# The Mathematics of Change Ringing

Spring 2011

Mathematics

## Abstract/Artist's Statement

Change Ringing is the art of ringing a set of n tuned bells in a series of distinguished sequences of permutations of those bells. Those sequences are called “changes”. Change ringing can be found all over the world, but it remains most popular in the context where it developed: 17th century English church towers. Bells for change ringing are hung in sturdy frames that allow the bell to swing through 360 degrees. The bell is usually bronze and ranges in weight from a few hundred pounds to several tons. A ring of bells consists of four to twelve bells. Each bell is attached to a wooden wheel with a handmade rope running around it. The harmonic richness of a swinging bell cannot be matched by the same bell hanging stationary, and each swinging bell requires one ringer’s full attention. The bells are arranged in the frame so their ropes hang in a circle in the ringing chamber below. With a pull of the rope, the bell swings through a full circle to the “up” position again. With the next pull it swings back in the other direction. Because of their great momentum, bells take about two seconds to rotate, so they cannot be used to play ordinary “melodic” music, hence the emergence of permutations as an organizing principle. We label the bells 1,2,...,n; 1 being the treble, lightest and highest-pitched bell and n being the tenor, heaviest and lowest-pitched bell. A specific change is then a permutation of the full sequence 1,2,...,n.
We present a full study of change ringing compositions and how to generate those according to specific rules given in the introduction, and an inquiry of what happens when one or more of the traditional rules are changed. Among the structures found include cosets, groups, Cayley graphs and Hamiltonian cycles.

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