7.4 Add a touch of nonlinearity

Some stringed instruments give the impression that they should be approximated well using linear methods, but in fact crucial details for the sound involve nonlinear effects. In this section, we dip into two examples. Both of them were included in the set of plucked-string example sounds in section 7.1: the harp and the lute. In both cases the nonlinear effects are fairly weak, and we will be able to see roughly what is going on by simple approaches. But a full discussion of nonlinear effects needs more care, and after this introductory “toe in the water”, we will give a more systematic description in the next chapter.

If you were able to pick out the harp from the set of instruments in Section 7.1, you will know that there must be something characteristic about the sound, even of a single note. Probably, though, you find it hard to put the distinctive quality into words. So we can start by looking for physical features that distinguish a harp from other plucked-string instruments. First, the physical configuration is unusual. In the vast majority of plucked or struck stringed instruments, the strings lie approximately parallel to the soundboard. However, the strings of a harp meet the soundboard at an angle: Fig. 1 shows the small harp used in this study, where the string-soundboard angle is 35$^\circ$.

The other conspicuous physical feature that distinguishes the harp from instruments like the guitar is that it has a separate string for each note. This means that there are usually a large number of sympathetic strings contributing to the sound of a given note, and the layout of strings also influences many aspects of performance style. Two of these will be particularly important here. Harpists normally pluck their strings closer to the centre than is usual in other instruments. They can also impose a much higher initial displacement than a guitarist is able to use, because there is no equivalent of the danger of a “fret buzz”. The amplitude is only limited by the spacing between strings, about 15 mm for the harp shown in Fig. 1. As we will see shortly, string motion with large amplitude is likely to bring nonlinear effects into play.

Figure 1. The small harp used in this study, and also for the sound demonstration in Section 7.1.

A harp string, like all other instrument strings, can vibrate in two polarisations. One polarisation, in the plane of the strings, generates a component of force that can excite soundboard vibration efficiently. String vibration in the perpendicular plane, however, will drive the soundboard less strongly. The player will normally excite string vibration with some mixture of both polarisations, which means that harp notes might exhibit the double decay phenomenon discussed in section 7.3. However, the effect does not in fact seem likely to be a very strong one: Fig. 2 shows a plot similar to Figs. 9–11 of section 7.3, using the string properties and measured bridge admittance of the small harp. It can be seen that the red curve rarely rises above the green curve. This means that the decay rates for vibration in the two polarisations will not be significantly different. We must look elsewhere for an explanation of “harpish” sound.

Figure 2. Loss factor comparison for the harp in the same format as Figs. 9–11 of Section 7.1, based on the string to be used for tests later in this section. This is a nylon string with length 390 mm and diameter 1.045 mm, tuned to E$_4$ (329.7 Hz). It is the 20th string from the top of the instrument.

One candidate might involve longitudinal string vibration, something that is often ignored in discussions of plucked instruments. The geometric arrangement of strings and soundboard in a harp means that there is a direct coupling between transverse and longitudinal motion in the string. The harp soundboard will present a high impedance to in-plane forcing, whereas it can execute out-of-plane bending vibration more readily, so the soundboard will tend to move in a direction that must involve both transverse and longitudinal string motion.

This coupling can be modelled using linear methods, similar to the synthesis models described earlier (see section 5.4). The conclusion is that this effect, too, seems to be rather minor: unimportant enough, in fact, that we needn’t go into details here: see [1] for the full story. Our quest for the distinctive source of “harpish” sound seems to be thwarted at every turn. However, we must not be too hasty. What has been investigated and dismissed here is longitudinal string motion driven via linear vibration. But this is where nonlinear effects start to appear: longitudinal string vibration, able to drive the soundboard efficiently because of the string-soundboard geometry of the harp, is also generated by another mechanism.

There are two aspects of nonlinear excitation of longitudinal string vibration, and both have been shown to give audible effects in some musical instruments. They both have their origin in the fact that when a string vibrates transversely, the length of any small element of the string must increase a little as a simple consequence of Pythagoras’ theorem: this is called geometric nonlinearity. The first effect is that the small increase of the total string length causes an increase in mean tension. Higher tension raises the pitch of the transverse string vibration, of course, so during the early part of a vigorously-plucked note a “pitch glide” can occur, starting a little higher than the expected note. The pitch moves rapidly downwards as the string motion decays, allowing the tension to fall back towards its nominal level.

The second effect is more subtle. Dynamic axial forces are generated in the string, leading to frequency components in the sound sometimes described as “phantom partials” and also to direct excitation of the longitudinal string resonances. We will show some examples from the harp in a moment. Conklin [2] has demonstrated the importance of longitudinal string resonances for the tone of the piano, to the extent that a piano designer needs to take the effect into account to achieve the best tonal balance: some convincing sound demonstrations from Conklin’s work can be found in this web link. Both effects of nonlinearity in string vibration are usually discounted for non-metallic strings like nylon or gut harp strings, but we will show in a moment that both can appear strongly in plucked harp notes. They may well play a significant role in the distinctive character of harp sound.

Some preliminary examples involving a harp string, measured and synthesised.

Sound 1. Soundboard acceleration from a harp string, plucked with a breaking wire in the plane of the strings
Sound 2. Soundboard acceleration from a harp string, plucked with a breaking wire perpendicular to the plane of the strings
Sound 3. Soundboard acceleration from a regular fingertip pluck of a harp string
Sound 4. Synthesised soundboard acceleration from harp string pluck
Sound 5. Simplified linear synthesis of the plucked harp string
Sound 6. Approximate nonlinear synthesis of the plucked harp string, assuming that the string has a pinned boundary at the soundboard
Sound 7. Approximate nonlinear synthesis of the plucked harp string, assuming that the string has a clamped boundary at the soundboard