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Grandsire Triples

Singles-only

Is a full peal composition of Grandsire Triples possible with just singles and plain leads - no bobs at all?

If so, what would the building blocks for such a peal be? The normal blocks for Grandsire composition are the P-Blocks (Plain Courses, five leads long) and B-Blocks (Bob Courses, three leads long.) Some compositions are built up from the 72 P-Blocks, and some are built up from the 120 B-Blocks. Others are, I think it is fair to say, built up from mixtures of the two.

Firstly, is it possible to use singles alone to link up the 72 P-Blocks, and get a peal that way? ('Singles', in this discussion, refers exclusively to ordinary Grandsire singles.) No. The 72 P-Blocks have to all be rung In-Course, and the use of singles would instantly shatter that. This is because a P-Block is asymmetrical and cannot be rung backwards. For singles to work truthfully in a composition they have to alter the direction of a round block, but as a P-Block is monodirectional and can only be rung forwards, they do not work in this context. B-Blocks are bi-directional, because they are symmetrical. For example, the B-block starting 235476 is the same as the B-Block starting 324567, the only difference being that they have been rung in opposite directions to each other. This is why singles can function in B-Blocks. But as a B-Block is made up from bobs, this doesn't help very much with our singles-only quest!

 

Perhaps we could build up a peal from S-Blocks. An S-Block looks like this:

 

 s 572634

 s 645372

 s 326745

 s 753426

 s 467253

 s 234567

 

As can be seen, this is six leads long, entailing that 60 S-Blocks would be required to make a peal. Is there a set of 60 mutually true S-Blocks that could do this? Yes there are! An examination of this particular S-Block shows that it is bi-directional (good news) and there is no internal falseness possible except when it is rung backwards (i.e: the S-Block starting with 64352 is false against this one.) This means that 60 mutually true S-Blocks for Grandsire Triples are the equivalent of 60 mutually true P-Blocks for a peal of Plain Bob Triples.

The question now is how we can join up these 60 S-Blocks of Grandsire Triples together to form a peal, and this is where we start to hit problems. The most obvious starting point would be to use Q-sets of plain leads. But we stumble into falseness straight away: if we call SSSSSP x 5, the resulting touch is false, and we have only got as far as one Q-set of plain leads! But as we have stuck to the Q-set law, where has the falseness come from?

The fact is that singles in Grandsire Triples affect bells in both the handstroke and the backstroke treble leads. You get the 'bob' part of the call at handstroke, and the single itself is made at backstroke, allowing repeated changes to leak into the treble's handstroke lead head.

This means that we must have a Q-set for the handstroke lead head and a separate Q-set for the backstroke lead end. But these Q-set members do not close into a group, so it grows larger and larger until all of the 360 lead ends are Q-set members! As all of these 360 lead ends would then have to be plain leads, we have abstractly regressed back to 72 P-Blocks.  So a singles-only peal of Grandsire Triples is impossible. For more information about these S-Block q-sets, click here for a thread on the ringing theory mailing list in 2008 that briefly went in to this matter.

 

     

More about S-Blocks

We now know that a singles-only peal can't be had, but perhaps we can still build up a peal predominantly from S-Blocks, with a small number of bobs. (Interestingly enough, S-Blocks have been incorporated into some peal compositions already. Compositions that have large numbers of calls can do this by coincidence. One example was my first ever peal composition of Grandsire Triples, a six-part that had a block of five consecutive singles in each part.)

To start this off, we can come up with the following five part touch:

                                           840 Grandsire Triples

 - 752634  - 723546  - 734265  - 746352  - 765423

 s 647352  s 567423  s 257634  s 327546  s 437265

 s 326547  s 435267  s 642357  s 563427  s 254637

 s 573426  s 274635  s 376542  s 475263  s 672354

 s 465273  s 652374  s 523476  s 234675  s 346572

 s 234765  s 346752  s 465723  s 652734  s 523746

   273546    374265    476352    675423    572634

 s 562473  s 253674  s 324576  s 436275  s 645372

 s 435762  s 642753  s 563724  s 254736  s 326745

 s 724635  s 736542  s 745263  s 762354  s 753426

 s 657324  s 527436  s 237645  s 347562  s 467253

 s 346257  s 465327  s 652437  s 523647  s 234567

This touch is certainly true and self-contained. Being 840 changes long, we would have to find 6 mutually true sets of this calling to make a peal. However, this calling doesn't seem to be at all co-operative for lending itself to mutually true counterparts. For example, an obvious searching point would be to find any three bells of 2, 3, 4, 5, 6 to rotate, tripling the 840 to a 2520. But there isn't a single trio from 2, 3, 4, 5, 6 that allows this to happen without repetition somewhere. The only other option is to look for possibilities which involve moving the 7th (perhaps even a 'bicycle three part'), but as the little bells have been so unco-operative, it doesn't seem any more likely that this will get us nearer to finding a peal.

 

An example of a peal composition that uses singles as an integeral part of the composition, and considerably fewer bobs, is the old five-part by Thomas Thurstans. This peal has 75 bobs, but I have no idea whether this is the minimum possible, or not. If you know, please get in touch.  

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