Bobs-only Stedman Triples made easy

The latest composition is in 10 parts, based on block B illustrated below. The part-ends are rotations and reversals of rounds on 5 with 67 behind, so are easy to spot, especially as there is always a bob on 67 just before the part-end, and all the reversals come first, then all the rotations of rounds. As 6 and 7 are fixed at the part-ends the conductor has a choice of easier bells to call the composition from, rather like 7 or 8 in Major.

The other parts are minor variations of block B. Block D is block B with a course repeated. The remaining parts are block B with this course removed and replaced by 1, 3 or 9 bobs with 76 behind. Block E is identified for use in quarter peals (see below). The last three parts are the same, which aids the conductor. There are no more than 4 bobs in a row, apart from one run of 9. This is of great musical benefit, as it avoids the monotony of the same pair of bells dodging behind for long periods.

The average number of bobs per run is less than 2, which is lower than for any twin-bob composition. Bells quite often go up to the back and are unaffected by the bobs before returning, which is preferable to having so many bobs that bells are always affected when behind. There are not many runs of an odd number of bobs (of more than one) which helps in working out which way to go in.

Easier compositions tend to be made up of identical parts with part-ends as members of a group. A group is a set of transformations of the order of bells such that if any two transformations are combined the result is also in the set. The simplest group is the cyclic group, for example the rotations of rounds on 5 which is equivalent to rotations of a pentagon. Brian Price has been using various groups for Stedman compositions for some years, and Hudson's 60 courses are a group. Nigel Newton reported that he had investigated the dihedral group D5 (rotations and reflections of a pentagon) and had produced five mutually true round blocks, the crucial odd number for a bobs-only composition, but could not link them into a peal.

I investigated cyclic groups of order 5 or 7, and found several touches, but none long enough to divide the extent into 5 or 7 blocks. I also investigated D5, which was much easier to search as it divides the extent into 10 shorter blocks, of only 504 changes. I found many blocks of 504 changes, but only one linked to the start of another block and so was remotely useful. These blocks happen to join in pairs to form 5 round blocks of 1,008 changes. Unfortunately the Q-sets (where three bobs can be changed into three plains, or vice versa) present do not allow the blocks to be linked. Other linkages such as pairs of bobs did not seem to help. I think I have shown that there is only one such block (with its rotations and reversals) using D5. Using 10 singles in total this block can be arranged to give a perfect five-part peal.

I noticed that the sixes ending 67 and 76 occurred in the same place in each part as they were the fixed bells at the part-end. If a block could be found without 67 at a six-end, then in the 10 blocks 67 would never occur at a six-end, and 67 would never occur behind at backstroke. These missing 20 sixes could then be arranged into two bobbed blocks, which would still leave an odd number of blocks in total. Similarly sixes ending in 76 could be excluded from the main block, and included as bobbed blocks.

Block B is such a block of 492 changes, excluding the two sixes with 76 behind, with a suitable part-end to link in pairs to give 5 mutually true round blocks of 984 changes. There are then two bobbed blocks of 60 changes with 76 behind, containing the other changes in the peal. These bobbed blocks can be chosen as one of six possible complementary pairs, and with a suitable choice enough Q-sets are available to link the seven blocks together. There are four basic arrangements of the peal. The Q-set linkages were chosen to minimise the number of bobs (438 rather than 441), the appropriate reversal of the whole peal was chosen to give cyclic part-ends and the order of the parts and start was chosen to give 3 identical parts at the end and all the "reverse rounds" part-ends in the first half of the peal.

The blocks can also be arranged into a quarter peal, which might be useful to give a conductor some practice. For example, calling EEC gives a touch of 1,308 changes.

Block B was discovered using a computer program written in 'C', capable of processing about 500,000 sixes per second of the graph linking sixes together. This search detected falseness against sixes in other parts and against sixes with the front three bells in another order. A complete search of a group D5 takes about 6 hours on an IBM 486/50MHz machine. By running the program under the OS/2 multi-tasking operating system this can go on in the background without interfering with normal operation of the computer. Searching for Q-sets, proving peals and formatting data was done using utility programs written in REXX, making use of content-addressable arrays and the translate function for easy transposition.

Other simpler peals might be possible. A perfect five-part would be nice, though unless it had some ten-part type structure it may not be any easier to learn. Other irregular ten-part peals could have 67 and 76 both excluded from the main block, but might result in more complex linkages of the blocks. Block B has the advantage of at most 4 bobs in a row, and the course omitted or repeated in other parts is only 12 sixes long, making the parts quite similar, which might not be the case with another block. A perfect seven-part, though an attractive concept, is probably not a conductor's first choice as there are no fixed bells to call it from. There are probably other peals out there — whether they are any simpler to call is an open question.

5,040 Stedman Triples

by Andrew Johnson

2314567 1 2 3 4 5 6 7 8 9 0 1 2
2147536     - - -       - -   - | |
2531647   - - - -         - -   | |
7152436 - -   - - - -     -   - |A|
1465732 -   -     -       - -   | |B
4561327 - -     - -   -         | |
5473216 - -     - -   - - - -     |
5413267   -     -   - -   -:      |
3261457           A             |
1534276 -   - -:                |C
1524367 -       -   - -   -:    |
4362517           A             |
3251476 -  :                    |E
3241567 -       -   - -   -:    |
4352167           C            
2165347           A             |
1273456 - -     - -   - - - -   |D
2175436         - -   - - - -   |
2135467   -     -   - -   -:    |
4531267           B            
1263547           A            
1243576 -   - - - - - - - -:   
1253467 -       -   - -   -:   
2314567          3D            

First published in The Ringing World, 11 August 1995, p841, updated 2022.

Now available at ajohnson1.github.io