouncing on the string so that the bow force falls to zero between each pair of notes. This is a case studied by Guettler and Askenfelt [1], and we will represent the bow velocity and force by the same simple model that they used. In this plot, the modulation of bow speed takes place at 2 Hz (giving 4 notes per second).
Finally, the lower right-hand plot shows a sequence of tremolo notes, in which the bow is “scrubbed” rapidly back and forth, while holding the bow force constant. In this plot, the bow speed varies at 4 Hz (giving 8 notes per second).
All the simulations to be shown in this section were computed with the enhanced thermal model, assuming a single-point bow on the cello D string as studied in earlier sections.
A. Uniting Schelleng and Guettler
It will turn out that we can map all these different cases into versions of the Schelleng diagram, to allow us to compare them in terms of the feasible range of player parameters. Before we look at anything new, though, we first need to see how to map the previous Guettler results into the Schelleng diagram. Each Guettler transient, whether measured or simulated, has a fixed value of bow force and bow position $\beta$, so that it can be placed in the Schelleng plane. But Guettler transients also involve a third control variable, the bow acceleration.
We can take a set of measured or simulated Guettler results at different $\beta$ values, and for each individual pixel corresponding to a case that produced Helmholtz motion, we can plot a point in three-dimensional space. An example is shown in Fig. 2, constructed from the simulation results with a single-point bow that were shown in section 9.6.3. The Schelleng axes of bow force and position are shown in the base plane, while the acceleration is shown on the vertical axis. For consistency with the usual Schelleng diagram, bow force and position are both plotted on logarithmic scales. The acceleration is plotted on a linear scale. The length of the pre-Helmholtz transient is indicated by size and colour of each plotted symbol: large red symbols for the shortest transients, fading to smaller symbols in blue and then green for longer transients. It is hard to visualise the resulting three-dimensional region from a single image, so the animated figure shows it rotating to give a clearer impression.
Vertical slices from this region at individual values of $\beta$ are the Guettler plots shown before, looking slightly distorted because of the logarithmic scale for bow force. On the other hand, horizontal slices at fixed values of acceleration give snapshots of the Schelleng diagram. Some examples of such slices are shown in Fig. 3. The ranges on the axes are exactly the same as the earlier plots of measured or simulated Schelleng diagrams. The green lines show the bow force limits fitted to the simulations in section 9.7 (Fig. 6 there), with slopes matching Schelleng’s predictions for minimum and maximum bow force. A second pair of lines with the same slopes are plotted in magenta, simply to help you see the relatively small changes between the cases plotted in Fig. 3.
The points appearing in these plots almost never appear within the wedge marked by the green lines. Those lines correspond to the permitted region for Helmholtz motion with the rather low bow speed chosen for the Galluzzo measurements: 50 mm/s. What is seen instead is a region of points moving progressively towards the top right-hand corner of the plot as the acceleration rises. Comparison with the magenta lines reveals that this region more or less follows the slope of the dashed line, corresponding to Schelleng’s formula for maximum bow force.
This pattern makes some intuitive sense. For a transient of a given length, a higher bow acceleration automatically means a higher bow speed at the end of the transient time. Schelleng’s formulae suggest that both bow force limits will be proportional to bow speed, so a tendency to move upwards in the Schelleng plane as acceleration increases might be expected. But it still seems a bit surprising that it is almost impossible to use a Guettler transient to achieve Helmholtz motion within the limits of the measured Schelleng diagram. (It should be emphasised that although these particular plots were generated from simulated string motion, the same pattern is seen in corresponding plots generated from the measured Guettler transients.)
B. Modified Guettler transients
To investigate this question a bit further, we can look at bowing gestures of the kind suggested by the dashed lines in the top left-hand plot of Fig. 1: gestures in which the accelerating bow speed levels off after a while to a steady speed. Figure 4 shows the five cases we will consider. Figure 5 shows a scan of the Schelleng plane, in which each transient flattens off at the speed used in the measurements (50 mm/s) with a time constant 50 ms. This eventual speed and time constant correspond to the lowest of the three dashed lines in Fig. 1, and to the solid red line in Fig. 4. Some computational details are described in the next link.
Figure 5 shows that a sprinkling of these transients led to Helmholtz motion. Reassuringly these lie within the green lines, clustered towards the maximum bow force line at the top. Figures 6 and 7 show similar scans, in which the bow speed profile corresponds to the other two red lines in Fig. 4. The asymptotic bow speeds are respectively twice and four times the value for Fig. 4, while the time constant is unchanged.
It is far from clear whether we expect Schelleng’s analysis to apply in any way to these cases, but if it did we would expect the maximum and minimum bow force lines to move upwards by the same factors, to the positions marked by the dashed lines. It can be seen that the coloured pixels do indeed move progressively upwards, although falling somewhat short of the dashed lines. The other obvious change is that each plot shows more coloured pixels (i.e. successful transients) than the preceding one. In other words, it is easier to get Helmholtz motion going with this kind of transient if the steady bow speed is relatively high. But it also pays to use a fairly high bow force, not too far below the maximum allowed.
In Figs. 5—7, the eventual steady bow speed was varied while the exponential decay time stayed fixed. Figure 8 show what happens if the converse thing is done. With the same eventual bow speed as in Fig. 7 (0.2 m/s), the time constant was reduced to 20 ms (left-hand panel) and increased to 100 ms (right-hand panel). The resulting bow speed trajectories are shown by the blue and green dash-dot lines in Fig. 4. Comparing the results to Fig. 7, we can see that the changes appear to be relatively small, compared to the differences between Figs. 5—7. The tentative conclusion is that the eventual bow speed is more important than the time constant.
C. String-crossing transients
Next, we turn to another type of idealised bowing gesture, a string crossing in which the bow speed remains constant while the bow force ramps up from zero to a steady level, with an exponential trend like the one used for bow speed in the previous subsection. The top right-hand panel of Fig. 1 illustrates the two cases we will examine, with two different time constants (10 ms and 50 ms).
Again, we can represent results in the Schelleng diagram, using the eventual steady bow force to locate the points. Figure 9 shows results for transient length, for the case indicated by the solid line in Fig. 1. The bow speed is set to the value 50 mm/s, as in the measured Schelleng diagram, so we are not surprised to see the coloured pixels confined to the wedge between the two green lines — outside this wedge, steady Helmholtz motion is simply not possible.
But we should be surprised by another obvious feature of this plot. Up to now, we have been used to all plots concerning transients showing a “speckly” texture, indicating sensitive dependence of the detailed string motion on any small changes. But this plot shows no such thing! On the whole, the colours graduate smoothly from one pixel to the next. It appears that this kind of bowing transient leads to rather benign behaviour, which would no doubt be appreciated by a player. Notice that a wider range of transient lengths has been included in the colour scale here, to make the resulting pattern more clear. This finding is intriguing, and leads us to wonder whether there might be other bow gestures that have a similar effect.
To see what is happening, it is useful to look at some waveforms. Two columns are marked by blue rectangles in Fig. 9, and Figs. 10 and 11 show every second waveform in those columns. Sounds 1 and 2 allow you to listen to them. Figure 10 shows the left-hand marked column: working up from the bottom, the first four waveforms show multiple slipping (“surface sound”), waveforms 5—8 show Helmholtz motion, while the top two show raucous motion. Figure 11, for the right-hand marked column, shows Helmholtz motion in the first 5 waveforms, then progressively raucous motion above that. The orderly progression of transient lengths, for cases that led to Helmholtz motion, is clear. As we might have anticipated for bow gestures of this kind, there is a strong tendency to show multiple slipping early in the transient. This may or may not evolve into Helmholtz motion later.
Figure 12, 13 and 14 show the same information for a set of simulations using a time constant equal to 50 ms. Sounds 3 and 4 allow you to listen to the results. The pattern of Fig. 12 is rather similar to the one in Fig. 9. The same orderly progression of colours is seen, but perhaps there are slightly fewer really short transients. The sound files illustrate an effect that might have been predicted: with the slower build-up of force, the notes start in an audibly more sluggish manner.
D. Spiccato transients
The next type of bowing gesture to be investigated is the spiccato stroke. The bow moves back and forth, while bouncing in and out of contact with the string, as shown schematically in the lower left-hand panel of Fig. 1. If things are working in the way the player intends, the result is a clean burst of Helmholtz motion each time the bow bounces onto the string. The combined bow motion has been represented with a combination of sinusoidal waveforms, exactly as was done by Guettler and Askenfelt [1]. The simulations were run for 1 s with the bow-speed modulation taking place at 2 Hz, so that four spiccato notes are produced. Some details are given in the previous link.
Figures 15 and 16 show four examples of the resulting string motion — this choice of examples will be explained shortly. Figure 15 shows the bridge-force waveforms, and Sound 5 allows you to listen to them. Figure 16 shows the corresponding waveforms of string velocity at the bowed point. These velocity waveforms are included for comparison with the simulations and measurements of Guettler and Askenfelt. The sinusoidal modulation of bow speed can be seen directly in these plots. The string motion shows major slip pulses in opposite directions, depending on whether the bow speed is positive or negative. What these plots and sounds reveal is that the lowest waveform shows mainly double slipping, resulting in “surface sound”. The two in the middle are more or less acceptable, while the top one is decidedly “scratchy”.
To interpret these findings, we need to know how exactly these examples were obtained. Guettler and Askenfelt highlighted the fact that the string motion can be very sensitive to the relative phase of the bow speed and bow force waveforms — in other words to the relative timing of two components of what the player is doing with the bow. Where exactly in the sinusoidal bow speed cycle does the bouncing bow make contact with the string? Figure 17 illustrates four different choices for this relative phase.
For each of these phase choices, a scan of the Schelleng plane has been computed. The bow force is time-varying in all these cases, so the peak value of the force waveform has been used to define the vertical axis in the plots (see the previous link for details). For each simulated transient the bridge force was segmented into the four separate notes, then the automatic processing was applied to each one separately. The Schelleng plots then show a colour for each pixel representing the average transient length of the four notes. The results are shown in Fig. 18.
The green lines in these plots are the same as those shown earlier, marking the approximate positions of the minimum and maximum bow force as measured with a bow speed of 0.05 m/s. In these spiccato simulations the bow speed was time-varying, up to a peak value of 0.2 m/s. In view of this higher bow speed, it is perhaps not surprising to see the coloured pixels tending to fall in a region higher than the band marked by the green lines. In all four cases, the upper and lower limits of the region of coloured pixels seem to lie approximately parallel to the green lines, so something of Schelleng’s analysis seems to be still relevant. The white lines in these plots give an indication of the effect of bow speed. The solid white lines show the effect of adjusting the green lines according to Schelleng’s formulae, based on the maximum values of both bow force and bow speed. (The upper line falls outside the plotting range.) The dashed white lines show the effect of using the maximum bow force together with a bow speed of 0.1 m/s. The coloured pixels seem in fact to be confined between the solid white line for the minimum, and the dashed white line for the maximum.
The top two panels of Fig. 18 show the results corresponding to the two solid lines in Fig. 17. The lower two panels show results corresponding to the two dashed lines: both show fewer coloured pixels than the plots in the top row. This conclusion about the best phase relation to give clean spiccato notes is broadly in agreement with what Guettler and Askenfelt found. They reported that a skilled player could reliably produce a “perfect” sequence of spiccato notes with the phase relation within a certain range (but Guettler reported verbally that it took him a year to perfect this technique). They supported their conclusions with measurements and simulation results, but they do not give a lot of detail about either so we can’t make a direct comparison here. To see the influence of the phase relation in our simulated results more clearly, Fig. 19 shows the number of transients shorter than any given length, for the four cases shown in Fig. 18. The line colours and types match those in Fig. 17. It is obvious that the two solid lines lie well above the two dashed lines.
What we should be looking for in Fig. 18 to support the findings of Guettler and Askenfelt is a region of the Schelleng diagram showing uniformly pale colours, so that a “perfect” sequence of spiccato notes could reliably be expected. The closest approach to this desirable state of affairs is found in the top right-hand panel, around the middle of the right-hand edge. The four cases marked with blue squares straddle this region: they are the cases selected for Figs. 15 and 16, and Sound 5. The clear implication of these plots is that crisp spiccato should be easiest at relatively large values of $\beta$, in other words with the bow quite far from the bridge. We should also note that all four panels of Fig. 18 show the now-familiar speckly pattern, indicating sensitive dependence.
E. Tremolo transients
The final family of bowing transients to be investigated here relates to tremolo bowing, illustrated by the lower right-hand panel of Fig. 1. The bow speed is sinusoidally modulated as it was for spiccato transients, but now the bow force is held constant. A selection of simulated transients is shown in Figs. 20 and 21, in which the bow “scrubbing” frequency was 4 Hz so that there are 8 notes in the 1 s simulation interval. You can hear the four bridge-force waveforms from Fig. 20 in Sound 6.
There is no equivalent of the phase variable to explore in these transients, so results can be encapsulated in a single Schelleng diagram. The processing was similar to the spiccato case: the 8 notes in each simulation were segmented, and each one was processed with the usual automated analysis. The average length of transient was used to colour the Schelleng diagram, with the results shown in Fig. 22. The annotation in this figure is similar to what was done in Fig. 18: the green lines show the maximum and minimum bow force fitted to the simulated Schelleng diagram with a bow speed 0.05 m/s, the solid white lines show how these lines would move if based on the maximum bow speed (0.2 m/s), and the dashed white lines show the effect of using a bow speed 0.1 m/s. As was found with the spiccato transients, the coloured pixels are largely confined between the solid white line for the minimum bow force, and the dashed white line for the maximum bow force.
The band of coloured pixels is somewhat broader than it was for the spiccato case, but the colours are less bright, connoting slower transients. This is shown explicitly in Fig. 23, which is similar to Fig. 19. The blue line in this figure is the same as the blue line in Fig. 19, while the red line shows the corresponding results deduced from Fig. 22. It is immediately clear that the red line lies well below the blue line: at least based on these simulated results, it is much easier to get a rapid transient with a spiccato gesture than with a tremolo gesture. The four cases selected for Figs. 20 and 21, and Sound 6, are marked by blue squares. They include the brightest pixel in Fig. 22.
[1] Knut Guettler and Anders Askenfelt, “On the kinematics of spiccato and ricochet bowing”, Catgut Acoustical Society Journal 3, 6, 9—15 (1998)