Maths That Rings a Bell
If you have spent any time in the English countryside, you may have heard the sounds of bellringers merrily chiming from a nearby church tower – a peculiarly British pursuit in which, given a set of bells tuned to different notes and sufficiently skilled bellringers, a variety of tunes can be performed. But is there a mathematical aspect to bellringing too? Given that I am writing about it in this column, I imagine you have probably deduced the answer already.
Alongside beautiful melodies and religious songs, bellringers often like to put on more methodical performances, called change ringing. This is a process which involves playing the whole set of bells one after the other, over and over again – but each time changing the order in which the bells are played.
I imagine you can already see how this might quickly become mathematical – I have written about permutations on this blog before, and they can be used to record mathematically the different possible ways to order a set of objects. For example, if my church tower has four bells, I can play any one of the four bells first, giving me four choices; then I can choose from the three remaining bells which one to play second; then I have two choices for the bell to play third, and my choice of final bell is fixed by the others – giving 4 × 3 × 2 × 1 = 24 possible orderings, also written 4! (‘four factorial’).
Church ringers commonly face sets of four, five, six or even seven bells to play with, and in each case the number of possible orders in which they can be played gets larger: 24, 120, 720 and a whopping 5,040 different possible orderings for a seven-bell set. Some performances are created by picking a subset of these orderings, called changes, as a form of musical composition, and a caller will indicate to the bellringers what to play each time.
To a Great Extent
But a more lofty goal is to complete the full set – all the possible changes for a given set of bells, called an extent. For the bellringers, memorising this full list would be a feat beyond possibility: but mathematics allows us to work out an ordering which means the bellringers only need to memorise a simple set of rules, and keep following them until they have completed all the changes.
This technique is called method ringing, or scientific bellringing, and uses mathematical permutations to visit every change in order. The ultimate challenge in bellringing is a peal: the full 5,040 changes on seven bells, which takes around three hours.
There are some restrictions on the order in which changes can be played: for example, a large heavy bell that was rung last in sequence on the previous change will not be available to ring again by the start of the next one – the bells are rung by pulling on ropes wrapped around wheels at the top of the belltower, and need time to wrap and unwrap from the wheels before they are ready to be rung again. This means changes played one after the other must be carefully chosen.
There are many methods for doing this, but one of the oldest and simplest is called a Plain Bob. (There is an endless variety of enjoyable terminology involved in bellringing.)
Bob’s Your Uncle
To perform a Plain Bob, we start with a hunt: a simple set of swaps which mean we take adjacent pairs of bells and swap them to get the next change. If there were an odd number of bells (say, 7), we would swap the first three pairs and leave the seventh bell where it is, so it is rung in the same position on the next change. For an even number of bells, we just swap all the pairs.
For the next change, the first bell stays in its current position, and the two after it swap, making pairs all the way to the end of the row (and in the case of an even number of bells, there will be a singleton at the end).
These two steps are repeated over and over to produce a hunt – each permutation will be a new one that has not been heard before, and each time the bellringers just need to remember to swap with the person before or after them as needed.
This diagram shows examples of 6- and 7-bell hunts – with the swaps in each row starting from the first bell, then the second bell, then the first again and so on. This leads to a pleasing pattern in which each bell seems to ‘hunt’ across the row and back again.
However, this sequence will not visit all possible changes before we get back to the initial pattern: the ‘1’ bell will make its way all the way across and back in 2n steps, where n is the number of bells. At this point, we need to add in one extra change: a dodge, which involves swapping a single pair of bells (usually the right hand pair) or keeping one pair of bells the same while the others perform their usual swaps, and depending on the number of bells there are different patterns of dodges which will achieve the full extent.
In the case of four bells, a simple 3-4 dodge every time the ‘1’ bell gets back to the start will suffice. Below is an extent on four bells, called Plain Bob Minimus (‘Minimus’ being the term for 4 bells), with the dodges indicated. Equivalent patterns exist for other numbers of bells, with slightly different sets of dodges to make the maths work: 5 bells (‘Plain Bob Doubles’), 6 (‘Plain Bob Minor’), 7 (‘Plain Bob Triples’), 8 (‘Plain Bob Major’) and so on.
If you are interested in finding out more about the mathematics behind bellringing, there is a long YouTube video of a fascinating lecture given by Dr Sarah Hart – a former maths lecturer of mine – as the Gresham College visiting lecturer in 2021. As well as the sequencing of permutations, Sarah goes into plenty of other interesting aspects of maths and bells, including the shape of the bells themselves and the frequencies of sound waves they produce, as well as the group theory of permutation groups.
For me, it is fascinating to see yet another mathematical structure pop up in something that is not just a real-world hobby that non-mathematicians enjoy, but also a very old traditional activity that has roots back in the 17th century. And if you ever get the chance to visit the British countryside, make sure you wander past a church on a Sunday morning to catch some for yourself!