Three-part bobs-only peals of Stedman Triples

A new bobs-only peal composition of Stedman Triples was rung by a Cambridge University Guild of Change Ringers band on 7th October 2017 at St James Garlickhythe. New bobs-only peals of Stedman Triples are rare and this composition with 603 bobs is on a new plan, the first exact three-part.

The first compositions were those of Colin Wyld, complex one-parts composed in 1994 with 705 and 597 bobs (RW 24/2/1995 p.197). The first composition to be rung (RW 3/3/1995 p.227), by Philip Saddleton and me, has 579 bobs and was rung on 22nd January 1995. I then composed an irregular ten-part peal in June 1995 (RW 11/8/1995 p.841, updated article) with four arrangements having 438 or 441 bobs, and extended them in 2012 with a further 148 variations with between 438 and 456 bobs including 15 exact two-part arrangements. The ten-part peals are easier to call and have been rung more than 180 times.

For a peal of Stedman Triples there are 840 places where a bob could be called, so 2⁸⁴⁰ possibilities, a 253-digit number. This dwarfs the number of atoms in the universe so trying everything is impossible. A common idea in peal composition is to arrange the rows into round blocks, each starting with a different row and ending back at that start, such that all the round blocks cover all the rows in an extent without repetition. To find a peal, the blocks must then be joined by adding or removing calls to link all the blocks into one big round block, the peal.

It is not possible to arrange the 5040 rows of Stedman Triples into 60 round blocks with 14 consecutive plain sixes — into a set of plain courses. But the rows can be arranged into 84 round blocks of 60 changes, each with 10 consecutive bobs, called B-blocks. The first bobs-only peals are based on B-blocks, linked by Q-sets of three omits. A Q-set is a set of rows before a call such that if all the members of the Q-set are present in a composition then all the member sixes must be bobbed or all must be plain. For example, in Stedman Triples if the six-ends 2314567, 2314675 and 2314756 are present then all must be bobbed or all plain. Those Q-sets link three round blocks into one; doing this iteratively reduces the number of blocks by two each time, but keeps the parity of the number of blocks. This means that given an even number (84) of blocks to start we cannot reduce the number of blocks in this fashion to one, the peal. In Grandsire Triples, with common bobs, there is only one way a row can appear which leads to W H Thompson's proof in 1886 of the impossibility of bobs-only extents in that method.

Fortunately, in Stedman Triples, a row can appear in a different place in a six; for example, six-end 2314567 could be bobbed and 2314675 plain provided 2314756 occurs elsewhere in a six such as the six with six-end 3124756. The first peals used an additional technique called the magic block linkage which takes the rows of ten of the 84 B-blocks and arranges them in pairs linked by omits at, for example 1,4;1, into five larger blocks, breaking the parity barrier. The three sixes between the omits at 1 and 4 in each of the five larger blocks contain the same rows, but different six-ends from the three sixes of another of the original B-blocks, so with a five-way shuffle the five larger blocks and the ten B-blocks contain the same 600 changes. The five large blocks and the remaining 74 B-blocks can then be be linked in two at a time to make the peal. The result is a complex one-part with little discernible pattern.

The latest composition is in three identical parts using block A illustrated below. This circumvents the Q-set parity law as it is not based on linking all 84 B-blocks. Instead it directly arranges the sixes into an odd number of round blocks. Multi-part peals are a common idea in composition, and a branch of mathematics called Group Theory provides the basis. A multi-part search for peals is much restricted compared ot a one-part search, but automatically divides the extend into mutually true blocks as the search proceeds. A suitable choice of group and so part-ends can permit an odd number of round blocks. A successful three-part search could find three one-part blocks, possibly linkable with a Q-set, or the blocks could directly link to form an exact three-part peal, as in this composition.

Although there are a great many bobs, other features simplify the calling of the composition. The part-ends (which occur two rows before a course-end) are (double) rotations of rounds on six with 7 behind, i.e. 5612347, 3456127 and 1234567, so are easy to spot, and the 7 does the same work in each part, so is a convenient observation bell. Block A is divided into four sections, marked with lines in the composition. In each section the 7 enters the slow after two bobs 6/7 down, remains on the front for the entire section until it exits the slow, does two bobs 6/7 up, six bobs 6/7 down, bobs in and out quick, then six bobs 6/7 up for the section end. This work at the beginning and end of each section is simple to learn and comprises one third of the peal. For the body of each of the fours sections, when the 7 is on the front, each course starts with bobs at 1,4,5 or 1,2,3,4. The exact half-way point of each part is the end of section 3 with 2143567, 6521347 and 4365127 which are easy to check. The other section ends have bells in order: two even bells, two odd bells, even bell, odd bell, 7: 4613257, 2451637, 6235417, 2653417, 6431257 and 4215637. All of the courses in section 1 are repeated elsewhere, which reduces the amount of learning required. Section 2 is courses 1,2,6,7 of section 1. Section 4 is long, but includes courses 4,5,2–4 and 6–7 from section 1 and courses 5 and 7 of section 4 are the same.

There are 972 variations of this peal giving exact three-parts excluding rotations and reversals. These are formed from 12 basic blocks each with 603 bobs, with 81 minor variations each generated by adding extra Q-sets of three bobs. The number of bobs varies from 603 to 639. This variation has the minimum number (603) of bobs, the minimum number (13) of different courses, a four-section arrangement with the two repeated courses at the end of each section, and a nice half-way point in each part. If the parts are called differently by mixing up the basic blocks then there are millions of possible peals. Section 1 rung three times gives a quarter-peal of 1260 changes. Section 1 with course 4 called 1,4,5,6,7,8,9,10 (same as course 11 of section 4) rung three times gives a quarter-peal of 1260 changes with the same part-ends as the full peal. A musical quarter-peal of 1260 changes is section 2 with course 1 called 2,3,6; section 3, section 4.

One point of comparison between peals is the distribution of bobs and how they come in runs. For this peal the average number of bobs per run is 3.8, compared to about two for the ten-parts. Although there are many bobs there is one run of six consecutive plain sixes per part; all previous bobs-only peals have at most five plain sixes in a run. There are only 11 runs per part of an odd number of bobs which helps in working out which way to go in, especially as some of those runs are followed by another run of an odd number of bobs, so most of the time the front work alternates as normal between quick and slow. The 8- and 9-bob runs always have an odd bell (1,3,5) pair or an even bell (2,4,6) pair behind which are slightly more musical pairs for a long run. The 6-bob runs always have the 7 involved. The 3- and 5-bob runs always have an odd and an even bell behind. The 4-bob runs have an odd and an even bell behind or else one run per part with either 64, 42 or 26. Traditionally, the more musical rows in Triples are those ending 67, 46 or 74. These occur mainly in parts 3 and 1. The Birmingham handbell peal was rung with a rotation of the composition starting 22 sixes later, with the treble as the observation bell, and Queens, Tittums and Rounds as the part-ends, so the dodging pairs at bobs were different.

There are three ways a course can begin: 1,4,5; 1,2,3,4; or 2,3. There are eight courses per part beginning 1,4,5 and each set of such courses is preceded by a course with a single bob at 6, the only occurrence of a single bob, then all the calls occur in pairs as a bell goes in and out quick. Calls at 6,7 and 8,9 vary per course while there are always bobs at 10,1; omits at 2,3; bobs at 4,5. About a third of the peal has bobs coming in pairs as a bell goes in and out quick and some of the variations of the peal come from changing pairs of calls at 6,7 and 8,9 in these courses. Once there is an omit at 10 then the next course starts 1,2,3,4 or 2,3.

There are eight courses per part beginning 1,2,3,4, but the calling of some of those courses is repeated so there are only four types of course beginning this way.

There are 12 courses per part beginning 2,3 but there are only four types. These courses occur at the beginning and end of each section.

I have been working on bobs-only Stedman Triples from 1994, particularly since finding further multi-part peals in 2012. This composition was found with the aid of a computer, but was not a brute force search of all possible three-parts. Computers have become faster since 1995 — the latest version of my search program running single-threaded on a 2.60GHz Intel Core i7-5600U CPU finds the ten-part blocks in about 6 minutes, compared to 6 hours originally. The intensive search programs are written in ‘C’, and utility programs in Java, Python, and Bash. After many failures of promising concepts, finding these three-part peals was a combination of thinking of different ideas of a restricted three-part followed by searches to see if these led towards a peal. Once I found a three-part arrangement dividing the extent into an odd number of round blocks I then fed these blocks back into a search program where the six types were fixed but the calling was not so that Q-set links could be used to join the blocks. I also tried removing complete B-blocks from the input fed back into a search program so that it would have more flexibility in choice of calls and six types to link the blocks together. I then reduced the number of bobs to the minimum and chose a peal and a rotation that looked neat on the page.

Conductors often prefer peals in multiple parts, preferably with few calls and called identically from a fixed bell. Time will tell whether this peal with the three identical parts and a certain regularity of calling within each part becomes more popular than the irregular ten-parts with fewer bobs. This peal is based on a cyclic group of order 3. Other exact multi-part peals might be possible. From my searches I do not believe there is an exact seven-part or five-part — the only five-part blocks loop back to themselves rather than link. An exact four-part, for example based on part-ends 2341657, 3412567, 4123657 and 1234567, might be possible. I have found some other irregular three-part based peals. There may well be some exact two-parts other than the ten-part arrangements. I feel there must be other peals out there — whether they would be any simpler to call is an open question.

2314567 1 2 3 4 5 6 7 8 9 0 
2354761   - -   - - - -   - |
2654713 - - - -   - - - - - |
2153764 - - - -   -   - - - |
5162743 -     - -     - - - |
6143725 -     - -     - -   |
6143572   - -   - - - - - - |
6143257   - -   - - - - - - |
6123754   - -   - - - -   - |
6523741 - - - -   - - - - - |
6523174   - -   - - - - - - |
6523417   - -   - - - - - - |
1423756   - -       - - - - |
1423675   - -   - - - - - - |
1423567   - -   - - - - - - |A
1526743   - -     -   - - - |
2541736 -     - -     - - - |
4612753 -     - -         - |
1653724 -     - -     - -   |
1623745 - - - - - - - -   - |
5426713 - - - -     - - -   |
5416732 - - - - - - - -   - |
5316724 - - - -   - - - - - |
5214736 - - - -   -   - - - |
1235764 -     - -     - - - |
4265731 -     - - - - - - - |
6152734 -     - -           |
6152473   - -   - - - - - - |
6152347   - -   - - - - - - |
2314567          2A