How Do Church Bells Work? Tuning a Lump of Bronze to Five Notes at Once

A vibrating string produces a fundamental with harmonics at whole-number multiples, which is why the ear fuses them into one clear pitch. A bell is a three-dimensional shell and its vibration modes are governed by its geometry in ways that produce partials at ratios that are not simple. The named partials of a well-made bell are the hum, roughly an octave below the note the ear assigns, the prime or fundamental, the tierce a minor third above it, the quint a fifth above and the nominal an octave above. The perceived pitch is not any of these directly but the strike note, which the ear constructs from the spacing of the upper partials, principally from the nominal, and which is therefore a psychological rather than a physical frequency. That minor third is why a traditional bell sounds faintly mournful, and it is inherent in the classical profile rather than a choice, which is why attempts to cast major-third bells required a redesigned shape and modern computer modelling.

The manufacturing sequence has been essentially unchanged for centuries:

• •Bell metal, an alloy of roughly four parts copper to one part tin, chosen because it is hard, resonant and long-sustaining, and because it is also brittle, which is why bells crack rather than dent
• •A mould built in two parts, an inner core and an outer cope, with the space between them defining the profile, which is the whole acoustic design
• •Casting in one pour, after which the bell is cooled slowly over days to avoid internal stress
• •Tuning on a vertical lathe, cutting metal from specified places on the inner surface, since each partial responds to material removed from a different region
• •That process is subtractive only, so a bell can be lowered in pitch and never raised, which means bells are cast deliberately sharp
• •The clapper, whose weight and the point at which it strikes substantially affect the sound, and which wears a flat spot that must be periodically turned

Two quite different traditions produce different sounds and different music. Continental practice generally swings bells through a limited arc or strikes them with hammers, which allows melodies and permits the carillon, an instrument of dozens of tuned bells played from a keyboard with fists and feet, developed particularly in the Low Countries. English full-circle ringing mounts each bell on a wheel so it rotates through a complete circle, which gives the ringer precise control over exactly when it sounds and produces the characteristic open sound, while making it impossible to play a tune, because a bell rotating through a full circle can only be slightly speeded or delayed and cannot be held or repeated at will. That constraint produced change ringing, in which a band rings the bells in a continuously changing sequence of permutations according to strict rules: every bell sounds once in each row, no bell moves more than one position between rows, and no sequence repeats until the composition ends.

Change ringing is a permutation problem worked out in practice long before the relevant mathematics existed. With seven bells there are five thousand and forty possible orderings, and a full extent on seven bells rings every one of them exactly once, which takes around three hours of continuous concentration with no written music, since the sequence is memorised as a method. With eight bells the number exceeds forty thousand, requiring around eighteen hours, which has been achieved rarely. The rules mean each ringer must know where their bell moves next relative to the others rather than following a score, which is why the skill takes years. Mathematicians recognised in the twentieth century that ringers had been working with concepts from group theory, and that the standard methods correspond to systematic ways of generating permutations, with the Plain Bob and Grandsire methods being particular solutions arrived at empirically in the seventeenth century. The first published treatise on the subject appeared in 1668, which predates the mathematics it anticipates by a considerable margin.

Between rings, a bell used for English full-circle ringing rests in a surprising position: upside down, with its mouth pointing at the sky. The ringer swings it up from hanging to this position and then balances it just past the top, where a wooden stay fixed to the headstock rests against a slider and holds it. Each pull of the rope, alternately on the woollen grip called the sally and on the tail end, sends the bell swinging through a full turn to balance on the other side, sounding once on the way. Because a bell weighing a tonne or more is balanced so finely, it can be controlled with modest effort.

Bell metal's brittleness has shaped some famous sounds. The great bell of the Palace of Westminster, known as Big Ben, cracked within months of being brought into service in 1859, apparently because the hammer that struck it was too heavy. Rather than recast it, engineers turned the bell so the hammer struck a fresh spot, fitted a lighter hammer and cut a small square into the metal at the end of the crack to stop it spreading. The crack is still there, and it gives the bell a tone slightly different from the note it was cast to sound, which is the note millions of people now recognise.

The rules are easiest to see on four bells, numbered from the highest, 1, to the lowest, 4. Rounds, 1234, is the starting row. In the simplest pattern, called plain hunt, the ringers alternate between two moves: first the bells in places one and two swap while those in places three and four swap, then the two middle bells swap while the outside pair stay put. The rows run 1234, 2143, 2413, 4231, 4321, 3412, 3142, 1324 and then back to rounds. Each bell works its way steadily from the front of the row to the back and down again, a path ringers call hunting.

Plain hunt reaches only eight of the twenty-four possible orders of four bells before repeating. To ring all of them, a method changes one move at a set point. In the method called Plain Bob Minimus, whenever bell 1 comes back to the front, the middle swap is replaced by a swap of the last two bells, so the row after 1324 becomes 1342 instead of rounds, and the hunting starts again from a new order. Three such sections of eight rows produce all twenty-four orders exactly once before rounds return, and longer methods on more bells build on the same idea of hunting plus a small, regular disruption.

Change ringing is organised largely through local bands and societies, and some are remarkably old. The Ancient Society of College Youths, based in London, was founded in 1637 and is one of the oldest ringing societies in the world, with membership open by election to accomplished ringers. Its records chart the development of ringing methods over nearly four centuries.

Today the Central Council of Church Bell Ringers coordinates ringing across Britain and abroad, and long performances called peals, which ring through thousands of changes without repetition and typically take around three hours, are recorded in detail. There are thousands of towers with ringable bells in England alone, far more than in any other country, which is why change ringing is often described as a distinctively English art.

A bell vibrates in several patterns at once with partials that are not whole-number multiples, and the pitch a listener hears is a strike note constructed by the ear from the upper partials rather than any single frequency. The characteristic minor third is inherent in the traditional profile. Tuning removes metal from inside on a lathe and can only lower a partial, so bells are cast sharp. Full-circle ringing prevents tunes and produced change ringing, which works through permutations anticipating group theory.