There is a church which insists on ringing its bells every 15 minutes, and for hours every Saturday next to my new apartment. I remember hearing that there is some maths behind the music they play, and indeed there is! Say they have 4 bells, then ‘ringing the changes’ involves ringing all 24 possible sequences of these four bells one after the other- with some rules. The first rule is that you must start and end with the sequence 1, 2, 3, 4. The second rule is that each sequence must only be obtained from the previous one by swapping bells that were played next to each other. So the second one could be 2, 1, 3, 4- but not 4, 2, 3, 1.
The animation above shows how this can be done. Possible sequences are shown as rings, which are played clockwise from the top. Two rings are joined together if the transition is allowed, and the thick red line shows the solution. This song is synthesised in the recording above. There is clearly a lot of symmetry and mathematical structure in the problem and its solution, but it doesn’t make it any less annoying. [more] [code]