Ringing the changes

One rule

In most of the world a church bell is swung through a small arc, or struck with a hammer while it hangs still. Early in the seventeenth century English ringers found that if a bell was hung on a whole wheel and swung almost a full circle, it would come to rest mouth up at the top of each swing. There a ringer could hold it a moment longer, or let it go a moment sooner, and so decide exactly when it spoke.

That is control, but not much of it. A tenor can weigh as much as a small car, and nobody can hurry it or hold it back by more than a fraction of a second. So from one row to the next a bell can move only one place earlier or later. Everything else follows from that rule: between one row and the next, a bell may change places with a neighbour, and nothing more.

Rounds and changes

The bells are numbered from the lightest, the treble, to the heaviest, the tenor. Rung in that order they sound a falling scale, and that row is called rounds. Every piece of ringing starts in rounds and ends in rounds.

A row rung by every bell once is a change. The gentlest kind of change ringing is call changes: the conductor names a pair of neighbours, they swap, and the band rings the new row over and over until the next call.

Plain hunting

Method ringing changes the order at every stroke, and every ringer works out their own part. In plain hunting, the simplest method, every bell moves one place at every change: out from the front to the back, a pause there, and in again. Each bell follows the same zigzag, started at a different point, and together the paths plait into a braid.

Plain hunting soon runs out. On five bells it gives ten rows and then it is back in rounds. Everything that follows is a way of going further.

How many rows there are

On n bells there are n × (n − 1) × … × 1 different rows. Ringing all of them, each exactly once, is called an extent.

On seven bells the extent is 5,040 rows, about three hours of ringing, and that is why a peal, the ringer’s great test of skill and stamina, is at least 5,040 changes on up to seven bells (5,000 on more). The extent on eight bells, 40,320 rows, has been rung by a single band only once: Plain Bob Major at the Loughborough Bell Foundry on 27 July 1963, in 17 hours and 58 minutes. On twelve bells it would take decades of ringing without a pause.

Writing a method down

Nobody carries 5,040 rows in their head. A band carries a method instead: a short pattern of changes, called a lead, rung over and over until the bells come back to rounds. Ringers write a lead in place notation. For each change you write only the places where bells stay put; every other pair of neighbours swaps, and x means that every pair swaps.

So Plain Bob Minor is x16x16x16,12. Everyone crosses; then the bells in front and at the back stay while the others cross; and so on, the comma meaning “and back again”, until the last change, which is 12 where plain hunting would have 16. That single difference is the whole of Plain Bob. When the treble leads, the bell in 2nds stays put while the bells above it dodge, so each lead begins from a new order, and the bells reach five times as many rows before they come round.

Ringers do not read notation while they ring. Each learns the path of their own bell through the lead, its blue line, and rings by it, while watching the moving ropes to see which bell they should follow. That skill, picking out the right rope from the others, is called ropesight.

Calls, courses and truth

Rung plain, Plain Bob Minor comes back to rounds after 60 rows, and the other 660 are never heard. To reach them the conductor makes calls at the lead ends. At a bob, the bell in 4ths stays put instead of the bell in 2nds; at a single, the bells in 2nds, 3rds and 4ths all stay. Either one sends the band into a different course.

Every one of the 720 rows belongs to exactly one lead, so ringing them all is the same as finding a way through all 60 leads that never meets a lead twice and finishes where it began. A touch that rings any row twice is called false; one that never does is true.

The map shows something ringers knew long before anyone drew it. Every row is either in course or out of course (a mathematician would say even or odd), and a plain lead or a bob always leaves you on the same side, while a single always takes you across. Bobs alone can never reach more than half of the extent, so every extent of Plain Bob Minor needs singles, and an even number of them, to get back to rounds.

Proofs in the belfry

Questions like these are about permutations, and ringers met them first. Whether Grandsire Triples could be rung to its full 5,040 rows with plain leads and bobs alone was a question its composers had puzzled over for a long time. In 1886 W. H. Thompson proved that it cannot, in a short book called A Note on Grandsire Triples. In 1948 the mathematician R. A. Rankin turned the argument into a theorem about groups.

The same question for Stedman Triples stayed open until 1995, when it was answered the other way. Colin Wyld, and separately Andrew Johnson and Philip Saddleton, found peals of Stedman Triples that use bobs alone.

A band, and a board

The oldest peal with a full record was rung at St Peter Mancroft in Norwich on 2 May 1715: 5,040 changes of Grandsire Triples. A board in the church records it as the first whole peal ever rung “to the truth”. Boards like it, painted with the date, the method and the name of every ringer, hang in towers all over England. When you finish ringing in Ropesight’s tower, it paints you one.

Take hold of a rope See the methods