A. Differential slipping and “spikes”
One thing missing from the simulation models used so far is the influence of the finite width of the ribbon of bow-hair — the models have all assumed a single point of contact between bow and string. The most obvious effect of finite width is best introduced by an example. Listen to Sound 1: this is a recording using a bridge-force sensor on the open G string of a violin. The player gradually slides the bow towards the bridge, getting very close at the end of the sample. In the sound, you should be able to hear a growth in harshness of the sound, ending with a definite “crunch”. This example is intentionally extreme, but players sometimes make deliberate use of this harsh sound at moderate level.
Figure 1 shows two extracts from this measured waveform, from early and late in the sequence. Both show the familiar Helmholtz sawtooth waveform, but they both have something extra superimposed. The upper trace shown occasional small spikes in the waveform, which differ in detailed placement from cycle to cycle. The lower trace shows a similar thing in more extreme form: it is not surprising that this irregular waveform might sound harsh or noisy.
We get a clue about what might be happening from Fig. 2. The upper diagram shows the moment in an ideal Helmholtz motion when the travelling corner has just gone past the bow, setting off towards the player’s finger. The ribbon of bow-hair is indicated by the yellow stripe: deliberately shown very wide, to make the effect clear. The lower diagram shows what might happen later in the cycle, shortly before the Helmholtz corner gets back to the bow. In the ideal version of Helmholtz motion, the string shape would follow the dotted line. But if the portion of string in contact with the bow has been sticking throughout this time, over the entire width of the bow-hair, the string would need to take up a zig-zag shape more like the solid line.
There are sharp corners at the edges of the bow-hair. These produce concentrated forces, in opposite directions on the two edges. This feels physically unrealistic: surely we might expect the string to have slipped in some region near the edges of the bow, before things got this extreme? Those localised slips, at the edge of the bow facing towards the bridge, are responsible for the “spikes” we saw in the bridge-force waveforms of Fig. 1. The effect becomes more extreme when the bow moves closer to the bridge.
We can confirm this idea by extending the computer simulation model to allow for a bow of finite width. Finite-width bowing has been most thoroughly investigated by Roland Pitteroff [1,2]. He included effects such as the elasticity of the bow-hairs, but unfortunately the approach he used cannot readily be combined with the simulation model we have used earlier (developed by Hossein Mansour [3]), for reasons of numerical stability. So instead, we use a simpler model which harks back to the earliest work on this topic [4], around 1980. It assumes a rigid bow, but one which contacts the string at several points rather than a single point. This model will allow us to make direct comparisons with the earlier simulations and the Galluzzo measurements, both based on bowing a cello D string. Some details are given the next link: the model is a straightforward extension of the point-bow model used earlier, so we can make direct use of the same friction models introduced in section 9.6.
Figure 3 shows one example of such a simulation, using 6 “bow-hairs”. This particular example was made using the enhanced thermal friction model, initialised with ideal Helmholtz motion and then allowed to run for a while for the motion to settle down. The bridge force waveform shows an irregular pattern of spikes superimposed on a more-or-less steady Helmholtz sawtooth, qualitatively similar to the results in Fig. 1.
Figure 4 shows the corresponding pattern of sticking and slipping across the width of the bow. The time axis matches Fig. 3, and the top edge of the plot corresponds to the edge of the bow closest to the bridge. Black pixels indicate sticking and white ones indicate slipping. The main Helmholtz slips show up as top-to-bottom white stripes, while the partial slips causing the spikes in Fig. 3 show up as white streaks in the upper part of the plot. There are a small number of red pixels, indicating slipping in the reverse direction: these occur occasionally near the bottom edge of the plot, as Fig. 2 might have led us to expect.
B. The effects on Schelleng’s diagram
The same computational approach can be used to explore the effect of finite bow width on the behaviour in the Schelleng diagram. Simulations initialised with ideal Helmholtz motion can be made at a grid of points in the Schelleng plane, with different values of bow force and bow position distributed on logarithmic scales to match the Galluzzo measurement. The result of that measurement was discussed back in section 9.3, and the main plot is reproduced here as Fig. 5 (with the addition of some mark-up in green, to be explained shortly).
The measurements were made with the rosin-coated rod — unlike the Guettler measurements, they were not repeated using a normal cello bow. This means that direct comparisons with simulations will need to be with a point-bow model, but we can then use the finite-width simulations to indicate how things might change with a real bow. Figure 6 shows a Schelleng diagram simulated using a single-point bow and the enhanced thermal model of friction. Figure 7 shows the corresponding plot using 6 “bow-hairs” spaced over the 10 mm width of a cello bow.
It is probably useful to start with a reminder of what we expect the Schelleng diagram to show, and what the different colours signify in these plots. All three plots show a central region of red pixels, connoting Helmholtz motion. Below the red pixels is a band of orange colour, connoting “double-slipping motion”, often described by players as “surface sound”. Below these orange pixels are some yellow ones, indicating a region where long-lasting steady stick-slip motion seems not to be possible at all — the string motion simply dies away.
On the opposite side of the region of red pixels, in the top right corner of all three plots, some black pixels can be seen. These connote non-periodic motion, described by Schelleng as “raucous”. Finally, there are some white pixels. These connote approximately periodic motion that is neither Helmholtz motion nor double slipping. We will see some examples shortly — they are all what Raman described as “higher types” of string motion, and are something players usually try to avoid unless they are deliberately seeking an unusual tone colour.
Schelleng predicted that the upper limit of Helmholtz motion, the maximum bow force, should follow a straight line in these log-log plots, with a slope of $-1$. The lower limit, the minimum bow force, should also follow a straight line, but with a steeper slope with the value $-2$. The two green lines marked in all three plots show what these slopes look like. The specific lines were fitted to the simulated results in Fig. 6, then the same lines were shown in the other two plots.
We can learn some interesting things from these lines. First, it is by no means obvious that Schelleng’s argument will really apply to our thermal friction model, and even less obvious that it will apply to the finite-width simulations. But in fact Figs. 6 and 7 both show the red-orange boundary tracking the lower green line quite closely, and the red-black boundary tracking the upper green line moderately well. It looks as if the pattern of Schelleng’s limits continues to apply, and that a finite-width bow makes rather little difference to this aspect of behaviour. The regions of white pixels cut across the upper green line in Figs. 5 and 6, but that is to be expected. Versions of Schelleng’s calculation can be constructed for the various “higher types”, and this is the predicted behaviour: each variety of “higher type” has its own maximum and minimum bow force, different from those for Helmholtz motion.
Comparing Fig. 5 with Fig. 6, we can see some strong similarities but also some differences. The red-orange boundary follows a similar trend to the green line, but it is displaced upwards a little. The precise shape of the red-black boundary cannot be seen very clearly, but it seems to be in a very similar position to the green line deduced from the simulations. Looking back to section 7.3.1 where Schelleng’s limits were derived, this pattern makes good sense. The maximum bow force only depends on the bow speed and position, and the properties of friction. Those quantities are all supposed to be well matched by the simulation. However, the minimum bow force also depends on an additional property, related to the damping of the string vibration. It would not be at all surprising if the simulations do not quite match the damping of the experimental string and cello. One component of that damping relates to energy loss into the cello body at the bridge, and the simulations are based on measurements of a different cello body (see Fig. 20 of section 9.6.3 for details).
One obvious difference between Figs. 5, 6 and 7 is in the number and positions of the white pixels. Figures 5 and 6 show a generic similarity, although the details do not exactly match. Both show two vertical columns of white, at high values of $\beta$. Figure 7 shows far fewer white pixels, and they do not form obvious columns in the same way. To see what lies behind these differences, we need to look at some waveforms in detail. Six pixels have been chosen, marked (a)—(f) in all three plots. Matching extracts of bridge-force waveforms for these cases are plotted in Fig. 8—10. In each plot, the waveforms are in the same order as Figs. 5—7: the measurement at the top, the single-point simulation in the middle, and the finite-width simulation at the bottom.
Figure 8 shows cases (a) and (b). Both points have been chosen to fall in regions with the same colour in all three Schelleng diagram. As expected, the left-hand panel shows Helmholtz motion, the right-hand panel shows double-slipping motion. For both of these, all three waveforms agree rather closely.
Figure 9 shows cases (c) and (d), placed in a region where white pixels appear. On the left, the measured waveform shows a “higher type” while the point-bow simulation shows a different one. The finite-width simulation shows Helmholtz motion, with a fairly well-developed pattern of Schelleng ripples. On the right, this figure shows a higher type in the measurement which is very similar to the one seen in red curve on the left. The middle plot shows slightly irregular Helmholtz motion, and the lower plot shows yet another higher type, this one a variety of “double flyback motion”. Figure 10, showing cases (e) and (f), explores a different region of white pixels: it reveals a different mixture of higher types, along with Helmholtz motion and non-periodic raucous motion. This time, the measured waveform in the left-hand panel shows well developed “S-motion”, as does the middle waveform in the right-hand panel.
The conclusion from these plots is that the measurement shows a pattern of white pixels that is qualitatively similar to the single-point simulations. The precise position of the vertical columns is different in the two cases, but Figs. 9 and 10 suggest that the same repertoire of higher types is seen in both cases. With such a rich array of possible string motions, it is perhaps not surprising that there are detailed differences between measurement and simulation in terms of which regime occurs where.
The finite-width simulation (Fig. 7) shows far fewer white pixels, and thus a Schelleng diagram which is more orderly and closer to Schelleng’s original prediction. Somehow, the finite width is having an effect that tends to suppress higher types, which all involve large oscillations at the same frequency as the Schelleng ripples. Probably, players would welcome this difference. They are less likely to stray into unwanted S-motion or other higher types. Perhaps this is one reason for the use of a conventional bow, although there are many other reasons, some of them to be explored in the subsequent discussion.
C. Tilting the bow
The finite-width bowing model allows us to make a preliminary exploration of something important. We have already discussed the effects of several variables a player can control during bowing: the bow force, speed and acceleration, and the position of the bow on the string. But there is one more variable, which we have ignored up to now: players do not necessarily press the bowhair flat against string. Instead they usually tilt the bow, so that the distribution of normal force is not uniform across the width. The tilt usually varies in the course of a bow stroke, and in extreme cases it may be so large that the hairs only contact the string over a small part of the width. Manipulating bow tilt is a major ingredient of the player’s armoury, when striving to achieve subtleties of tone.
We will not attempt to model such subtleties here: instead, we will explore some of the effects of bow tilt by comparing three idealised cases. The bow width in contact with the string will stay the same, but three different distributions of normal force will be used: uniform (the “flat bow”), or tapering linearly across the bow from zero at one edge to a maximum at the other edge. This taper could go in either direction. Conventional tilt involves reducing the force on the side facing the bridge, and players do not ordinarily tilt in the opposite direction, so we will label these two cases “correct tilt” and “incorrect tilt”.
Figures 11, 12 and 13 give a first impression of the consequence of a tilted bow. Figure 11 repeats the plots from Figs. 3 and 4, with a flat bow. It shows Helmholtz motion, with some irregular “spikes” in evidence. Figure 12 shows the “correct tilt” case: all parameters are the same, except that the normal bow force varies linearly across the width of the bow, falling to zero at the edge nearest to the bridge and rising to double the mean level at the opposite edge. Figure 13 shows the converse case, with “incorrect” tilting so that the force is higher at the edge nearest to the bridge.
Comparing Figs. 11, 12 and 13, the bridge force waveforms look quite similar but the stick-slip patterns are significantly different. With the flat bow, the white streaks denoting partial slipping at some of the bow-hairs extend across almost the whole width of the bow. For the correctly tilted case, the edge of the bow nearest to the bridge is slipping for most of the time (occasionally in the reverse direction). For the incorrectly tilted case, the converse pattern is seen. The section of the bow facing the bridge is sticking most of the time, while the other edge is usually slipping, mostly in the reverse direction indicated by red points in the plot.
[1] R. Pitteroff and J. Woodhouse, “Mechanics of the contact area between a violin bow and a string. Part II: simulating the bowed string”; Acta Acustica united with Acustica, 84, 744—757 (1998).
[2] R. Pitteroff and J. Woodhouse, “Mechanics of the contact area between a violin bow and a string. Part III: parameter dependence”; Acta Acustica united with Acustica, 84, 929—946 (1998).
[3] 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).
[4] M. E. McIntyre, R. T. Schumacher and J. Woodhouse, “Aperiodicity in bowed-string motion”, Acustica 49, 13—32 (1981). See also Erratum, Acustica 50, 294—295 (1982).