'Bicycle' is a term that I have used quite frequently on the pages about Grandsire Triples, and I decided that this very interesting type of composition deserved a page of its own.

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As you will probably know, three-part peal compositions are very common on many different numbers of bells. In Major for example, the part ends 14235678, 13425678, 12345678 are very popular for three-part peals, as are 12356478, 12364578, 12345678. I also happen to be a fan of part ends 15243678, 13542678, 12345678. What these groups of part ends have in common is that they all involve a single trio of bells rotating at each part end. In the first set of part ends, we have bells 2,3,4 rotating until they get back to Rounds, while bells 5,6,7,8 remain fixed. In the second set of part ends, bells 4,5,6 are the ones that rotate. And in the final lot of part ends, bells 2,3,5 are the rotating trio, while bells 4,6,7,8 remain unmoved.

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However, it would be interesting if we could have a three-part composition in which two trios of bells are rotating at the same time. The most clear example would be this set of part ends: 14237568, 13426758, 12345678. A closer look and you can see that this time, not only are bells 2,3,4 rotating at the part ends, but bells 5,6,7 are rotating too, at the same time. The particularly special thing about this is that the two trios of bells do not affect each other at all.

And you can get more variety of part ends. 14236758, 13427568, 12345678 is actually a different set of part ends to the previous one - despite the similarity - because the two trios have different starting points. You can then combine the two groups for the maximum permutations of part ends, and have a nine-part composition.

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For several years I have referred to these special types of transpositions as 'Bicycle three-parts,' because the simultaneous rotating of two trios of bells is like two wheels going round. There are other kinds of Bicycle part ends on higher numbers of bells where the two groups of rotating bells are of different sizes, taking longer to get back to Rounds because it takes different times for each group to complete a revolution. Here is an example of such a 'Penny Farthing' group for ten bells: 1352469780, 1543268970, 1425367890, 1234569780, etc.

Bells 2,3,4,5 are rotating while bells 7,8,9 are also rotating.

 

But back to Grandsire Triples, can a peal composition be had for this method on the Bicycle three-part plan? Yes, fortunately the 360 leads of a peal can be divided this way, allowing for a greater variety of compositions. J. J. Parker produced a palindromic six-part on this plan, and also a bobs-only three part with the missing bob course added in to one of the parts. Another early peal composition on this plan for Grandsire Triples was by E Bankes James. It was claimed as the first exact three-part in the method to use normal calls (I seem to remember that a normal unicycle three-part had been composed by J. W. Washbrook and published in Bell News, but that this had to use unusual calls.) As E Bankes James was an ingenious composer - like his brother, H Law James - I'm sure that this would be a particularly special example of a bicycle three-part. However, it is a shame that the part ends for this peal are difficult to use and not very aesthetically pleasing, being based on trios 2,3,7 and 4,5,6. However, a more suitable version can be found in the 1903 odd-bell book, using part ends like the ones I'm about to describe.

 

At some point after this, A. J. Pitman came up with a composition based on the much more sensible group 13426758, 14237568, 12345678. I have the figures for this composition, so do get in touch with me if you'd like to see them (email address at the top of this page.)

 

How is it that peal compositions of Grandsire Triples are possible on a bicycle three-part plan? If you look at the B-Block starting from Rounds (i.e: the three-lead touch of three bobs in a row) you will see that this is in fact a bicycle three part touch. So the smallest possible building blocks for Grandsire Triples peal compositions are themselves bicycle three-parts - how charming! Because of this, a composer can choose the set of bicycle part ends in advance and then find out which particular B-Blocks rotate the two trios of bells required. These B-blocks are then split across the three parts, to produce the part ends and stop the bells from coming back into Rounds at the first part end. There are two ways to split the B-Blocks. You can either sandwich an isolated lead between two consecutive lead ends (this the equivalent of the well known block SBBS, and becomes SS - the SBBS is split equally across the parts.) Consequently, many bicycle three-parts include blocks of SS, and sometimes SSSBS, etc. The other way to split the part B-Blocks is by a plain lead. Once again, these special leads occur in the equivalent place in each of the three parts. If any of this is getting unclear, feel free to drop me an email and I'll show you some diagrams.

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After Pitman's bicycle three-part, the next one to be composed was probably the composition by Brian Price that is explained on the cyclic page. However, this is really a 12-part that can divide into a bicycle three-part, so doesn't quite have the same "no strings attached" feel of authenticity that some of the other peals have.

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And now for the extremely neat peal compositions by the late Edward W Martin, who was a master Triples theoretician. But first a further digression from me.

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If we choose to rotate trios 2,3,5 and 4,6,7, we can get the following set of part ends: 15263748, 13572468, 12345678. Tittums and Queens! Amazingly, these two famous named changes are linked in the form of a mathematical group of transpositions. This is the case on all numbers of bells - indeed most stages also incorporate Reverse Rounds, Reverse Queens, and Reverse Tittums into the group too. How amazing that you can get an exact three-part of Grandsire Triples where the part ends are Queens and Tittums (the fact that these changes are transpositions of each other makes it certain in my mind that it is no coincidence they both have names.) And Eddie Martin has done just that. An article in the Ringing World about his novel and ingenious contribution to this genre was published in the 1980s. I think that four of his compositions were published in that article, of which three are now online: his no. 4, no. 13, and no. 14. 

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Despite the technique of splitting the special part B-Blocks across the three parts, it can still be the case that the composition comes round at the first part end. To prevent this happening in peal no. 4, the bob that would have been replaced by SS in all three parts, instead has to be replaced with SBBS in just one part. This is why no. 4 is an inexact three-part. No. 13 is probably the best peal of the three, having only 150 calls (Eddie questioned in his article whether this was the fewest possible on this plan, but Pitman's peal may have fewer), Queens being saved for the second part end, and the fact that it is an exact three-part with no substitutions required. No. 14 should be fun too - I wonder if any band has rung it? Some real masterpieces left for us by Eddie, a testament to a truly great mind.

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In 1997 Richard Pearce published some compositions that produced 357 leads of a peal by bobs alone (like Holt's Original), which are made up to peal length by replacing the final bob with SBBS. These four peals were all bicycle three-parts, though non-standard singles have to be used. Earlier compositions on this plan had been published in Bell News by the non-ringing mathematician Professor W. H. Thompson, and J. J. Parker. If you would like a transcript of Richard Pearce's compositions, get in touch and I'll send you a copy.

 

My own 18-part peal composition (my no. 18) was composed in 2012, and this forms an inexact bicycle three-part. The 18 part ends are the 18 possible in-course permutations of 2,3,4 and 5,6,7. It is quite pleasing that the A Blocks (the natural parts) rotate 2,3,4 and leave the back bells fixed. However, this composition would be much improved if it had fewer calls.

 

I think bicycle extents are fascinating, and greatly extend the repertoire. They are not so easy to call in Grandsire Triples, as there is no observation bell possible. But peals of Stedman Triples also exist on this plan, and these do have an observation bell. A fine example is the brand new bobs-only composition produced by Andrew Johnson, which can be transposed to have Queens and Tittums part ends. I had longed for such a bobs-only composition of Stedman Triples for some years.