In order to generate plots of Guettler diagrams, it is necessary to have a reliable automated way to analyse the sets of transient responses, whether those are measured or simulated. This turns out to be a surprisingly tricky task, for several reasons. For a start, we need to make a decision about the exact purpose of the analysis. One possible aim would be to identify the detailed physics behind each transient, for example in terms of Raman’s description of different possible regimes of vibration (see section 9.1). As we will see shortly, that would lead to a rather fine-grained discrimination between different waveforms.
But in some cases those waveforms may not sound very different to a player or listener. This leads to the second possible aim: to classify transients that sound “acceptable”, versus those that are in one way or another undesirable. This would be in the spirit of Guettler’s original study. He was interested in how to control the bow to obtain the best possible crisp start to a note, but he was approaching as a player and teacher of the double bass, so what mattered to him was the sound of the note. Subtleties of the physics would not be of interest if the consequences were inaudible.
We can illustrate with a gallery of examples. These plots are all of extracts of measured bridge force, drawn from the same Guettler scan in which the D string of a cello was “bowed” with a rosin-coated rod in the Galluzzo bowing machine at a bowing position with $\beta = 0.0899$. Figure 1 shows a typical example of Helmholtz motion: a sawtooth wave, with a characteristic pattern of “Schelleng ripples”. This particular value of $\beta$ is approximately equal to 1/11, and if you count the ripples you will find about 11 of them in each period of the vibration. The pattern is exactly as we would expect from Fig. 13 of section 9.2.
Figure 2 shows an example of double-slipping motion for this cello string. Recall from section 9.3 that this type of string motion is expected when the normal bow force falls below the minimum bow force for Helmholtz motion. The result is often described as “surface sound”, or “not getting into the string properly”: those terms make it plain that such motion is ordinarily found unacceptable by string players and teachers. Notice, by the way, that you can still count a total of about 11 “Schelleng ripples” in each period-length of the motion.
A different kind of unacceptable string motion is illustrated in Fig. 3. The two panels show two different varieties of “multiple flyback motion”: both look a little like the Helmholtz sawtooth waveform, but the single abrupt flyback of the sawtooth is replaced by a cluster of up-and-down swings: three of them on the left, five on the right. The consequence is a sound quality that one can learn to recognise: in fact, whenever you hear a good violinist make a slightly odd sound, this regime is a likely culprit.
Figure 4 illustrates the tricky nature of the difference between analysing for physics and analysing for sound. Both panels show waveforms that look quite like the Helmholtz sawtooth, except for some details that are important for the physics. The left-hand plot shows a steady pattern of rather large “ripples”, interrupted about 1/3 of the way up the ramp by a downward skip. This is probably a combination of “S-motion” (see Fig. 6 of section 9.1) and double slipping. But is it audibly different from Helmholtz motion? To answer that would need a careful psychoacoustic investigation, but one might guess that the difference is rather subtle, and that most string players would find the sound acceptable, at least in the context of a few cycles during a transient.
The right-hand panel of Fig. 4 shows a different issue. Here, there are small but clear differences between each cycle and the next. The second cycle looks like Helmholtz motion, but the others have a clear trace of some kind of change occurring about halfway up the ramp. Is this low level of non-periodicity audible? Again, to give a definitive answer would require careful psychoacoustical investigation. For the moment, we can only guess that in the context of a transient leading eventually to Helmholtz motion, it might not be noticed.
Figure 5 shows two examples of more complicated string motion. Both have a hint of “sawtoothness”, but there is obviously something complicated going on. It is hard to be sure from these waveforms how many times the string is slipping on the bow in every cycle, or how exactly these cases should be fitted into Raman’s scheme.
From all these examples, we can deduce that we would have a hard job if we tried to develop a procedure to satisfy the first aim, to describe each bridge force waveform in a way that gave a clear description of the underlying physics. But if we adopt the second aim, to classify transients in terms how they might sound to a player, we can get away with a much simpler strategy. This is the approach we will take. The process will be illustrated with a particular transient from the same Guettler scan as the earlier examples. It is shown in Fig. 6.
Figure 6 also shows three indications of possible choices of “effective transient length”. To the right of the black line, it seems clear that the string has settled into Helmholtz motion. Between the blue line and the black line, there is already a regular-looking sawtooth waveform, but it is clearly not quite the same as the later waveform: the rising ramps are more “fuzzy”. A similar description applies between the green and blue lines, but now the ramps are even more “fuzzy” and also varying more from cycle to cycle. But before the green line the string motion is seriously irregular, and this is unambiguously part of the initial transient. So the transient runs at least up to the green line, and it is certainly over by the black line, but the choice between those limits is a matter of judgement.
We are now ready to describe the procedure adopted for automated classification. As a first step, it is convenient to compensate for the varying bow speed that is a characteristic of any Guettler transient. The bow speed is growing linearly at a known rate, and the bridge force should vary in proportion (approximately) to this bow force. So the bridge force can be scaled with the inverse of the bow speed, giving the result shown in Fig. 7. All the ramps indicating periods of sticking now have the same slope. If the motion settles into a regular Helmholtz sawtooth, simply growing as the bow speed increases, the successive cycles should now be identical in the compensated waveform. This will simplify the subsequent processing.
The next step is to look at the final few cycles, and test whether they show a good approximation to periodic Helmholtz motion. First, the autocorrelation function is calculated for the last 5 period-lengths, giving the result shown in Fig. 8. For this example, we see a strong peak after about 7 ms with peak height about 0.94. This time lag gives the period of the motion, and the peak height indicates how periodic the signal is. If the height is above some chosen threshold, and if the period is acceptably close to the nominal fundamental period of the string, the signal has passed the first test.
A steady periodic signal does not necessarily indicate Helmholtz motion, though: it might be one the regimes shown in Figs. 2 or 3, for example. The next step is to best-fit an ideal sawtooth to the final cycle of the motion, as illustrated in Fig. 9. The correlation coefficient between the data and the ideal sawtooth needs to be above a chosen threshold, to exclude blatant cases of S-motion (see Fig. 6 of section 9.1). In this case, the two match very well.
Finally, the gradient of the final period-length is examined. For the example case, it is shown (inverted) in Fig. 10. This case really is Helmholtz motion, so that plot shows a single strong positive peak. But if it had been double-slipping (like Fig. 2) or multiple-flyback motion (like Fig. 3), there would have been other strong peaks. Yet another threshold value is chosen, such that the number of times the gradient plot rises above this value is an acceptable estimate of the number of “sawtooth jumps” in the waveform. If this number is 1, as in Fig. 10, we finally accept that the Guettler transient has produced Helmholtz motion by the end.
We are now ready to estimate the length of initial transient before an approximation to this Helmholtz motion was established. We simply work backwards from the end one period-length at a time, repeating the correlation of each cycle with an ideal sawtooth. The first time that correlation falls below the relevant threshold value is deemed to be the end of the initial transient. The process is illustrated by Fig. 11. A red star has been inserted for every cycle that passes the test, on a plot of the original measured bridge force. The chosen threshold values are fairly forgiving, and the transient length identified by the algorithm falls between the green and blue lines in Fig. 6.
When this procedure is applied to the Guettler scan from which the example case was drawn, the result is shown in Fig. 12. The example case is marked by the green circle. The pixel is coloured orange, denoting a transient length of about 10 periods. This length is defined from the time of the first slip in the measured waveform, which is found by taking a slightly smoothed version of the bridge force and looking for the first time it exhibits a negative gradient exceeding a threshold.
Finally, Fig. 13 shows a plot of the number of “sawtooth jumps” detected, for cases that passed the tests for periodicity in the final few period-lengths. The palest yellow shade denotes the value 1, corresponding to Helmholtz motion. In the lower right-hand portion of the plots we can see quite a few darker yellow pixels, corresponding to 2 jumps. These are cases of double-slipping, like Fig. 2. It is not surprising to find them in this region, below the “Guettler wedge” of Helmholtz cases. For any given bow acceleration, these are points having a bow force which is too low to support Helmholtz motion, exactly the behaviour we expect to see below the Schelleng minimum force.
