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.

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.