5.2 “Can you make it louder?”

Not all stringed instruments are intended to be loud. Some instruments were developed in a context where music was performed for the private enjoyment of the influential few, whether in courtly ceremonies or aristocratic dinner parties. However, there must always have been outdoor music requiring loud instruments, and over the last couple of centuries the development of public concerts and large concert halls has motivated evolution of instruments to make them louder and louder.

For a plucked or struck instrument, there is a balancing act to be performed. For a given amount of energy put into the string by the player, you can either let it leak out to the body quickly or slowly. This leakage rate is determined entirely by the magnitude of the scaled admittance, as was shown in section 5.1.2. The plot of this scaled admittance from the previous section, comparing three different instruments, is reproduced here for convenience. Compare the results for the guitar and the banjo. The curve for the banjo lies well above the one for the guitar, over virtually the entire frequency range. The banjo will be louder (no surprise there), and it will inevitably have a sound that dies away quickly because the energy is used up fast. If a long sustain is wanted, to allow long legato lines, loudness must be sacrificed. To get both, there must be a lot of energy in the string in the first place: this is the background to the evolution of the modern piano, with strings so heavy and tight that players ordinarily use mechanical assistance (from the piano mechanism) to excite them: they don’t play with their fingernails like a guitarist.

Figure 1. A repeat of Fig. 7 from section 5.1: scaled bridge admittance magnitude, for three different stringed instruments. Red curve: a violin; purple curve: a classical guitar; blue curve: a banjo. The frequency scale is expressed in semitones, starting from the lowest tuned note of each instrument. Successive octaves are indicated by ticks.

Bowed instruments are different: the scaled admittance still governs the rate at which energy is extracted from the string into the body, but now the player is constantly feeding energy back into the string with their bow. The trade-off between loudness and sustain does not apply: a bowed note can be sustained as long as the player’s bow arm permits. The different designs of the violin and the guitar have come about to exploit this key difference.

Compare the curves for the violin and guitar in Fig. 1. At the lowest frequencies, a few strong resonances of the guitar produce peaks higher than the corresponding low-frequency peaks of the violin. (The reason that the violin cannot tolerate peaks as high as those of the guitar lies in a phenomenon of bowed-string instruments called the wolf note: we will come back to this in Chapter ?.) But at higher frequencies, from about semitone 24 on the axis scale here, the violin curve rises steadily above the guitar curve. By semitone 40 or so, it reaches very similar levels to the banjo. This suggests that the violin has been designed to extract energy quickly from the string in this higher-frequency range, allowing it to be quite loud. We will find out in section 5.3 about a particular feature of the design of the violin, that is mainly responsible for this strong divergence from the behaviour of the guitar. But no-one chooses a pizzicato violin to play legato lines of the kind possible on the classical guitar or the harp.

We will return yet again to this plot and the comparison between these three instruments in section 5.3. But before that we digress a little to enquire what, if anything, an instrument maker might do to increase loudness, and what are the associated hazards and pitfalls. If the energy from the string is to ‘leak’ into the body more rapidly, strings must be chosen with the highest possible wave impedance, while the admittance of the body must be made as large as possible. In other words, you want to put heavier and heavier strings on lighter and lighter instruments. The limit on that process might be set by the sound getting too loud, but usually something else goes wrong first: the structure of the instrument body fails under the stress of the string tensions.

Instrument designers need to exercise ingenuity to get round this problem. One extreme example is the modern piano. Unlike earlier instruments like the harpsichord, a substantial metal frame is used to carry the string tensions, separate from the wooden soundboard that serves to convert the energy from the vibrating string into sound. Piano strings have become extremely heavy as well: indeed, the typical strings of a modern concert grand are not what you would ordinarily call ‘string’ at all: imagine trying to tie a parcel with piano wire 1 mm in diameter.

The violin gives an example that is less extreme, but arguably more ingenious. A number of features of the violin play a role in helping to withstand the string tension. That is not to say that this is the only reason for these design features: as in other highly-evolved designs, many details manage to fulfill multiple functions. But for the moment we will concentrate on this aspect. First, it is useful to see how far down the line of “making it louder” the violin has been pushed. The total tension of a typical set of violin strings is around 20 kg (or around 200 N). The strings turn over the bridge through an angle around $25^\circ$, resulting in a downbearing force on the violin’s top plate around 10 kg. That much weight is supported on a panel of softwood that may be less than 3 mm thick.

Figure 2. A violin by Antonio Stradivari, labelled to highlight some design features discussed in the text

Figure 2 highlights a few features of the violin that help it withstand the stresses. Number 1 is rather general: notice that all the component parts of the violin body are curved in some way, rather than flat. The top and back plates are arched, the thin pieces of maple forming the ribs are curved. As is familiar from structures like bridges, an arch is much better than a flat beam or panel when it comes to withstanding a downwards load. We already know the reason for this, from the discussion in section 3.2 of curved shells compared to flat plates. A flat plate can only resist loading through bending, but a curved shell automatically brings in forces of compression or tension. This tends to make it stiffer: recall the discussion of corrugated roofing panels.

Feature number 2 directs attention to the shape of the f-holes. In common with many other stringed instruments, the violin body has “sound holes” in order to create a Helmholtz resonance. As discussed in section ?, this produces an extra resonance at low frequency to help the bass response of the instrument, and also improves the sound radiation efficiency of low modes of the body. Furthermore, there are advantages in having slot-shaped holes. This creates an “island” between the holes, exactly where the bridge sits. The slots make this island more flexible, and we will see in section 5.3 that this is implicated in a major difference in acoustical behaviour between the violin and the guitar. But putting slots close to the bridge is dangerous. If simple rectangular slots were used, the sharp corners would create stress concentrations, and cracks would soon form in the softwood of the top plate, starting from these corners. To avoid this problem, both ends of the slot are curved over, avoiding the sharp corner. There are two choices: you could bend both ends the same way, producing C-holes as seen in some earlier instruments like viols, or you bend the two ends in opposite directions, producing the familiar f-holes of the violin.

Feature number 3 shows two blue rectangles, and if you don’t know what the inside of a violin looks like, these may be rather mysterious. They are meant to indicate two important structures inside a violin. One is the soundpost, a wooden rod wedged between the top and back plates, near one foot of the bridge by the highest string. This is indicated by the vertical blue rectangle. The second internal structure is the bassbar, a wooden beam running approximately 3/4 of the length of the top plate, passing beneath the other foot of the bridge. Both soundpost and bassbar have an important acoustical consequence, to which we will return in section ?. But they also help with the problem of withstanding the stress from the strings. The downbearing force is transmitted through the feet of the bridge, so that placing a reinforcing beam under one foot and a supporting pillar near the other will obviously help.

Feature number 4 is the purfling around the edge of the top and back plates. The black lines of the purfling are not painted on (except in very cheap factory-made instruments): instead, a three-layer sandwich of dark-light-dark wood is inlaid into a slot painstakingly cut around the edges of the plates. Purfling has an important aesthetic effect, and violin-makers put a lot of effort into making it look just right. But this apparently decorative feature of the violin also serves a practical function. The edges of a violin, especially on the top plate, are vulnerable to being knocked, which can start a crack running in from the edge. Inlaying strips of wood across the grain close to the edge gives some protection against this.

The features numbered 5 and 6 go together. Some of the most famous and valuable violins are several centuries old already. Despite the best efforts of their makers, most of them have suffered damage during that time. But these instruments look in good condition, and are still being played regularly. They have all been repaired over the years, and there are two features of the construction that have allowed repairs to be effected without leaving eye-catching scars. When a violin maker has to make a major repair, they will remove the top plate to open the box up. This can be done without causing much damage because the box is held together with gelatin-based animal glue which is strong enough, but slightly less strong than the surrounding wood. A violin repairer can remove a top plate by carefully inserting a flat-bladed knife like a table knife into the glue layer, and working it around the edge to crack the joint apart. (The label in Fig. 1 marks the joint of the back plate rather than the top plate, simply because it is more easily seen in the picture. But it is very rare to remove a back plate: nearly all repairs of this kind involve the top plate.)

Having done the repair, the top plate will be glued back on. Another ingenious feature is now revealed. Even with care and skill, the act of opening a joint like this will probably cause some damage, so that the joint does not fit back together quite as snugly as was originally the case. But look again at Fig. 2: the plates fit to the ribs with overhanging edges, so that the glue join is in a rather dark and obscure corner. So a tiny amount of damage near this joint will not show as much as it would have done without that overhang. Instruments like the guitar or lute do not have this feature, and it is very much harder to remove the top plate and effect an invisible replacement.

Finally, feature 7 is another very general one. It concerns the choice of material out of which the violin is made. Material choice is such an important topic, for all stringed instruments, that we will defer a discussion for now, in order to do it justice in a section of its own later on: see section ?.

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