[r-t] 40320 Spliced Major (3m)

[Philip Earis]

I am keen to ring a challenging but achievable extent of major. MBD has now put together a new composition of spliced that has many attractive qualities.

As discussed at various times on here, extents of regular symmetric plain methods are easy to obtain, whilst extents of treble dodging major methods are relatively rare. There are methods like Derwent with "a" falseness that can easily generate an extent. Alternatively, there are published extents where other surprise methods can be hammered into an extent, such as Colin Wyld's Yorkshire composition. In either case, though, the thought of ringing 180 courses of the same single method has limited appeal to me.

Ander Holroyd has a conceptually very conceptually neat extent of half-lead spliced, but this involves a change of underwork every halflead and a change of overwork every leadend, which is unlikely to give the requisite stability.

So what are the other options? Extents can be easily obtained from "helixoids" (like Double Helix) - as these have a cycle of 3 bells ringing in every position relative to each other, offset against a cycle of 5 bells, you simply need to join together all the 24 possible courses of the 5-cycle. Double Helix (or similar helixoids) would better retain a ringer's interest, but is perhaps not intrinsically stable to be rung for 18+ hours.

Indeed, for long ringing I felt some "blocks of stability" was needed. Plain bob major is a very simple and familiar method, which could help here.  Moreover, I was especially drawn to the property that plain bob has strong helixoid properties.  The rung extents of plain bob major have three courses of bobs at V throughout - this gives a block of 336 changes with 3 bells (here 1,7 and 8) in every possible relative position.

This can be simply rotated to three middles, putting 1,2,3 in every relative position. Splicing works best when mixing apples with apples, and my thought was therefore that an interesting yet achievable extent could hopefully exist where a 336 change block of plain bob would be a course splice against one or more worthwhile helixoids.

MBD took up the challenge, and after much clever thought, intensive computing and marvellous work has produced the gem copied below. This composition is a real step-change development.  I'll leave it to MBD to describe in more depth both what he has done (including the various pitfalls and blind avenues along the way), and how he got there (using an interesting new programming language that is very well-suited to the task).

As you'll see, the composition uses the two magical helixoids, H and J, introduced in my recent email. A fair number of candidate helixoids were thrown up by MBD's searches, but H and J, with their double nature and attractive features, seem the best.

The plain bob can be interchanged with double bob (or reverse bob) - indeed, ringing an extent of 4 spliced including both plain and double has strong attractions.

I think this approach and MBD's composition has mileage, and I'll start trying to organise a date for an attempt.  There's nothing like the pressure of competition to spur progress, though, so if anyone else wants to take up the challenge to ring this then please feel free...

There remain some key unanswered questions resulting from the composition development, including why many helixoids could be found that were congruent to a 336 change block of plain bob, but no helixoids could be found that were congruent to 336 change blocks of other plain methods (eg Double Norwich).  MBD can introduce these, though, and hopefully some answers will be forthcoming.
 
Of course, there are other (potentially simpler and very attractive) ways an extent of major might be arranged. Having the feature of 336 change blocks where three bells ring in all relative positions can certainly be exploited in other ways.  For example, something Richard Smith has recently been looking at is getting an extent of 21 spliced regular treble-dodging major methods. A block of 21 methods gives 672 changes (ie 2 * 336). If these changes can be arranged in such a way that three bells (1,7,8) ring in every relative position twice, once with each parity, then an extent should easily drop out.  

I would love to see such an example composition (though would also appreciate any party-pooper saying why such an approach might not work).

Many great possibilities, though.  The future is bright.


===
40320 3-Spliced Major comp. MBD (no.3)
(20160 Plain Bob, 10080 each Helixoid H, Helixoid J)

234567
------
234657 P0s
237546 HHH H-HH HHH HHH HHHs
235674 P1s
234756 HHH H-JJ JJJ JJJ JJJs
234576 P0s
236745 JJJ J-HH HHH HHH HHHs
236475 P0s
235764 HHH H-HH HHH HHH HHHs
237456 P1s
236574 HHH H-JJ JJJ JJJ JJJs
236754 P0s
235647 JJJ J-H-H HHH HHH HHHs
------
235467 P0s
237654 HHH H-JJ JJJ JJJ JJJs
237564 P0s
234675 JJJ J-JJ JJJ JJJ JJJs
234765 P0s
236457 JJJ J-H-H HHH HHH HHHs
------
236547 P0s
237465 HHH H-JJ JJJ JJJ JJJs
237645 P0s
235476 JJJ J-JJ JJJ JJJ JJJs
235746 P0s
234567 JJJ J-H-H HHH HHH HHHs
------

This peal is built from 24 regular blocks, each the length of a Helixoid plain course. 
The P0s and P1s blocks are comprised of 15 courses of Plain Bob, built up as described below. 

The basic calls are:

- = 16 in Helixoid
s = 1256 in PB, 1456 in Helixoid
x = 123456 in PB

The methods are:

P = lead of Plain Bob
H = &-16-58-14-18-14-18-58-38.56.38-14-18-14-1458-36-1458-58-18-58-16.34.16-14-18-58-18-58-14-38-58, +14 
J = &-16.58-58-18-58-16.34.16-14-18-58-18-58-14.58-36-14.58-14-18-14-18-58-38.56.38-14-18-14-14.38-58, +14

Note that the Helixoid methods are both Double, and are course-splices of each other.

The basic 3-course part of PB is:

part = PPP[P]XPP PPPXPSX, XPPPPSP

This gives a "cyclic" part end 12367845, and is repeated 4 times to make the 15-course touch P0.
P0s is then P0 with the final plain of the last part replaced with a 1256 single.

P1s is the same as P0s, with the bracketed plain replaced by another X=123456 in the first part only.

Mark B Davies 23 June 2012
===



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