Section 9.9 explores some transient bowing gestures other than the constant-acceleration Guettler transients on which we have concentrated so far. This subsection gives some details of the procedures and analysis of these new gestures.
A. Modified Guettler transients
The first step is a simple modification to the Guettler approach: instead of the bow speed increasing without limit, it tends to a steady value with an exponential trend:
$$v_b = v_0 (1 – e^{-t/t_v}) \tag{1}$$
where $v_0$ is the chosen asymptotic speed and $t_v$ is the time constant. The initial acceleration is
$$a_0 = \left[ \dfrac{d v_b}{dt} \right] _{t=0} = \dfrac{v_0}{t_v} . \tag{2}$$
These transients behave quite similarly to the original Guettler transients, and the automatic classification routine is unchanged from the one described in section 9.5.4.
B. String-crossing transients
A simple simulation of a string-crossing transient can be made by using a similar exponential profile for the bow force, while holding the bow speed constant. Bow force $F_b$ varies as
$$F_b = F_0 (1 – e^{-t/t_f}) \tag{3}$$
where $F_0$ is the asymptotic force and $t_f$ is the time constant. The plots of Schelleng’s diagram for this case use $F_0$ as the variable on the vertical axis.
The results have been processed using the same automatic classification routine as for the Guettler transients, but it should be noted that the interpretation of the absolute transient length is slightly different. The time reference is taken from the first detected slip, based on the bridge-force waveform. For Guettler transients this is generally quite clear-cut, at least for cases with relatively high bow force where there is a definite jump in the bridge force to mark this time. However, the string-crossing transients always begin with a rather smooth growth in bridge force, as shown by Figs. 10, 11, 13 and 14 in section 9.9. Exactly what moment the automatic routine detects is then somewhat moot — but it will be consistent from one point to the next in the Schelleng diagram, so the picture that emerges will capture the variation in transient length, which is the main thing we want to see.
C. Spiccato transients
Following Guettler and Askenfelt [1], we represent a sequence of spiccato gestures by the combination
$$v_b(t) = v_0 \sin (\Omega t) \tag{4}$$
and
$$F_b(t) = F_0 [1 + 0.5 \cos (2 \Omega t ~-~ \alpha)] \mathrm{~~~for~positive~values,} \tag{5}$$
where $\Omega$ is a chosen frequency, 2 Hz in the examples shown here, and $\alpha$ is a phase angle. The velocity amplitude $v_0$ was set to 0.2 m/s throughout, while the force amplitude $F_0$ was varied in order to scan the Schelleng plane. To avoid numerical issues, rather than setting $F_b$ to zero when the bow is out of contact it was set to a very small positive value. The variable $F_0$ is used on the vertical axis in the Schelleng diagram plots, but note that in consequence of eq. (5) the peak force during each note is actually $1.5 F_0$.
The earlier simulations, for the Guettler diagram, had a length of 0.3 s. In order to accommodate several spiccato notes, the length of the simulations for this subsection was increased to 1 s. This increase revealed a small snag in the thermal model with the parameter values used previously: the mean contact temperature continued to rise, reaching implausibly high levels by the end of each run. This was prevented in an ad hoc manner, by adding an extra cooling term to the heat balance calculation described briefly in section 9.6. Additional heat loss was assumed to occur at a rate proportional to temperature above ambient (often known as “Newton’s law of cooling”), with a coefficient that was chosen so that there was little effect during the first 0.3 s, but sufficient to cause the mean temperature to level off soon after that. All the simulations shown here, and in the subsequent subsection on tremolo bowing, incorporated this extra cooling term.
The post-processing of these simulations was based on the approach described earlier, except that the bridge-force signal must be segmented into the four separate notes in each simulation. When the bow speed was negative, the bridge force waveform of a segment was inverted. An example of the resulting processing is shown in Fig. 1. This shows the first 0.2 s of the four separate segments of one of the most successful spiccato simulations, corresponding to the third blue square from the bottom in the upper right-hand panel of Fig. 18 in section 9.9. Red stars mark cycles of “acceptable” Helmholtz motion according to the settings in the processing routine.
D. Tremolo transients
The final type of bowing gesture studied here is an idealised version of tremolo bowing. The bow speed is sinusoidally modulated as for the spiccato case, but at a higher frequency:
$$v_b(t) = v_0 \sin (\Omega t) \tag{6}$$
where the frequency $\Omega$ was set to 4 Hz here, and the velocity amplitude $v_0$ was again set to 0.2 m/s. The bow force was held constant throughout.
The post-processing of these simulations was similar to that used for spiccato notes. The bridge-force signal must be segmented into the separate notes in each simulation. When the bow speed was negative, the bridge force waveform of a segment was again inverted. An example of the resulting processing is shown in Fig. 2. This shows the first six separate segments of the most successful tremolo simulation, corresponding to the third blue square from the bottom in Fig. 22 in section 9.9. Red stars mark cycles of “acceptable” Helmholtz motion according to the settings in the processing routine.
[1] Knut Guettler and Anders Askenfelt, “On the kinematics of spiccato and ricochet bowing”, Catgut Acoustical Society Journal 3, 6, 9—15 (1998)