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An Introduction

Welcome to mathematical music

Change ringing developed because bells hung in towers, and weighing several hundredweight had wheels attached and were swung higher and higher until they turned a full circle. With a limited number of notes (often just the six lower notes of a diatonic octave) and the fastest repeat of a note being about 2 seconds, music as we know it is impossible.

On the continent this developed in two ways, the first being a small number of bells, chimed (swinging through maybe 30 to 45 degrees, and allowed to chime (i.e. clash and clang) at the same time, in smaller churches. The second, in the larger Abbeys and Cathedrals the installation of a larger number of bells, often with a chromatic scale, and hit by hammers operated from a keyboard. In this case musical tunes are possible, and are uniquely tuneful owing to the percussive nature and slow fade of each note.

In the England we found that a bell could be swung just past the vertical, and the ringer could balance the bell at that point thereby delaying how long it would be before the note was repeated. Also, by watching other ringers you could ensure that no two bells rang at the same moment. So it was then possible to ring the bells for example in the order down the scale from highest pitch to the lowest pitch without discordant clashes.


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In the belfry the ropes normally fall in this order, and visually the ropes fall and rise in order around the circle. Hence ringing 1-2-3-4-5-6 became known as ringing “rounds”. The sound is musical without being music, rhythmic, repetitive, and after a short while, boring.

Written out this would look like:




The separation into pairs of rows is in recognition of the normal rhythm of ringing where a short pause (roughly the space for one bell note) is inserted to quantify the rhythm.

Change Row

Writing out the sequence of bell notes as a series of numbers, and then changes to the order of those numbers, leads logically to the idea of a change row.

Changes into the order could be introduced by holding one bell a fraction longer and ringing another fractionally quicker. So a call of 2 after 3 would result in the sequence 1-3-2-4-5-6.

1-2-3-4-5-6 call 2 after 3

1-3-2-4-5-6 call 4 after 5


Simple Change Ringing

It is a small step from there to get to change ringing where the sequence alters at every pull of the rope. So simple change ringing on 4 bells would look like:

1 2 3 4
2 1 4 3
2 4 1 3
4 2 3 1
4 3 2 1
3 4 1 2
3 1 4 2
1 3 2 4
1 2 3 4
8 sequences before repetition occurs

Change ringers then developed more and more complex patterns and techniques for extending the number of sequences (or changes) without repetition. The patterns and techniques are, today, known as methods.

On 7 bells the maximum possible number of changes is 5,040 and this became a standard length. So to ring 5,000 or more changes became known as ringing a “peal”.

Ringing changes on handbells then uses the same method principles as on tower bells to avoid repetition of sequences. However, whilst the challenge on towerbells for the novice is initially physical because it takes a level of skill to control such a heavy musical instrument, on handbells the challenge is more cerebral, you have two bells to navigate through the complexity of the method of ringing.

Various strategies are used to reduce the mental challenge of ringing a pair of bells but nothing replaces the need to study the structure of the ringing methods in great detail. This study pays dividends in both handbell ringing and towerbell ringing. Anyone thinking “I might be interested in learning” needs a) a good sense of rhythm, and b) some ability to think logically.

Please go to the Links Page for more information, and a great set of YouTube recordings.