11.3.0 Summary of bowed-string behaviour

This summary is intended for readers who want to understand section 11.3 without previously having read Chapter 9 on bowed strings. In order to understand some rather surprising analogies between bowed strings and reed instruments like the clarinet or saxophone, you need to know a little about bowed strings — the effects we are interested in were studied first in that context.

The bowed-string story begins back in the 1860s with Hermann von Helmholtz, who first observed and described the way that a violin string usually vibrates. Figure 1 shows an animation of this “Helmholtz motion”, reproduced from Fig. 2 of section 9.1. At any given moment, the string has a rather unexpected triangular shape: two straight portions separated by the “Helmholtz corner”. This corner shuttles back and forth between the bridge and the player’s finger.

Figure 1. The “Helmholtz motion” of a bowed string. The upper plot shows the string motion (with an exaggerated vertical scale), the lower plot shows the corresponding waveform of force exerted on the violin bridge, which excites body vibration and creates the sound of the instrument.

If you watch the point where the string crosses the moving bow in the upper animation of Fig. 1, you can see that all the time the corner is making the long journey to the finger and back, the string is moving at the same speed as the bow: the string is sticking to the bow throughout that time. But when the corner travels to the bridge and back, the string is slipping across the bow hairs. The motion of the string is a stick-slip vibration, and the frequency is governed by the time-keeping role of the travelling corner. Since that corner travels at the normal wave speed on the string, this explains why the frequency of a bowed note is the same as the frequency of a plucked note on the same string.

The lower plot in Fig. 1 shows the developing waveform of the force exerted by the vibrating string on the bridge of the violin. This “bridge force” is important for bowed-string studies because it is something we can easily measure. However, for the purpose of understanding the analogy with reed instruments a different waveform is more useful. This is the velocity waveform of the string at the point where the bow interacts with it. Figure 2 shows a simulated example of this waveform, alongside the bridge-force sawtooth wave. The stick-slip vibration results in a pulse-like waveform. Most of the time the string velocity is equal to the bow speed, producing the flat top part of the waveform. The short episodes of slipping result in pulses of negative velocity. Pulse waveforms rather like this will appear when we look at the waveform of pressure inside the mouthpiece of a clarinet or saxophone.

Figure 2. A simulated example of Helmholtz motion in a bowed string. The upper plot shows the bridge force, as in the lower part of Fig. 1. The lower plot shows the waveform of string velocity at the bowed point.

However, something everyone knows about the violin is that it doesn’t always produce the musical sound you wanted. A bowed string is capable of vibrating in many other ways, and a beginner on the violin has to learn to control their bow so as to create Helmholtz motion, rather than any of these other, less desirable, types of string motion. In a famous study of the bowed string in the early 20th century, the Indian physicist C. V. Raman (who later became famous for work on spectroscopy) gave an ingenious argument that allowed these other types of bowed-string motion to be described and classified. He argued that they can all be described in terms of travelling “Helmholtz corners”. The difference was that Helmholtz motion had a single corner, but the undesirable types of motion had more than one corner.

The simplest example would have two corners, and lead to string motion with two episodes of slipping per cycle, rather than a single episode. This kind of “double-slipping motion” is often described by violinists (or critical violin teachers) as “surface sound” or “not getting into the string properly”. The corresponding string velocity waveform has two pulses in each cycle. We will see in section 11.3 that an instrument like the saxophone can exhibit a very similar type of pressure waveform. This is no coincidence: Raman’s ingenious argument can be applied equally well to reed instruments.

Double-slipping motion played a crucial role in the next stage of our exploration of bowed-string behaviour. In section 9.3 we introduced the “Schelleng diagram”: a schematic version is shown in Fig. 3, reproduced from Fig. 1 of section 9.3. When a violinist is trying to control the sound of a single, steady note, they have three main parameters to think about: the bow speed, the force with which they press the bow against the string, and the position of the bow’s contact point on the string. Forget for the moment about the bow speed, and think about the interaction between bow force (sometimes called “bow pressure” by violinists who are a bit shaky about physics) and bow position.

Figure 3. Schelleng’s diagram, reproduced from Fig. 1 of section 9.3. In order to sustain a steady Helmholtz motion, a violinist must control their bow force and bow position so as to lie within the wedge-shaped region between the two red lines.

In the 1970s, John Schelleng pointed out that if we represent these two parameters pictorially by a point in a plane, then you can only sustain steady Helmholtz motion if that point lies within a wedge-shaped region of the plane. For a given bow position, there is a range of allowed bow force. Below a minimum bow force, you get double-slipping motion, “surface sound”. Above a maximum bow force you get some kind of raucous “crunch”. But these bow force limits vary with the bow position, leading to the wedge-shaped region as sketched in Fig. 3. The Schelleng diagram neatly encapsulates an important aspect of the “playability” of a violin: a player has to learn to control the combination of bow force and bow position so as to stay within the wedge.

Well, we will find somewhat analogous diagrams when we come to think about the playability of wind instruments. They will not involve bow force and bow position, of course, but wind players have their own control variables such as blowing pressure. We will often be able to learn something useful by looking at a plane defined by a pair of these control variables, and finding that the player has to remain within some region of this plane to achieve a particular note and tone quality. Indeed, we will often find that it is a wedge-shaped region, reminiscent of the Schelleng diagram.