Why Are Those Bells Not Playing a Tune? They Are Working Through Permutations

Large tower bells are enormously heavy and swing through a full circle, which means the interval between one strike and the next is fixed by the physics of the swinging bell and cannot be shortened or lengthened much at will. Playing a tune requires control of rhythm and duration that such bells simply do not permit. What can be controlled is order. A ringer can hold a bell slightly or let it come round slightly sooner, which swaps its position with a neighbour, so the art became the systematic permutation of the order in which the bells sound rather than the arrangement of pitches in time.

The constraints are strict and produce the character of the sound:

• •Bells start in order from highest to lowest, called rounds
• •Each change is a new ordering of all the bells
• •A bell may move only one position between changes
• •No ordering may be repeated within the performance
• •The whole thing returns to rounds at the end
• •Nothing is written down during ringing and nobody conducts the notes

The threshold is defined numerically rather than by time. On seven bells the total number of possible orderings is five thousand and forty, and ringing all of them became the standard of a complete performance, so the requirement for larger numbers of bells was fixed at a minimum of five thousand changes. That takes roughly three hours on an average ring of bells, during which no ringer may stop, be replaced, or make an uncorrected mistake, and any error means the attempt is abandoned and does not count. The first recorded full performance was rung in Norwich in 1715.

The physical demands are easily underestimated by anybody who has only heard the result. A bell may weigh anything from a couple of hundred kilograms to several tonnes, and it is controlled by a rope wrapped round a wheel, with the ringer standing below and pulling in a rhythm set by the swing rather than by choice. The bell rests mouth upward between strokes, balanced past the vertical against a stay, and holding or hurrying it by a fraction of a second is how position is changed. Learning to control a bell safely takes months before any method can be attempted, and a bell that gets away from a ringer is genuinely dangerous.

The practice is an applied study of permutation groups that predates the mathematics describing it, which is why mathematicians take an interest. A method is a rule for generating each successive ordering from the last by swapping adjacent pairs, and ringing all possible orderings without repetition is the problem of finding a path visiting every permutation exactly once using only adjacent swaps, which is a well-studied question in graph theory. Ringers solved instances of it in the seventeenth century by trial and by practical ingenuity. The person calling the performance introduces occasional variations at set points to prevent the sequence closing early, which requires holding the structure of thousands of changes in memory.

Heavy swinging bells fix the interval between strikes, so the only controllable variable is order, and the practice became the systematic permutation of that order with each bell moving one position at a time and no ordering repeated. A complete performance is at least five thousand changes, taking about three hours with no stopping and no uncorrected error. The underlying problem belongs to graph theory.