Ding dong merrily on high: the maths of Christ Church belfry

Long perplexed by the melodic pandemonium of church bells, I follow bell ringer Ken Hartnett through a little door in the side of Christ Church Cathedral. We climb an ancient stone staircase, traverse a vertiginous roof-bridge and make our way into the bell tower, where a campanology rehearsal is in full clamour.

A friendly troupe perform exercises involving up to 10 bells at a time as conductors Nigel Pelow and Vyvyenne Chamberlain call changes to nurse novices along.

They are performing a piece that bell ringers call a “grandsire”, says Hartnett. He shows me the notation the ringers are following: long columns of rows of numbers like computer code.

Hartnett calls over a ringer, who confirms that, although chiming patterns called “queens” or “tittums” could be produced, bells are generally rung in complex non-repeating patterns rather than melodies. It is all part of keeping things interesting for the ringers, he asserts, rather than mathematically attesting to the glory of God.

So while melodic pealing church bells are a staple part of our shared Christmas experience, there is another, entirely different side to bell-ringing, one so complex that it helped create a form of advanced mathematics called group theory.

The carols are nice, but the bells are generally rung in non-repeating permutations, says Gary McGuire, a professor and head of mathematics at University College Dublin.

It first arose as a very English, very Anglican invention called “change ringing”, after the Reformation sacking of medieval church bells.

In the early 1600s, bells were recast and remounted on wheels and rung in 360-degree circles. They could be held with wooden stays to keep a bell in a “ready” position, allowing ringers more control over the timing of when an individual bell might clang.

This innovation triggered a rampant craze for ringing long series of non-repeating permutations of bells. It got so complex over time that a new type of maths was created to map these out across algorithmic compositions called “methods”.

As bell-ringing became increasingly popular, so too did the complexity of the pieces produced in bell towers across Britain. Methods were composed by the first local ringing societies in Lincoln and Bristol in the 1610s, but also spread to far-flung Victorian outposts of empire.

A method for all 24 permutations of four bells was published in 1621. The full 120 “plain changes” on five bells were rung in 1642.

In Restoration London, Fabian Stedman and Richard Duckworth published the bestselling Tintinnalogia in 1667, while Stedman's Campanalogia (1677) firmly expressed elementary group theory a century before Lagrange, and three centuries before the Steinhaus-Johnson-Trotter algorithm of the 1960s.

Complexity upon complexity

As towers gained more bells, Stedman’s methods were elaborated. In 1741, John Holt constructed a method for “grandsire triples” involving seven bells.

However, he only managed to deliver 4,998 permutations instead of the 5,040 expected from seven bells. This mathematical problem in turn was solved by imperial civil servant WH Thompson in India in 1886, and more generally in 1948 by Scottish number theorist, RA Rankin.

And the research into the maths behind bell-ringing methods continues today. A similar 18th-century problem remained unsolved until 1995.

Ringers exercise their ensemble skills on what are known “extents”, the full set of permutations possible from a given number of bells.

Three bells generate 3x2x1 or 6 “rows” (123, 132, 213, 231, 312, 321) that provide the resultant ringing patterns. Five bells create 120 rows, which take five minutes to ring. Six bells produce 720 rows (half an hour); and seven bells, 5,040 (a rigorous 3 hours 20 minutes). A full extent for eight bells (40,320 rows) would take 23 hours; 10 bells, three months solid – so, in practice, for abbreviated “peals” of larger sets of bells, ringers cap the rows at 5,040.

Rules of the game

Extents could be rung in many possible orders, but this is curtailed by the physical limitations of the bells, the convention of starting and finishing with “rounds” (simple descending scales), and the rules. From one row to the next, each bell can change by only one position in the sequence, by swapping with a neighbour. No row can be repeated, and no bell occupies the same position for more than two successive rows.

Any set of rows obeying these rules is called a “method”, essentially an algorithm to guide ringers without needing them to memorise long number lists, although coloured lines are often drawn over methods to plot the course of individual bells.

So listen a bit closer next time you walk past the local cathedral. The bell ringers may be accompanying Christmas carols, but you might also hear the bell patterns inspired by mathematics.

BELLS AND WHISTLES: McGUIRE'S FEATS

Prof Gary McGuire heads the maths department at UCD and got interested in bell-ringing in 1999, joining after reading an article by then ringing master Leslie Taylor about the seven new millennium bells going into Christ’s Church Cathedral.

McGuire published a paper, Bells, Motels and Permutation Groups, which delved into the symmetries of group theory.

McGuire was lauded in 2012 for solving a long-standing sudoku problem: what is the least number of clues needed to produce a unique solution? It was long conjectured to be 17 (although the maths does not exist to prove it), and McGuire demonstrated it by astronomical calculations that, even on supercomputers, could have taken millennia.

Last year, McGuire and three colleagues set a world record in cracking an encryption code for online credit card transactions, exposing the weaknesses of public-key cryptography.

He has also found time to compose a unique new method for five bells, just 60 rows and three minutes long. Did he pay any heed to melodies? “No, it’s really about the maths.”