Focus on...change ringing
Bell ringing is a context in which students can explore connected mathematical ideas such as the concepts of order, change and system. They can make and test hypotheses, and generalise. And there are opportunities for crosscurricular links with music and physics.
Change ringing is a team activity in which bells are rung one after another rhythmically according to patterns that the ringers memorise. It originated in England around the end of the sixteenth century as a way of ringing church bells that was compatible with the design and construction of sets of bells in church towers, and that fulfilled particular social functions.
Bells at Howden Minster photograph by Neil Donovan
courtesy of the Central Council of Church Bell Ringers
This traditional activity has survived – as this school student explains (if you use this video in a lesson you can decide whether or not to show your particular students the last part – When it went wrong).
Change ringing is also done on hand bells, either by church bellringers when practising,
or in groups that are unrelated to churchbell ringing.
Change ringing bells are tuned to a normal diatonic scale. Pythagoras, it is believed, discovered that pleasant musical intervals are related by small integer ratios. He may have used a singlestringed instrument with a bridge that could be moved to divide the string into two parts in any proportion. Pythagoras found that plucking strings with lengths in the ratio 2:1 produced notes between which the interval is what the ancient Greeks called diapason, and what we call an octave. The Greek word dia meant between, through or across.
This website was created for students by students (aged about 15) while they were working on a project. It provides more explanation about relationships that Pythagoras is thought to have investigated and established between musical pitches and tones. Students will also find information about the ancient Greek origins of the western musical scale at Midicode.
The C major scale is a diatonic scale.
In a change ringing room, the ringers stand in a circle, one behind each bell rope. Students can see in this one bell animation how a bell is rung ‘full circle’ by pulling on, and then releasing, the bell rope.
Animation by
Barney Stratford.
Exactly how the clapper is made to strike the bell, and how the bell is prevented from over rotating is clarified in this swing bell animation.
Animation by
Chris Billinge.
It is usual to start with ringing down the scale starting with the lightest bell – which rings the highest note. This sequence of rings is called a ‘round’, and is demonstrated in this five bell animation.
Animation by Ed Donnen. Animations used with permission of The Washington Ringing Society
A round is usually repeated several times. Then the order in which the bells sound is changed from the order that produces the descending scale – so that all the bells in the set are again rung, but in a different order. The order is changed again, and all the bells are rung in this second order. Each different order in which all the bells in the set are rung is called a ‘row’ or a ‘change’. A ‘method’ is any sequence of ‘rows’ that satisfies the following conditions:
 the sequence begins and ends with a ‘round’
 each bell sounds once in each row
 no bell may move more than one position from its place in one row to its place in the next row
 no row, apart from the first and last rows which are ‘rounds’, is repeated.
During ringing the ringers do not use any written prompts or music sheets. They memorise various methods, each of which has its own unique name, and they shift between methods when they are prompted to do so by ‘calls’ from their conductor. Methods are shown in diagrams that are collected in books, and on websites such as this introductory collection of methods for new bell ringers.
One wellknown method is called ‘Plain Hunt’. The changes in this method when it is rung on four bells are shown in these diagrams, in which each diagram highlights the ‘route’ of one particular bell:



















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Students could be challenged to describe in words this system that is followed when the ‘Plain Hunt’ method is rung on four bells, before investigating the same method on other numbers of bells, or before exploring other methods.
Students could also be challenged to work out the total number of possible orders of four bells (permutations of four objects).
It is possible to adapt the method shown above to create a method that includes every possible order. After seven changes, instead of following the system shown above by swapping the positions of the two central bells and returning to the original round, another pair of bells are made to swap positions. For example, instead of the change from the 8th row to the 9th row being this,
it becomes this,
and then the original system restarts.
There are many traditional change ringing methods that can be explored on different numbers of bells. Or students can be challenged to devise their own methods for particular numbers of bells – perhaps with the added condition that every possible row is included. The patterns traced out through method diagrams by the ‘routes’ of particular bells are interesting!
You may be able to obtain a set of hand bells from your music department. Students can then ring, and listen to, methods that they devise or investigate – my students greatly enjoyed doing this, staying in during lunch hours to experiment and practice!
Students can try their hands at ringing ‘Plain Hunt’ on the change ringing simulator at LearnToRing.Com.
The Bells Applet is brilliant! It shows the number of each bell as it rings, so students can see how the order changes – start it, stop it, and they can see the sequence of rows!
There is much general information about change ringing at the Changeringing Wiki, which is an online community encyclopedia for change ringers.
Video and sound recordings of church bells and hand bells can be found at the Library of Bell Recordings. 