Equal-call extents

 

An area of ringing theory not discussed much is extents that have an equal number of bobs, singles and plain leads. The most well-known example is probably Alan Burbidge's peal of Stedman Triples "the Impossible One-part" which has 280 each of bobs, singles and plain sixes. The composition was devised in the '70s or '80s with the intention of being difficult. For a long time only one person had called this peal, but it has had a resurgence of interest in recent years. The version found here has been rung a lot recently, but these figures are different to the ones on ringing.org.

 

Another peal of Stedman Triples containing equal bobs, singles and plains is this 7-part by R. W. Pipe.

 

Bob Minor

Recently I put together a one-part 720 of Plain Bob Minor containing 20 each of bobs, singles and plain leads. It was very tricky to put together, taking a few hours. You soon realize how non-trivial such a composition is to devise - every call counts! I started off by dividing all the leads into 10 blocks. Six of the blocks have course end xxxx6, and the calls sW,F. The other four blocks have the tenor kept on the front with the calls In,In,sT,Out,Out,sB.

 

This set out a few things to bear in mind:

 

1 - These blocks between them almost have equal calls, except for thee extra bobs in lieu of three fewer plain leads. However, I knew that this could be balanced out in joining the xxxx6 blocks to the oher blocks by omitting a q-set of three bobs

2 - It doesn't matter where the singles occur in any of the I,I,sO,O,sB blocks - they can be placed anywhere depending on the circumstances, which is helpful

3 - One or more of the I,I,sO,O,sB blocks can be replaced by all singles and still contain the same changes. This might be helpful

 

Even at this ordered early stage there was still a lot of tweaking that was required in joining all the blocks together. Any bob replaced by a single had to be balanced out by a single replaced by a bob somewhere else. The fact that bob q-sets come in multiples of threes - yet 20 bobs is not a multiple of three - meant that composite q-sets had to be used. Finally, to join two blocks together to form the 720, irregular q-sets are necessary (where the three q-set members are not all rung at the same stroke, so allowing for two blocks to be joined together. This is how the standard bobs-only 360 works. You couldn't join the two blocks of my 720 together with two singles, because then you would have two singles too many, and two bobs/plains too few.)

 

Though a one-part, there is clearly a pattern behind the blocks which can be seen in the notation. This is in contrast to Burbidge's seemingly random one-part. It seems, then, that there are two kinds of equal-call extents - ones which are totally random, and others where there is a pattern apparent.

 

Bob Triples

Soon afterwards I came up with an equal-call 5040 of Plain Bob Triples  For this I made use of the fact that there are six leads in a course. I came up with the block W,sB,M,sH. This block - which produces the course end 432657 - already has an equal number of bobs, singles and plains, and it is true to the 60 in-course course ends. It is even palindromic, and so would look the same to any in-course courses rung in reverse. So this is already the basis for a holistic equal-call 5040. I came up with a variation of this course for my 5-part peal: W,sB,M. This neatly joins up all of the 567 and 657 courses. However, it is one single down, so this had to be taken in to account. For a five-part the remaining challenge was to join up the 567/657 courses to the "other" group of courses, replace the missing singles, and successfully use irregular q-sets to get the right parity of blocks and prevent the first part end from being rounds. I used omits at W to incorporate the "other" courses, and added extra bobs at H in the "other" courses to balance them out.

 

The part went through several phases using irregular q-sets, and got twisted and turned all over the place. Bits of the part were rung forwards, then backwards, then forwards again, the first part end kept changing, until finally there were 24 bobs, singles and plains in the part.

 

Grandsire Triples (not possible)

An equal-call 5040 of Grandsire Triples had been an idea of mine for years, after seeing Burbidge's Stedman peal. The obvious choice - though not particularly appealing - is the 3-lead course plan. This has 120 plain leads and 240 calls. I came up with the block SPB x 4. 30 of these blocks contain 120 each of bobs, singles and plains. To join the blocks up you simply replace two singles with two bobs, and then balance this out by replacing two bobs with two singles when joining up another pair of blocks. However, before even starting to compose this I knew that an equal-call peal would not be possible. You will inevitably be left with the peal partitioned into two round blocks containing equal calls. In joining these two round blocks together you will invariably end up with two singles too many or two bobs too many. An example is this composition which has 118 bobs, 120 plain leads, and 122 singles.

 

I'm sure this problem applies to all compositions of Grandsire, not just those on the 3-lead course plan. I think that singles always come in multiples of 4 + 2. This is because you must have at least two singles in a 5040. From then on singles must come in multiples of 4 - not 2 - to get the right parity of blocks. It isn't possible to partition an extent of Grandsire Triples into an odd number of blocks, so the singles must progress in multiples of 4 to successfully join all the blocks together. 120 singles are therefore not possible - it has to be either 118 or 122. Irregular q-sets aren't possible in Grandsire, and can't come to the rescue as they can in Bob Minor and Bob Triples (equal-call extents presumably always rely on irregular q-sets.)

 

So stuffy old Grandsire does not allow equal-call 5040s for precisely the same reason that it does not allow bobs-only 5040s!

 

These three methods seem to be the most relevant for equal-call extents. St Simon's, etc, probably do not appeal as much, and Spliced compositions seem to somehow defy the point of having equal-call extents. I might update this page if new ideas come along.

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