Non-groups

Method ringing is essentially a branch of Group Theory mathematics. If you have a search on Google, you can find a paper called Was Fabian Stedman The First Group Theorist? by some American academic. But what is a group?

​

When we ring a touch, quarter or peal, it is a common feature for the touch to be divided into parts, with every part called exactly the same (or with minor subsitutions in some of the parts.) Touches and peals aren't composed in parts just to make things easier for the conductor - though it certainly does make things easier - but because the repetition creates a sequence of part ends that eventually loop back to Rounds.

​

Example: we call a touch, in parts, on 8 bells, for which the first part end is 13456278. If we call the next part exactly the same, we get 14562378 for the second part end. We continue the process, producing part ends 15623478, 16234578, 12345678. Essentially, the first part end - 13456278 - is our operator, so we always use that as our transposition.

​

Because the same operator is used to produce all five of the part ends, the five part ends form a closed group. Take any of the five part ends and transpose it by any of the others, and you will always get one of the other members of that group. So 14562378 transposed by 16234578 produces 13456278, and 15623478 transposed by 14562378 produces 12345678. This feature of closure is predominantly what defines a group.

​

This group is, in fact, only a sub-group of a larger group (though it is still a fully functioning independent group in its own right.) If you add the five part ends 16543278, 15432678, 14326578, 13265478, 12654378, we have a group of ten part ends. This group is known as the Dihedral Group in mathematics. Again, if you transpose any of the ten part ends by any of the other ten part ends, the result will always and only be another one of the ten part ends in the group. And this group can be extended to an even larger group of 20 part ends! The full group of 20 can be found on my page about 20-part peals of Grandsire Triples.

 

What is the importance of these groups, and why are they of use to composers? If we treat each part end as the heading of a column, we can arrange sets of course ends underneath each column. These course ends (or lead ends, or six ends, depending on the specific example) are technically known as cosets. Because of the group relationship between the part end headings, it is easy to arrange the cosets underneath to all be different, and so produce the foundations of a true composition. For example, in the five-part group above we can treat part end 12345678 as the first heading, with the course ends 14235678, 13425678, 12534678, 13254678, 15324678 underneath. Note that each of these cosets has bell 6 in 6ths place. The same cosets will be used underneath the other four part ends, but because each of the other four part ends has a different bell in 6ths place, there will be no repetition across the cosests, and the composition will be true. For example, the same five cosets under the part end heading 13456278 are: 15346278, 14536278, 13645278, 14365278, 16435278, which clearly do not clash with the other cosets: it is the same across all parts. Thus the use of group theory has ensured a true touch of 30 different courses.

 

That, in a nutshell, is the advantage of using groups in composition, and why not using groups is a recipe for disaster. For a far more complete treatise by someone who actually knew what they were talking about, see this fascinating paper by the late Brian D Price. The paper includes a diverse collection of groups that can be used up to 7 bells, many of which have been used for specific compositions.

 

However, this page on my site is meant to be about examples of compositions where a group hasn't been used! Another fascinating piece by Price has been written on this subject (scroll down to the section titled 'Sufficient but Not Necessary,'). Extents are the most interesting examples of such compositions, as groups are usually of the utmost importance in those cases. There is only limited scope for methods that can produce a non-group extent. Bob Triples is such a method, as can be seen in the intriguing examples offered by Price. As he points out, parity is a factor that makes non-group extents possible: an extent divided into an in-course half and an out-of-course half is divided into two mutually exclusive sets, so it is not necessary for the part ends in both halves to be members of the same one group. An old example of such an extent is this peal of Bob Triples by Henry Hubbard.

 

I used to look down my nose at these rare compositions that don't stick to a group system. Now I have come to respect them! In 2009 I produced this irregular 5-part of Original Triples. The parts (the 'C' sections) are theoretically all called the same, but the part ends do not form a group. But because there is a fixed-bell in 5ths place in each part, you get all the same course ends that you would anyway, so a group is not necessary. Unlike the Bob Triples examples, this composition doesn't need to rely on parity for its non-group constitution.

 

On reading Price's paper recently, I have become interested in producing some other non-group extents. First this 10-part 40320 of Bob Major, and then a 20-part 40320. It is great that you can get a 20-part peal of Major with all parts called the same (bar the halfway singles) - a feature not shared by 20-parts based on a group. Both of these extents rely on parity. My interest in this topic remains, so watch this space for more compositions!