There are no mainstream tuned percussion instruments based on flat plates, although when we come to stringed instruments we will find many examples of more-or-less flat soundboards. But there are certainly percussion instruments based on curved shells. In this section we will look at a kind of “percussion instrument” which develops the theme of vibration engineering to align overtones into near-harmonic patterns: church bells. Bells must be among the oldest tuned percussion instruments, both in Europe and (with an entirely different tradition) in China. European bells are still cast from bronze in a very traditional manner: indeed, a bell foundry is a place that feels like a glimpse of an earlier century.
Molten bronze is poured into moulds to make each bell. After casting, the bells are tuned using a vertical lathe to adjust the thickness profile of the walls. The principle is similar to adjusting the undercutting of a marimba bar, but the technology is very different with such large objects. Once the bell is tuned, it will be installed in its church tower, with the associated equipment for ringing it.
British tower bells are rung by rope from a chamber further down the tower from the bells. They are traditionally rung “full circle”: the bells rest facing more or less vertically upwards, as seen in Fig. 4, then on each stroke the bell is made to swing a complete circle to come to rest in a similar position but with the clapper now resting on the other side, ready for the return stroke. Actually, during normal “change ringing” the bell does not quite come to the top of the travel each time: the ringer has to develop the skill of stopping it in the right place to swing back at the right moment for its next place in the complicated mathematical sequence that constitutes a change ringing “method”. Sound 1 gives a brief snatch of change ringing, for illustration.
There is some interesting science involved in this type of ringing, which lies behind how the bells should be hung and adjusted to give good sound and good handling from the ringers’ perspective. We will come back to this later (see section ?), but for now we concentrate on the natural frequencies of the bells and their tuning. There are no simple mathematical models that can cope with the complicated geometry of a traditional bell, so we begin with empirical measured frequencies. We will draw heavily on an extensive study of church bell tuning by Hibbert, described on his web site.
It turns out that the tuning of European church bells has a rather complicated history. Bell-makers settled on the classic form of bell, no doubt after much empirical experimentation, many centuries ago. As we shall see in a moment, this form more or less guarantees a sufficient level of near-harmonic relations between certain of the strongly-excited overtones that a listener can identify a pitch when the bell is struck. But some bell-makers use a more sophisticated pattern of tuning, resulting in a stronger set of harmonic relations and a correspondingly stronger pitch sensation. This approach, called “true-harmonic tuning” in the English tradition, seems to have been discovered, lost and rediscovered over a long period of time, perhaps more than once. No doubt the underlying reason was that the details of this tuning method were trade secrets, carefully guarded from competitors.
Combined with the fact that church bells can have a very long lifetime, this means that there are bells to be found around Europe spanning the centuries, and embodying a variety of styles and accuracy of tuning. Hibbert has measured the natural frequencies of a very large number of them. The key behaviour of a representative pair of bells is shown in Fig. 5. This is a synthesised example, based on measured frequencies of two bells: one true-harmonic tuned, and the other not. The frequencies have been scaled so that both give the same “strike note”, to be explained and illustrated shortly.
The red curve in Fig. 5 shows the frequency pattern and amplitude levels of a bell measured by Hibbert (from a church in Ipswich). The black curve shows the pattern of a true-harmonic bell. This is a synthesised example, using the strongest overtone frequencies from the measurements but disregarding various complicating factors. It is immediately clear that the two bells have very similar behaviour at higher frequencies, but that the first few modes fall at quite different frequencies.
The peaks that match in the two bells are annotated with numbers in the plot: these give the number of full wavelengths of deformation of the corresponding mode shape around the circumference of the bell. For a bell with a perfectly circular cross-section, an argument based on this symmetry behaviour shows that the circumferential variation must take the form $\cos n \theta$ or $\sin n \theta$ for each mode, just as we saw for the wineglass example in section 3.2. (Remember $\theta$ is the angle round the bell, pronounced “theta”.) The annotations give the values of $n$.
The geometry of the bell is more complicated than the wineglass, but Rayleigh’s argument that inextensional motion tends to govern the low-frequency modes still applies, at least approximately. The labelled modes involve motion that is approximately inextensional, and largely confined near the free rim of the bell. This has two important consequences: it is the position where the clapper strikes, so they are strongly excited during normal ringing; and their relative frequencies fall in a rather regular pattern that only depends on the geometry of the bell close to the rim.
This regular pattern is responsible for the “strike note” of the bell, through an interesting psychoacoustical phenomenon often called the “missing fundamental”. For both bells, the peaks labelled 4, 5 and 6 have frequencies in the approximate ratio 2:3:4. The higher labelled peaks continue this regular pattern: the two bells differ slightly, but not a great deal.
But at lower frequencies the two bells are very different. The true-harmonic tuned bell has been adjusted, using a lathe like the one seen in Fig. 4, to bring the first 4 mode frequencies into deliberate harmonic relations to the sequence just described: relative to the 2:3:4 frequencies, these 4 modes have frequencies 0.5, 1, 1.2 and 1.5. The first two are tuned to the “missing” fundamental, and an octave below that. The other two, with ratios 6/5 and 3/2, are less obvious at first sight, but they make up the notes of a minor chord and so are musically “harmonious” with the strike note and its harmonic series. The corresponding ratios for the non-tuned bell are 0.56, 0.89, 1.17 and 1.56: sufficiently different that any sense of harmoniousness is absent.
The sound of one note of the non-tuned “bell” with the red curve is given in Sound 2. The same thing for the tuned bell is given in Sound 3. I hope you agree that Sound 3 gives a more clear and definite sense of pitch, while Sound 2 is rather ambiguous. Indeed, hearing single notes like this does not give a very convincing impression that they have the same strike pitch, as I have claimed. But listen to the same pair of “bells” in Sounds 4 and 5, where scaled versions with different fundamental frequencies have been used to play a short passage at roughly the speed typical of normal change-ringing, as in Sound 1. Now it is perhaps more persuasive that the perceived pitch of the two tunings is essentially the same. The trouble with the long-ringing single notes in Sounds 2 and 3 is that you tend to concentrate on the slow-decaying lower frequency components, which are the ones most different between the two “bells”.
I have kept putting “bells” in quotes here, because these synthesised sounds aren’t in fact terribly convincing as church bell sounds. Perhaps they sound a little like tubular bells, used in orchestral performances. But there is something about the sound heard in Sound 1 which is surely not reproduced well here. That is an issue that will return again and again in this story: synthesised sounds based on what appears to be the main ingredients of the underlying physics often do not sound very convincing. This should not be interpreted as a failure of science, but as a challenge to do better. Something important for perception has been missed, and tracking that down requires persistence. But a convincing response should not simply involve arbitrary fudging of the sounds to be more “realistic”: we want to know what the missing physics really is. We will not forget this challenge, but we will not pursue it any further for church bells for the moment. It is too early in this story to get tempted into messy details of any single example.
 Neville H Fletcher and Thomas D Rossing; “The physics of musical instruments”, Springer-Verlag (Second edition 1998)