The Ninth Bell Needs Eight Days Without a Pause

A change, not a tune

Heavy tower bells are hard to start and stop, so English ringers do not play tunes on them. Each bell sounds exactly once in every row, and the full sequence across all the bells is called a row. The move from one row to the next is called a change, and it follows one mechanical rule: no bell may move more than one position from where it stood in the previous row. The rule comes straight from the physics of a several-hundred-kilogram bell swinging full-circle, which a ringer can only nudge earlier or later by a fraction of a second, right at the top of its swing.

With three bells, the number of possible rows is small enough to check by hand: 123, 132, 213, 231, 312, 321, six rows in all, or 3×2×1. With four bells there are already 24 rows, three factorial possibilities for each of the four positions the first bell could take. This multiplying rule is called a factorial, written n!: for n bells, the number of possible rows is n×(n−1)×...×2×1. A performance that works through every possible row exactly once, with no repeats, is called an extent.

How long a full extent takes

Drag the slider below to see exactly how long an unbroken extent would take, at the two-second-per-row reference pace English ringers themselves use to estimate a new performance.

Try it yourself, on four bells

Four bells have 24 possible rows. This game uses a simplified rule: swap one adjacent pair at each step. Real change ringing can swap several disjoint adjacent pairs at once, provided no bell moves more than one position. For example, 1234 → 2143 is one valid change in real ringing; here it takes two clicks. Try to visit all 24 rows without a repeat.

The rule kept in the head, not on the page

English ringers do not read sheet music. They memorise a method: an algorithmic rule that tells them, at every change, which disjoint adjacent pairs of bells to swap, so that the whole list of rows fills up without a repeat. The first book on the art, Tintinnalogia, was printed in 1668 by Richard Duckworth, with a contribution from the printer and ringer Fabian Stedman; Stedman's own second book, Campanalogia, followed in 1677. The methods described there correspond, in modern language, exactly to the operations of a mathematical permutation group, more than a century and a half before group theory was formalised as its own branch of mathematics. Historians of mathematics have called Stedman "the first group theorist."

Not every extent proved easy to construct. For seven bells, ringers spent nearly three centuries searching for a method built entirely from triple swaps, without ever proving one was possible. The problem, known as Stedman Triples, was finally solved in 1994, independently, by Colin Wyld and by Andrew Johnson together with Philip Saddleton; the Johnson-Saddleton composition was first rung in early 1995, by a Cambridge University band.

The novelist Dorothy L. Sayers, in her detective novel about bells, The Nine Tailors (1934), wrote that the proper use of English bells is "to work out mathematical permutations and combinations."

The Loughborough record

The first proven true performance, meaning one with no row repeated, took place on 2 May 1715, at St Peter Mancroft church in Norwich: 5,040 rows on seven bells, known today as Plain Bob Triples. The extent on eight bells, 40,320 rows, seemed impossible for a long time. The first team to complete it did so in relay, with ringers substituting in at the rope, over 27 hours, in 1761.

A single team, the same eight ringers from the first row to the last, with no substitution at all, did not manage it until 1963. Ringing began at 6:52 in the morning on 27 July, at the tower of the John Taylor bell foundry in Loughborough, and finished at 12:50 the following night, after 17 hours and 58 minutes of continuous ringing. It remains the only documented instance of a single band completing the eight-bell extent without a break, from the first of the 40,320 rows to the last.

The art remains almost entirely English. Romanian Orthodox church bells are rung by hand or by motor, in fixed rhythms, or replaced with the wooden semantron: no tower in Romania practises method ringing.

The same count, done for the 52 cards of a deck, gives a number 68 digits long. How many shuffles it takes for the deck to reach a truly random order: Three shuffles won’t hide a card.

Sources and method note

The full table used by the instrument above, from three to nineteen bells, with the exact row count and the duration at the two-second-per-row reference pace, is published by the primary source below. The historical dates for 1715, 1761, 1963, and Fabian Stedman come from the same lecture.

• Sarah Hart, "The Mathematics of Bell Ringing", Gresham College public lecture, 5 January 2021. Primary source for the factorial table, the historical timeline, and Stedman's place in group theory.
• Wikipedia, "Change ringing" and "Peal". Independent confirmation of the exact date and duration of the 1963 Loughborough performance.
• Plus Magazine (Millennium Mathematics Project, University of Cambridge), "Ringing the Changes". A third independent confirmation of the 1963 record.