An Introduction to Group Theory for Bell-Ringers
Leeds, January 2011
The art of change-ringing has been practised, particularly in England1, for over four hundred years2 but only recently mathematicians have taken an interest3 in the fact that this art can be described rather elegantly in mathematical terms. Surprisingly, the mathematical concept in question, group theory, is about a century4 younger than the applications of it in the ringing of changes, described thoroughly by Fabian Stedman in 1667.5
In this essay I will introduce groups ‘as a tool for exploring’6 the art of change-ringing and through bell-ringing introduce the mathematical concept of sets, functions and groups.
An algebraic structure such as a group, ring, module or field7 is an abstract collection of objects and operations between those objects. Many patterns and symmetries, many structures occurring in nature, in physics, chemistry or other sciences8 but also in music and art can be described by algebraic structures in a more abstract way that allows us to analyse their structure mathematically and prove results about them.
In order to be able to show that certain aspects of change-ringing can be described with group-theory, let me recall a few well-known facts and some
terminology from both change-ringing and mathematics.
The bells in a ring, when we talk or write about the order they are to be rung in, are usually numbered, such that 1 denotes the treble and the highest number, n, denotes the tenor. This way we have also established how many bells there are in the ring of bells in question: n.9
These n bells are usually rung in a specific order, the simplest of which is rounds, where they all sound in the order of pitch: 1 2 3 ... n. To make things interesting the order in which the bells are rung changes. ‘Each ringing of the n bells, once each in some order, is called a change ’10, rounds being a special change.
An extent, finally, is a sequence of a certain number of changes, depending on how many bells participate. According to Arthur White in his mathematical analysis of change-ringing11 the number of changes in an extent is [Abbildung in dieser Leseprobe nicht enthalten] The “+1” can be explained by the fact that we want every extent to begin and end with rounds.12
Following a few rules, an extent contains every possible order that can be rung with n bells. To determine possible orders for the changes in an extent, bell-ringers follow certain rules that have been established for musical, practical and technical reasons:13
1. An extent both begins and ends with rounds
2. No change, except for rounds at the beginning and end, is repeated. The order of ringing always changes.
3. From one change to the next, no bell moves more than one position.
4. No bell rests in one position for more than two consecutive changes, except for covering bells.14
With these rules, there are restrictions as to which row can follow a given one. In order to make sure that no bell moves more than one place in a
change, she only swaps places with one of the adjacent bells. For every num- ber of bells there is a certain number of possible swaps that can happen in one change. The number of bells determines how many changes there are in an extent15:
1 Morris, The History and Art of Change Ringing, Chapter 1, for full references see reference list on page 8 of this essay
3 White, Ringing the changes
4 Fauvel, Flood, Wilson, Music and Mathematics - From Pythagoras to Fractals, p. 118
5 Stedman, Campanalogia: or, The Art of Ringing Improved
6 Mirman, Group Theory - An Intuitive Approach, p. v
7 for further information see any book on algebra and linear algebra
9 White, Ringing the cosets
10 White, Ringing the changes, p. 203
11 White, Ringing the changes and White, Ringing the cosets
12 see also: Wikipedia.org, Wechsell ä uten
13 White, Ringing the changes
14 There are more rules than these four but for the purposes of this discourse these are enough. For further information see White, Ringing the changes and Morris, The History and Art of Change Ringing
15 White, Ringing the changes
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