The Mathematics of Bell-Ringing Combinatorial mathematics, in this case permutations, is the foundation of change-ringing, a method of bell-ringing that works through permutations of the bells in the collection. The result is the charming musicality one often hears in English villages.

Let's look at how this works. With five numbered bells, one can:

count down the scale: 1 2 3 4 5,

do neighbour swaps holding each of the numbers fixed:
2 1 4 3 (5),
(1) 3 2 5 4
3 (2) 1 5 4
2 1 (3) 5 4
2 1 5 (4) 3

etc.

Bell ringers learn the path that their bell makes through the sequence by remembering the line of the route, which is known as the ‘blue line’.

With more bells, more possible permutations.

A splendour of mathematics and memory is a 'peal', or a period of ringing, usually lasting about three hours and with over 5,000 changes, all rung from memory using the method of "the blue line", to commemorate special occasions.

To hear a beautiful example see [3] below (8 minutes) and explained in [4] (6 minutes).

References

[1] Change Ringing: A Guide
https://www.nagcr.org/pamphlet.html

[2] The mathematics of Change Ringing, explained
https://plus.maths.org/content/ringing-changes

[3] Devon Call change ringing:
https://www.youtube.com/watch?v=w2wWtOyBvfA

[4] University of Chicago - 6m video getting to the essence of change ringing as a team sport, and a deep source of satisfaction and meditation.
University of Chicago's bell tower is a replica of Maudlin Tower in Oxford, England.
https://www.youtube.com/watch?v=khc-iA0FZEY

[5] Discover Bell Ringing (Cambridge Bells)
http://www.bellringing.org/changeringing/