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[An] experiment, reported in a paper by D. A.
Papworth [“Computers and Change Ringing”],
involves programming a method
of composition called “change ringing,” a
permutational technique that
developed over the centuries in connection with the ringing of church bells,
particularly in England. The standard descriptive treatise on change ringing
was written by J. W. Snowden [sic] [Standard Methods in the Art of
Change
Ringing], and recent mathematical treatments of the subject have been
published by Rankin [“A Campanological Problem in Group Theory”]
and Fletcher [“Campanological Groups”].
Papworth programmed change ringing for a PEGASUS computer. As he notes, the problem of change ringing can be described mathematically as follows: Given the numbers, 1, 2, . . . , n, representing church bells of different pitches in descending order, find rules for generating in some order all n! permutations or “changes” or subsets thereof. However, the following restrictions must be observed: (1) the first and last permutations of any sequence or subsequence must be the original row, 1, 2, . . . , n, which is known as a “round”; otherwise, no two rows may be the same; (2) in any row, no number may occupy a position more than one place removed from its position in the preceding row; (3) no number may remain in the same position in more than two consecutive rows. Papworth solved the particular system of change ringing called “plain bob major.” Changes on eight bells are called “major,” and “plain bob” is the simplest kind of change, namely that in which consecutive rows differ by as many exchanges as possible. . . . Papworth was concerned, first, with proving that “plain bob major” is sufficient to generate all 8! or 40,320 possible permutations of eight numbers, and, second, with generating sample compositions starting with random numbers. . . . . . . Papworth’s work stimulated Bernard Waxman and me at the University of Illinois to write an analysis and generating program for change ringing for our own use. As we saw it, the basic problem involved in change ringing is the generation of permutations of n discrete symbols following some set of given rules. Using this basic idea and some of the conventional rules of change ringing, subroutine RING was written so that it would generate sequences of numbers that symbolize different permutations on a set of tones. In our first experiment the basic rule was followed that each sequence must be different from all others, with the exception of the first and last. There are also additional constraints which may or may not be used depending on how restricted one wishes the permutational set to be. . This program can be used to generate such things as sequence of rhythmic variations and changes in dynamics as well as sequences of tones. . . .
Lejaren Hiller
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