A beginner on the violin soon finds out that drawing a bow across a string doesn’t always make a satisfactory musical note. They must learn to control the parameters of bowing to stay within certain limits: those limits are the subject of this section. For the moment we will ignore transient effects, and concentrate on steady bowing with the aim of producing a steady note. Once the instrument, the strings, the bow and the particular note have all been chosen, the player has four things to control in a steady bow stroke: the bow speed, the bow force, the position of the bow on the string, and the angle of tilt of the ribbon of bowhair relative to the string. We will ignore tilt for now: we come back to it in section 9.5.
That leaves bow speed, force and position to consider. If the speed and the position are kept constant while the bow force is varied, a player quickly finds out that the force must lie within a certain range in order to achieve a satisfactory Helmholtz motion of the string. Press too hard, and instead of a note the violin will make a raucous “crunch” noise. Press too lightly, and a more subtle problem arises. A musical note is still produced, but the tone quality changes in a way that is likely to be criticised by your violin teacher: it is often described as “surface sound”, or “not getting into the string properly”. The vibration regime responsible for this “surface sound” is one we have already met: it is the double-slipping motion shown in Fig. 5 of section 9.1.
We deduce that there must be a minimum bow force and a maximum bow force for Helmholtz motion. We will soon see that both these force limits change, depending on the values of the other two control parameters: bow speed and bow position. The first steps in understanding these limits were taken by Raman, and then in the 1970s his work was refined and extended by John Schelleng. He presented the results in the form of a diagram, which we will see shortly. Schelleng, like Cremer, had a distinguished career in his “day job”: in his case, research work in the Bell Telephone Laboratory. But, like Cremer, he was a keen amateur string player, and in his retirement he devoted a lot of his time to researching violin acoustics as a founding member of a group of enthusiasts based in the USA who called themselves the “Catgut Acoustical Society”.
Schelleng’s condition for maximum bow force is the simpler of the two limits: it follows from the graphical construction relating to frictional hysteresis, explained in section 9.2. In order for Helmholtz motion to be possible under the constraints of the hysteresis rule, the Helmholtz slip speed must be outside the range of the hysteresis. The details of the resulting formula are given in the next link.
The condition for minimum bow force is a little more complicated. For an ideal textbook string, there would be no minimum bow force, because Helmholtz motion is a possible free motion of the string, not requiring any support from the bow. However, as soon as we allow the system to have some energy dissipation, steady motion is only possible if the bow somehow compensates for the energy loss in each cycle.
Schelleng’s condition is calculated by assuming that this energy loss all takes place at the bridge, by coupling to the body of the instrument. However, he did not allow for the detailed dynamical behaviour of the body. Instead, he used an approximation rather similar to Weinreich’s model for double decays in a piano (see section 7.3): he modelled the body as a mechanical resistance (or “dashpot”) with an impedance assumed to be much greater than the impedance of the string. We will return to this approximation in the next section, and see how the result is modified by using a more realistic model for the body vibration. But for the moment we will follow Schelleng’s approximate treatment.
The chain of logic runs like this. If we assume an ideal Helmholtz motion, with a given bow speed and position, we already know what the waveform of bridge force will be: it is the sawtooth we have seen several times now. That force will produce some motion in the dashpot representing the body, with a velocity waveform which will mirror the sawtooth shape. This body motion will in turn create some additional force back at the bowed point. For Helmholtz motion to be possible, this additional force during a sticking interval must not exceed the limit of friction, otherwise it would trigger a second slip. This argument gives the condition we are after: the details of the calculation are given in the previous link.
According to Schelleng’s formulas, both the maximum and the minimum bow force are proportional to the bow speed. So a faster bow speed means you need to press harder, but because both limits change in the same way, the relative range of possible bow force stays the same. The dependence of the force limits on bow position is more interesting. We will describe it in terms of the parameter $\beta$ (“beta”) introduced earlier, the position of the bowed point as a fraction of the string length. According to Schelleng’s formulas, the maximum force is proportional to $1/\beta$, while the minimum force is proportional to $1/\beta^2$.
These different dependencies on $\beta$ are the ingredients of Schelleng’s diagram. If we plot the two force limits in a graph of bow force versus bow position, and if we choose to use logarithmic scales for both these variables, we get something like Fig. 1. Both bow force limits appear as straight lines in the plot, and the line for the minimum bow force is steeper. Helmholtz motion is only possible within the wedge-shaped region between the two lines. As the bow is moved closer to the bridge, the player needs to press harder, and also to control the force more carefully because the available range decreases. The two lines meet at some point. At least according to this approximate analysis, it is not possible to produce Helmholtz motion if the bow is closer to the bridge than this meeting point.
Schelleng’s diagram is useful to beginning string players. The interaction between bow force and bow position is not intuitively obvious, and the graphical representation can help to fix the idea clearly in the mind. For example, the diagram gives an immediate explanation of a common beginner’s error, illustrated in Fig. 2. The player may keep the bow speed and the bow force under control, but if they do not concentrate on the bow position they may find themselves wandering backwards and forwards along a line like the blue one in this plot. This can result in going above the maximum force or below the minimum force, even though the force itself has not changed.
Schelleng’s diagram gives us a first tool to ask questions about “playability”. What makes one instrument “easier to play” than another? One possible answer to that question is that an instrument with a larger Helmholtz wedge in the Schelleng diagram might strike a violinist as being easier to play. So it is of interest to ask whether the two bow force limits would be expected to vary from instrument to instrument. Schelleng’s approximate formula for the maximum bow force does not depend on anything to do with the instrument body. The only things that appear in the formula (see the previous link for details) are the bow speed and position, the characteristic impedance of the string, and some information about the friction coefficients. So choosing a different string or a different brand of rosin for the bow might make a difference, but changing the body of the violin should not.
The story is different for Schelleng’s formula for the minimum bow force. The result depends upon the impedance of the dashpot representing the body, and hence its potential for energy dissipation. Choosing a different value for that impedance would result in shifting the line up or down, as indicated in Fig. 3. The bigger the body impedance, the further down the line is shifted. In the extreme case when the impedance becomes infinitely large, the line would be pushed off the diagram entirely. Helmholtz motion would then be possible for any bow force below the maximum. This is the case we already mentioned, of an ideal string with no energy loss, which has no minimum bow force.
There is a snag with trying to apply this idea to compare two different violins: a violin body does not really behave like a dashpot, so it is hard to know what value for the dashpot impedance we should be using in the formula. We will return to this question in section 9.4, when we will show how to extend Schelleng’s analysis to take account of the details of body response, for example from a measured bridge admittance.
In this section and the two previous ones, we have described several features of the physics of a bowed string, and made some predictions. We have by no means exhausted the list of physics-related questions, but this is a good point to look at some experimental results to assess how we are doing so far. We will look at detailed results from a measured version of the Schelleng diagram: they will shed some light on all the things we have discussed.
It is a rather laborious business to scan the Schelleng diagram experimentally. It can really only be done with some kind of bowing machine, which allows the speed and force of a bow stroke to be controlled in a precise and repeatable way. Two such experiments have been carried out, both as part of PhD projects: one by Paul Galluzzo, working in Cambridge , and the other by Erwin Schoonderwaldt, working in Stockholm .
Both give somewhat similar results, which is reassuring, but neither is absolutely ideal for giving an overview of normal bowing of a violin string. Schoonderwaldt’s experiment used a real bow on a violin D string, but the string was attached to a rather rigid laboratory monochord rig. As we have just seen, this might have major consequences for the position of the minimum bow force line. Galluzzo’s experiment used a cello D string mounted on a real cello, so the bow force limits should be realistic. However, he did not use a real bow. His experiment was designed primarily to give experimental data for probing theoretical models of bowed-string transients, and it is easier to make detailed comparisons if the dynamics of the bow and the finite width of the ribbon of hair are removed from the picture. So Galluzzo’s Schelleng diagram test was carried out using a rosin-coated perspex rod rather than a real bow.
I will show some results from the Galluzzo experiment. The bowing machine is described in the next link, together with the procedure used in the Schelleng diagram experiment. Results were collected for a $20 \times 20$ grid of points in the Schelleng plane: 20 values of $\beta$, logarithmically spaced between 0.02 and 0.18, and 20 values of bow force, logarithmically spaced between 0.1 N and 3 N. For each of these points, the bridge force was recorded and then analysed to determine what the string had decided to do.
The results are shown in Fig. 4, using different colours to indicate different vibration regimes. Helmholtz motion is shown in red, occupying a large region in the middle of the plot. Below the Helmholtz region, orange pixels indicate double slipping (or triple slipping, etc.). Yellow pixels mark cases where the string did not really seem to want to vibrate at all. The string was set into vibration by the initial bowing gesture imposed by the bowing machine, but for these points it seems to be decaying away. Above the Helmholtz region, black pixels mark cases where the motion was not periodic, so some kind of “raucous” motion had occurred. Finally, white pixels mark cases of S-motion.
To make sense of this classification, Fig. 5 shows extracts from the measured bridge-force waveforms for a representative selection of four columns of the Schelleng diagram (marked with blue arrows in Fig. 4). Each panel of this plot shows all 20 values of bow force, stacked up in the same way as in the Schelleng diagram. The left-hand panel shows the Helmholtz sawtooth in the top few cases, then some double or triple slipping waveforms. By the bottom of the stack, the waveforms look almost sinusoidal, as they slowly decay away. Near the middle of the stack are a few cases where it is really not at all clear how they should be classified: but they are certainly not Helmholtz motion.
The second panel of Fig. 5 shows a similar sequence, but it is clear that the Helmholtz sawtooth reaches further down the stack, just as indicated in Fig. 4. The third panel shows double-slipping at the bottom, then a block of Helmholtz motion cases, but at the top it shows something different. There is an abrupt transition to S-motion waveforms. The right-hand panel is different again. There are still some recognisable Helmholtz sawtooth cases in the middle, but moving upwards we see a mixture with S-motion cases and non-periodic “raucous” cases. At the bottom, nothing really looks like double-slipping, but the waveforms shapes get progressively peculiar and indistinct. For the purposes of Fig. 4 these have all been marked in yellow, but it is not clear whether they are really “decaying”.
Having seen these examples of the waveforms lying behind Fig. 4, we can give a general description of what that figure shows. Helmholtz motion is confined within a wedge-shaped region, with double-slipping motion found below the wedge, and raucous motion above it. So far, so good: all this is qualitatively as predicted by Schelleng. In some regions, especially when the bowing point is a long way from the bridge (i.e. larger values of $\beta$), the red wedge is interrupted by columns of S-motion. It appears that for certain values of $\beta$, the string prefers S-motion to Helmholtz motion, especially with higher values of bow force. This is consistent with the early measurements that inspired Raman’s study, as we saw in section 9.1.
This behaviour doesn’t contradict the Schelleng analysis, but we need to be careful about what that analysis does and does not show. Schelleng’s bow force limits were both found by asking the question “when does it become impossible for Helmholtz motion to happen?” But just because Helmholtz motion is possible within the Schelleng wedge doesn’t mean that it will necessarily happen. When more than one type of string vibration is possible (such as Helmholtz motion and S-motion), the question of which one you actually get from a given bow gesture can be very subtle. We will come back to this question in later sections.
In the lower part of Fig. 4 where the bow force is low, the classification of regimes becomes rather indefinite. Is the string capable of producing a version of Helmholtz motion in the bottom right-hand corner? Possibly a human player could use more subtle bow control than the bowing machine; but the patterns we have seen suggest that this would be a dangerously unreliable region for a player to try to use. Indeed, the whole of the right-hand third of the plot suggests that players would have a hard time in one way or another, if they were trying to produce Helmholtz motion. This is the regime known in violin jargon as sul tasto (over the fingerboard): this is a specialised style of playing used for deliberate colouristic effects. We can now see that those effects are probably associated with a kaleidoscopic selection of different S-motion regimes.
There is more we can do with the Galluzzo data: we can use it to test some ideas from section 9.2 about the effect of bow force and bow position on the spectrum of a note, and also look for the predicted pitch flattening effect. For this investigation, we concentrate on the cases flagged in Fig. 4 as Helmholtz motion.
The sawtooth waveform makes it relatively easy to detect individual cycles of vibration by looking for the flyback of the sawtooth then calculating the time at which each one crosses zero. Having identified single periods, we can use the FFT to calculate the frequency spectrum (in other words, the magnitude of the Fourier series coefficients for that cycle).
Figure 6 shows some examples, for all the Helmholtz cases in the column of Fig. 4 indicated by the left-hand green arrow. The colours of the plotted lines change from red through blue to green, in sequence as the bow force increases in that column. The spectrum level in dB is plotted against the harmonic number, but it is easy to convert this into a frequency scale given than the played note is the open D in every case, with a nominal fundamental frequency just below 147 Hz. So the 10th harmonic is around 1.5 kHz, and the upper limit of the plot at the 46th harmonic is a little below 7 kHz.
Because a logarithmic scale has been used for the horizontal axis, the pattern of harmonic amplitudes for an ideal sawtooth wave would appear as a straight line with slope $-1$. All the plotted lines follow this prediction up to at least the 5th harmonic, but then they start to spread out, in their colour order. The curves in redder colours, corresponding to the lower values of bow force, fall away first, while the blue and green curves stick with the straight line for longer. This is Cremer’s corner-rounding effect in action. We are seeing a systematic change in the spectrum with bow force, exactly as predicted by the discussion in section 9.2.
But then something changes at higher frequencies. The vertical dashed line shows the harmonic number corresponding to $1/\beta$, which is the expected frequency of Schelleng’s ripples (recall section 9.3). Around about that point in the spectrum plot, all the curves start to show peaks and dips, associated with the influence of these ripples.
Figures 7 and 8 show corresponding plots for the other two columns of Fig. 4 marked by green arrows. They show similar features to Fig. 6, but the dashed lines are progressively further to the left because $\beta$ is larger (i.e. the bow is further from the bridge). The result is that the disruption associated with the Schelleng ripples happens before the systematic spreading of curves due to the corner-rounding effect has really got started. There is still a clear tendency for the curves at higher frequency to be spread out in their colour sequence, but this is superimposed on the peaks and dips associated with the Schelleng ripples.
The combined message of these three plots is that the frequency spectrum of bridge force, and hence indirectly of the sound of the played note, is influenced in a systematic way by the two phenomena we have discussed. Bow force has an effect through the corner-rounding mechanism, and bow position has an effect through the pattern of ripples — recall also that the magnitude of ripples is influenced by the bow force. These two interacting effects are responsible for at least part of the “sound palette” available to a violinist .
Finally, we can use the same data to investigate the pitch flattening effect. Figure 9 shows some results for the set of Helmholtz motions examined in Fig. 7, indicated by the middle green arrow in Fig. 4. This plot shows the flattening, in cents, as a function of bow force. The plot looks a little “noisy”: the individual points have some uncertainty associated with them because of the rather crude way that the period of the sawtooth wave has been measured, but the rising trend is clear and convincing. By the highest bow force, just below the Schelleng maximum force, the pitch has flattened by some 10 cents relative to the value with low bow force. It looks as if the string was tuned slightly flat, by about 2 cents, so those points at low bow force do not fall around zero. Don’t forget that these results are for an open string: a finger-stopped note would have had more corner-rounding, and thus a larger potential for flattening.
The conclusion from all these comparisons with measurements is that the simple models we have looked at so far seem to give qualitatively correct predictions for several aspects of real bowed-string behaviour. This is encouraging, but we shouldn’t go overboard with excitement about this level of agreement. We will see in subsequent sections that some musically-important details of bowed-string sound are influenced by effects that we haven’t yet included in our discussion. Furthermore, when we come to look at bowed-string transients, in section 9.6, we will find that it is remarkably challenging to go from qualitative to quantitative agreement between measurements and computer models.
 Paul M. Galluzzo; On the playability of stringed instruments, PhD Dissertation, University of Cambridge (2003).
 Erwin Schoonderwaldt; “The violinist’s sound palette: spectral centroid, pitch flattening and anomalous low frequencies”, Acta Acustica united with Acustica, 95, 901–914 (2009)