9.5 Getting that perfect start: Guettler’s diagram

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So far in this chapter we have been following the “physicist’s agenda”, using simplified models to look at one phenomenon at a time to build up a picture. But what about the “musician’s agenda”? How much have we learned that will be interesting to a player of the violin or cello? If we are honest, the answer is “only a rather limited amount”. Schelleng’s diagram is relevant, particularly to beginners. Wolf notes can be of direct interest to all players, but they are somewhat of a niche interest since they only affect particular notes.

But what players really care about, and spend a lot of time on, relates to transients. Hours of practice are devoted to mastering different bowing techniques, each associated with a particular way to start a note or to transition between notes. If an instrument had some subtle feature which made a particular tricky bowing gesture just a little easier or more reliable, it seems a good guess that a player would describe that instrument as being “easier to play”. This possibility opens an interesting avenue of study, using computer simulation models of the kind we have been talking about.

The earliest computer models date from the 1970s. A decade later, the increasing power of computers allowed these early bowed-string simulation models to be used to start exploring transient effects systematically [1,2]. How quickly, if at all, is the Helmholtz motion established after a given bow gesture? How does the transient length vary if you change parameters in the model that are relevant to a player, or an instrument maker? This early work established some useful methodology, but we needn’t look at the detailed results because the studies suffered from major flaws that only became apparent later: we will meet them in the course of this section and the next.

The people we have mentioned so far, like Raman, Cremer and Schelleng, have all been scientists or engineers with an interest in music. But the hero of this section was a musician who developed a strong interest in science. Knut Guettler was a virtuoso player and teacher of the double bass. He got interested in whether theoretical models and computer simulations could tell him things that would be useful in his teaching. Double bass players have a particular problem: some of the notes they play have such low frequencies that they can’t afford a bowing transient that takes 10 or 20 period lengths to settle into Helmholtz motion: a short note may be over by then! So Guettler set himself the task of understanding what kind of bow gesture a player needed to perform in order to get a “perfect start” in which the Helmholtz sawtooth waveform was established right from the first slip of the string over the bowhair.

Guettler’s initial step was to point out the first of the major flaws in the early computer studies: the transients used in those studies were all physically impossible! In the computer, it is easy to simulate “switch-on” transients in which the bow speed or force suddenly changes. But any physical transient cannot have jumps in either quantity: it must start with either the bow force or the speed (or both) equal to zero. If the bow is already in contact with the string with non-zero normal force, the speed must start from zero. On the other hand, if the bow is already moving when the bow makes contact with the string, as in a string-crossing gesture, then the normal force must build up from zero.

This realisation led Guettler to study a more realistic family of transient gestures. The bow starts in contact with the string, and the force is held constant while the bow is accelerated from rest with a chosen value of acceleration. Guettler then used the simplest available model of bowed string motion to pursue his agenda of finding the conditions under which a “perfect start” was possible from one of these constant-acceleration gestures. He assumed an ideal “textbook” string, terminated in mechanical resistances or “dashpots”. This simple model is yet another thing that goes back to Raman, and it is essentially the same model that Schelleng used in his discussion of bow force limits for steady Helmholtz motion. Some mathematical details of this “Raman model” are given in the next link.

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A virtue of this simple model is that it allowed Guettler to follow how the string motion develops during the early stages of a transient. The string is initially sticking to the bow, and it is pulled to one side as the bow moves. The first interesting thing to happen is that, sooner or later, the string releases from the bow. As sketched in Fig. 2, this release will give a similar effect to plucking the string. As we saw back in section 5.4, a pair of corners will then be created, travelling away from the bow symmetrically in both directions. One of these (the one travelling towards the bridge) looks very much like the Helmholtz corner we want, but the other one has the wrong sign. As Schelleng first pointed out [3], in order for a perfect start to occur, the “good” corner must survive and develop into the Helmholtz corner while the other one needs to disappear.

In a tour de force of analysis, Guettler was able to track the behaviour through the first few period-lengths after the first release, and he identified four things that might go wrong with the desired sequence of events [4]. For each of the four, he was able to find a criterion that would decide success or failure. The details of the calculation are quite messy, and we need not go into them, but we can show the key results in graphical form. All four criteria take the form of a critical value of the ratio of bow force to bow acceleration. For two of them, the ratio must be bigger than the critical value, while for the other two it must be less than the critical value. As an aside, Galluzzo [5] has generalised Guettler’s result with an argument that suggests that any threshold relating to bowed-string transients within the Raman model would be expected to take a similar form, as a critical ratio of force to acceleration.

There is a simple way to represent the result, illustrated in Fig. 3. Each criterion corresponds to a straight line in the acceleration—force plane. The slopes of these lines are determined by the four critical values. So there are four radial lines in that plane, and for a perfect start we need to be above two of these lines (shown in blue), and below the other two (shown in red). Unless the criteria are inconsistent, the result is a wedge-shaped region in the plane (shaded yellow here) within which a perfect start might be possible. Such plots are now known, naturally enough, as “Guettler diagrams”. This particular example is computed using the frequency and typical impedance of a violin G string (196 Hz and 0.363 Ns/m respectively), and the string is assumed to be undamped. The assumed friction coefficients are taken from the measurement shown in Fig. 6 of section 9.2. The chosen bow position has $\beta=0.13$ (recall that this parameter “beta” specifies the position of the bowed point as a fraction of the string length).

All four of Guettler’s boundary lines move in a rather complicated way when the bow position $\beta$ changes. An example of the variation of the slopes of the four lines is plotted in Fig. 4, using the same line colours and types as in Fig. 3. Logarithmic scales have been used for both axes here, to highlight an intriguing parallel with the Schelleng diagram. The Schelleng diagram shows that for a given bow speed, there are limits on the bow force in order for Helmholtz motion to be possible. If $\beta$ is decreased, both limits increase, and they get closer together and eventually meet. The new diagram says that for a given bow acceleration, there are limits on the bow force in order for a perfect start to be possible. These limits, too, increase as $\beta$ decreases, and get closer together and eventually meet. The pattern is more complicated than the Schelleng diagram, because there are two criteria for each of the upper and lower limits (shown in solid and dashed lines), and the lines cross so that all four play a role in determining the allowed region for some values of $\beta$.

Just as we did with Schelleng’s diagram, we can compare Guettler’s predictions with measurements using the Galluzzo bowing machine described in section 9.3.2. For each value of $\beta$, the machine bowed the open D string of a cello 400 times, in a $20\times 20$ grid in the Guettler plane. The bridge force from each note was recorded, and analysed using an automated procedure that attempted to find the length of transient before Helmholtz motion was established (if it ever was established, of course). This automated analysis is fallible, there is no doubt about that, but the same routine has been used in all cases (and will be used again when we come to compare with simulated results) so the comparison between cases should be fair.

Figure 5 shows the results, for 6 particular values of $\beta$. It is immediately clear that the results give at least qualitative support for Guettler’s predictions. There are not very many perfect starts (which appear as white pixels), but the successful transients (shown in colours other than black) are confined in each case to a vaguely wedge-shaped region. As $\beta$ increases, the boundaries rotate downwards and the wedge tends to get broader (although in the bottom left-hand plot it seems to have got narrower again). For the smallest value of $\beta$ shown here, there are very few coloured pixels. For even smaller values of $\beta$ the results were not worth showing, because they show virtually no coloured pixels at all. The left-hand plot in the bottom row shows a lot of black pixels within the wedge region. The main reason for this is something we saw earlier: with this particular value of $\beta$ the string often chooses to vibrate with S-motion rather than Helmholtz motion.

We can see and hear a few sample waveforms. They are all drawn from one particular column of the right-hand Guettler diagram in the middle row of Fig. 5, corresponding to $\beta=0.0899$. Figure 6 shows three waveforms, corresponding to pixels 2, 9 and 17 of the 9th column of that figure, counting everything from the bottom left-hand corner. The value of bow acceleration is 1.39 m/s$^2$, and the three force values are 0.55 N, 1.58 N and 2.76 N respectively. At the top, plotted in black, is the waveform for the highest force. It appears as a black pixel in Fig. 5, but it looks as if it is close to settling into the Helmholtz sawtooth by the end of the time shown here. You can hear it in Sound 1: it has a rather “scratchy” sound. In the middle, plotted in red, is a “perfect start” which you can hear in Sound 2. At the bottom, plotted in blue, is a case with a slow transient, settling into double-slipping (“surface sound”). You can hear it in Sound 3. All the sounds are very short, only 1/4 s for each one. You should be able to hear noticeable differences between all three, although the quality difference between the second and third sounds may not come across very clearly. Bear in mind that these are not the actual sounds made by the cello when the body vibrates and radiates: they are just waveforms of force acting at the bridge. These waveforms are recognisable as a bowed-string sounds, but they lack the extra interest generated by the body response.

To harness the full potential of theoretical bowed-string models for exploring issues of playability, qualitative agreement with measurements is not enough. We would like to use computer models to find out how transient lengths change as a result of changing parameters relevant to players and instrument makers. Well, in order for that agenda to be possible, the model must be sufficiently complete and realistic that it contains all those parameters. It must also be reliable enough to capture the influence of changing them. A first step would be to demonstrate quantitative agreement with measurements. As we will see, this proves to be a tall order.

What might we need to include in such a model? We have already met several factors that proved to be significant when looking at plucked strings, and it seems a fair bet that those will all be relevant to bowed strings too: the effect of the string’s bending stiffness and its intrinsic damping; the effect on frequencies and damping of coupling to the instrument body; the influence of the second polarisation of string motion, perpendicular to the bowing direction. These factors can indeed all be included [6]: the next link gives some technical details.

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In addition, there is a new factor that was not relevant to plucked strings. Transverse string vibration is not the only thing to be driven by the force that a violin bow applies to the string. That force is applied tangentially to the surface of the string, so it can also excite torsional motion of the string, as sketched in Fig. 7 and explained in more detail in the next link. Such torsional string motion probably isn’t directly responsible for a lot of sound from the instrument, but it can still be very important because of the way it interacts with transverse motion. It can be incorporated in the simulation model with no difficulty, although at some cost in complication [7].

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To see an important example of this interaction, think what happens when the string is sticking to the bow. If we only consider transverse motion, as in the discussion up to now, that means that the string just under the bow must be moving at the speed of the bow. But when torsional motion is also allowed, the string can roll on the sticking bow. This means, for example, that Schelleng ripples which were previously “trapped” on the finger side of the bow (see Fig. 13 of section 9.2) can now “leak” past the sticking bow and show up in the bridge force: see the previous link for more detail.

There is just one context where torsional string vibration is directly relevant to a violinist. Some violins are prone to a phenomenon called the “E string whistle”. Occasionally, when playing the open E string, a high-pitched sound is obtained at a pitch unrelated to the expected note at 660 Hz: it occurs at a frequency more like 5 kHz. This is the expected fundamental frequency of the first torsional mode of an E string, and Bruce Stough [8] has convincingly demonstrated that the whistle is indeed caused by torsional motion. Most violin strings have rather high damping of torsional modes, and the player’s finger will contribute further damping in a stopped note. But the E string of a violin is usually a steel monofilament, with no over-wrapping. So for an open E string, neither of these factors is present, and torsional modes can have very low damping. This allows torsional vibration to be directly excited by bowing. Some string manufacturers now offer E strings with a layer of over-wrapping, specifically to add torsional damping in order to suppress the whistle.

There is one more layer of potential complications in a realistic simulation model, associated with the details of a conventional violin or cello bow. The ribbon of bow hair has a finite width in contact with the string, rather than the single-point contact we have been assuming. Both the hair and the stick of a bow have vibration behaviour of their own, and those might influence the behaviour of the bowed string. But we can duck all those issues for the moment: the Galluzzo experiment deliberately used a rosin-coated rod in place of a conventional bow, and we will first try to match those results. In section 9.7 we will have a careful look at how things might change with a real bow.

For the particular cello string used in the Galluzzo measurements, we have a fairly complete set of measured properties relating to its transverse and torsional vibration behaviour. This means that we can formulate a computer simulation based on reliable, calibrated values of all the relevant parameters [5,6]. We can also incorporate a reasonably realistic model of the cello body, by extending the approach used for the wolf note in section 9.4 to include more body resonances.

Putting all this together, we can assemble a simulation model that includes a good representation of the behaviour of the cello and its string. We can “bow” this simulated string with the same friction curve we have used before, derived from the steady-sliding measurements and shown in Fig. 6 of section 9.2. We can then run a set of Guettler transients, with the same set of bow forces and accelerations as the measured set we saw in Fig. 5. We can process the result using the same automatic classification routine that we used for the experiments.

A typical result, compared with the corresponding measurement, is shown in Fig. 8, and it is very disappointing! No aspect of the simulated pattern gives a convincing match to the measurement. This simulated string would be far harder to play than the real cello: there are fewer coloured pixels, and many of them are scattered around in a speckly pattern rather than coalescing to give broad areas of bright yellow such as we see in the measurement. The only continuous patch of pale-coloured pixels is a narrow wedge running diagonally across the plot. This wedge is indeed reminiscent of Guettler’s prediction, but we see no trace of a directly corresponding feature in the measurement. We will explore some of the reasons for this in the subsequent discussion.

To see a little more detail of how the simulation model behaves, Fig. 9 shows the results of some model variations. The top row shows results without including the effect of the string’s bending stiffness, while the bottom row has it included. The left-hand column omits the effect of torsional motion in the string, while the right-hand column includes it. So the top left plot is without bending stiffness or torsion, while the bottom right plot includes both, and is the case shown in Fig. 8.

To glimpse what lies behind these results, Fig. 10 shows a set of simulated waveforms for one particular pixel of these Guettler plots. Counting from the bottom left-hand corner, it is the 9th pixel in the horizontal direction corresponding to acceleration 1.39 m/s$^2$, and the 6th pixel in the vertical direction corresponding to force 1.14 N. This is a bright yellow pixel in the bottom right-hand Guettler plot of Fig. 9 and a slightly darker colour in the upper right-hand plot. These correspond to the waveforms plotted in black and red respectively. It appears as a black pixel in the two left-hand diagrams, and we can see why in the two waveforms plotted in blue and green. The blue curve, corresponding to the case with neither torsion nor bending stiffness, settles into regular double-slipping motion rather than Helmholtz motion. The green curve, with bending stiffness but no torsion, does something similar, but the waveform shows more persistent irregularity from cycle to cycle.

Figure 9 tells us several interesting things. First, none of the plots look anything like the measured case. They all look too “speckly” for comfort: even for cases where a particular transient gave a satisfactory transient leading quickly to Helmholtz motion, there are often neighbouring pixels that are either black, or at least dark red. The interpretation is that if the player tried to do the same gesture twice, they are likely to get very different results because of inevitable tiny differences between the two gestures.

This “twitchy” behaviour should sound familiar, if you remember the discussion of chaotic systems in Chapter 8. It looks very much as if all four cases explored in Fig. 9 are exhibiting “sensitive dependence on initial conditions”, one of the hallmarks of a chaotic system. The measured Guettler diagram also has some evidence of “twitchiness”, with occasional black pixels in the middle of the bright yellow patch, but the effect seems far less extreme.

Indeed, sensitive dependence in the measured responses can be demonstrated directly. Paul Galluzzo made repeat measurements of the Guettler plane under nominally identical conditions, and 9 examples of these are illustrated in Fig. 11. Although the qualitative picture remains the same, individual pixels change between takes.

We can hazard a guess about where the twitchiness is coming from in the simulated results. When we looked at the phenomenon of chaos in section 8.4, we found that sensitive dependence was intimately tied up with the presence of saddle points in the phase space: two trajectories that are initially very close together can approach a saddle point, and be sprayed apart so that they end up in very different parts of the phase space. Well, our bowed-string model contains a mechanism that could do a similar job, separating two initially similar transients so that they diverge.

Figure 12 shows a copy of Fig. 8 from section 9.2, and it reminds us of the abrupt jumps that are inevitably generated by using this friction curve. We can imagine two transients which start out similar, and then come close to one of the critical points at which a jump is triggered. If one transient falls just short of the critical point but the other one passes it, one will have a jump and the other will not, and their subsequent development might be quite different. This idea points us towards the second major flaw in the early computer models: in the next section we will have a critical look at this “friction curve model” and discover that the actual frictional behaviour of a rosin-coated violin bow does not follow any single friction–velocity curve: it is all much more complicated!

Before that, we should note something else about Fig. 9. At least for these particular cases, we see that both bending stiffness and string torsion have a significant effect on the results: including torsion makes things better (more coloured pixels), while including bending stiffness makes things worse. The extent of these differences is perhaps another manifestation of “chaotic twitchiness”: as well as sensitive dependence on details of the bowing transient, we are also seeing sensitive dependence on changes to modelling details, by including or excluding various effects.

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[1] J. Woodhouse; “On the playability of violins, Part II minimum bow force and transients”; Acustica 78, 137–153 (1993).

[2] R. T. Schumacher and J. Woodhouse, “The transient behaviour of models of bowed-string motion”; Chaos 3, 509–523 (1995).

[3] J. C. Schelleng; “The bowed string and the player”, Journal of the Acoustical Society of America 53, 26–41 (1973).

[4] Knut Guettler, “On the creation of the Helmholtz motion in bowed strings”; Acta Acustica united with Acustica, 88, 970–985 (2002).

[5] Paul M. Galluzzo; On the playability of stringed instruments, PhD Dissertation, University of Cambridge (2003). See section 6.1.

[6] Hossein Mansour, Jim Woodhouse and Gary P. Scavone, “Enhanced wave-based modelling of musical strings, Part 1 Plucked strings”; Acta Acustica united with Acustica, 102, 1082–1093 (2016).

[7] Hossein Mansour, Jim Woodhouse and Gary P. Scavone, “Enhanced wave-based modelling of musical strings, Part 2 Bowed strings”; Acta Acustica united with Acustica, 102, 1094–1107 (2016).

[8] Bruce Stough, “E string whistles”; Catgut Acoustical Society Journal (Series II), 3, 7, 28—33 (1999).