Gray Change Ringing

This article describes change ringing sequences that consist entirely of singles, meaning exactly one pair of adjacent bells swaps at each change. We call such sequences Gray because they're related to Gray Code, which orders the permutations of a set so that successive values differ by only one digit. Put another way, these patterns are as granular as possible and thereby maximize smoothness. The author is a composer and appreciates these patterns for their musicality.

For four bells, only five sequences exist that are both Gray and extents. An extent is a sequence that uses all the permutations of a set of bells, without omitting or repeating a permutation. An extent for four bells has 24 changes (4 factorial), hence the number of possible sequences is 24 factorial, or approximately 6.2e+23, but fortunately most of those possibilities can be eliminated. The compliant sequences were determined using a recursive crawler with branch pruning, coded in C++.

The four-bell sequences are identified by a three-digit number that expresses how many times each pair of bells is swapped. If we call the bells ABCD, the first digit is the number of AB swaps, the second digit is the number of BC swaps, and the third digit is the number of CD swaps. So for example [10 9 5] means AB swapped ten times, BC swapped nine times, and CD swapped five times. The ideal sequence would be balanced, meaning it would swap each pair of bells the same number of times, or as close as possible. This idea is borrowed from balanced Gray code. The best-balanced sequence is [9 6 9] and it's also related to a classic change ringing method called Plain Hunt, which is unlikely to be a coincidence.

For five bells, the number of permutations is 120 (five factorial), hence the number of sequences is 120 factorial, or 6.7e+198. This vast number precludes the use of exhaustive methods, so in this case a more specific method was developed: one bell plain hunts, and when it reaches one side or the other, the possible swaps are explored recursively. Other five-bell Gray extents may exist.

• Four bells
  • [10 9 5]
  • [6 12 6]A
  • [6 12 6]B
  • [9 6 9]
  • [5 9 10]
• Five bells
  • The only one we found!