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

[Mark Davies]

Yes, it seems the desire to get an extent of spliced Plain and Helixoid 
Major was the real motivation behind Philip's initial request for a list 
of all Helixoids. It struck me as an interesting challenge. It is in my 
experience rare to be asked for an extent of Major, and rarer still for 
there to be a real danger of any resulting composition being rung!

However, it quickly became obvious that the very different construction 
of the two method types made the task pretty difficult. As Philip has 
pointed out, it is easy to find a touch (3M) which puts the triplet 
(1,2,3) into every possible position, just like the first three leads of 
the Helixoid, but sadly this does not at all imply that the remaining 
five bells produce changes which allow the two 336-row blocks to be 
spliced together. It's easy enough to see why this might be: in the 
Helixoid, the triplet is treated symmetrically, with (1,2,3) falling 
into every possible position up to self-permutation within the 
half-lead. Successive half-leads simply run through all the permutations 
of (123). By contrast, in a Plain method the triplet is treated as a 
hierarchy of hunts, with the treble visiting every place within a 
half-lead, then (12) filling every one of their positions in the 
half-course, and finally the entire triplet completing the set only once 
the full touch is complete.

In fact, a little more thought showed that no Helixoid existed which 
could be course-spliced with the 3M touch of Plain Bob. I set out my 
reasoning to Philip as follows:

1. The PB8 touch certainly includes the plain course, so any
course-spliced Helix method must also include these changes.

2. Immediately after the Wrong, the PB8 plain course contains change
31254768. Where can this occur in the Helix?

3. It must be a leadhead, since its prefix is a positive 3-cycle on
(123), and all such are by definition leadheads in a
(123)(45678)-Differential.

4. But now we have a contradiction: the suffix 54768 is not a leadhead
in any method with a 1- or 5-cycle on (45678).

A month went by with no real progress, until I chanced upon the idea 
that a different touch of Plain Bob might work better. What it turned 
out I needed was a block where the triplet (123) followed a fundamental 
rule of the Helixoids: given arrangements of (123) always occur at the 
same place in the Helixoid lead, and so have the same sign. Or more 
precisely, they have the same sign if you rotate (123), but the opposite 
sign if you swap a pair from (123), since that gives you a row in the 
second half-lead. If I could find a touch of Plain Bob (or any plain 
method) where all positions of (123) occurred, and all obeyed this rule 
of signs, then I reasoned it might become possible to find 
course-splicing Helixoids.

I soon found touches which obeyed this property, but *only* in methods 
where the half-lead was a true plain hunt (that is, Plain Bob, Reverse 
Bob, Double Bob and Plain Hunt itself) and *only* if I used special 
calls such as 1278 or 123456. In fact, nothing else worked other than 
three full hunting courses in the positive coursing orders 32....., 
3..2... and 3....2. or their negative reverses. Can anyone explain why 
the "rule of signs" leads to these two results? We were surprised, for 
instance, that nothing was possible with a method like Double Norwich.

Despite this progress, course-splicing Helixoids were still not coming 
forward. This led me to think about weaker splices, and the obvious line 
of enquiry was the weakest/largest of all: if I added a further 
restriction, that a row with (123) in a given arrangement in the Plain 
Bob must not only have the same sign as its rotations, but must also 
match the sign of the same row in the Helixoid plain course, then I 
could partition the extent into two sets, with Plain Bob being rung from 
positive 123..... course ends, and the Helixoid from negative 123..... 
course ends. Truth would be guaranteed under these conditions.

It turned out that millions of Helixoids matching the Plain Bob touch in 
this way did exist, but unfortunately none of them appeared to satisfy 
Philip's other requirements: to be Double, and to have one or more 
"splice-sister" Helixoid methods which would go together in (for 
instance) a course-splice. Just exhausting the list of Helixoids was 
very difficult, since for every method I found in my initial searches 
there are maybe millions of trivial variations, with either the back 
five or front three permuted amongst themselves by alternative place 
notations.

Expanding all these TVs and carrying out expensive checks for congruency 
with the PB signs, plus searching for existence of splice sisters, 
looked intractable. However in the end I found it was possible to 
construct a processing pipeline which could tackle the job. First I 
searched the Helixoid quarter-lead to identify all Double methods up to 
"Trivial Variation". (Of course, I'd already done that bit). The second 
stage of the pipeline was to expand TVs which generated different signs 
for the (123) arrangements. For example, a method starting:

12345678
21436587 x

Has different six TVs in that change, because there are six place 
notations which keep the same pattern XX.X.... for the front and back five:

12345678
21435687 56

12345678
21435768 58

12345678
21435678 5678

12345678
12435687 1256

12345678
12435768 1258

12345678
12435678 125678

But of these, only three affect the sign of the (123) row:

-> place notations 56 and 58 both generate a negative row, hence 
different from the cross change, but with 123 in the same position;

-> place notation 125678 also generates a negative row, but here 
although 123 are in the same position, they have undergone a pair swap, 
meaning the sign of the overall row needs to be the opposite of the 
original configuration.

-> place notations 1256 and 1258 also have a pair-swap on (123), but 
generate a positive not negative row.

In practice I excluded more than two consecutive places, so 5678 and 
125678 were not considered. This means that in general I limited the TV 
expansion to at most three distinct types of place notation per change, 
whilst still checking every possible type of Helixoid-to-PB row-sign match.

Despite the pruning described above, stage 2 was the slowest part of the 
pipeline, and the full search has only just completed.

Once I had filtered down the expanded methods to that subset which was 
"sign-congruent" with my PB touch, I ran stage 3 of the pipeline. Here 
the remaining place notations were expanded (so 56 and 58 in the above 
example would become distinct methods to consider), and looked for any 
method for which a good set of splice-sisters existed. This immediately 
threw up the outstanding examples of my "H and J" methods, which you can 
see in the finished peal composition. Interestingly enough, nothing else 
particularly worthy appeared even after exhaustive pipeline searches in 
smaller spaces, and many days' running of the full pipeline.

Now I have the full pipeline results I will investigate to see if this 
conclusion is unaltered, and H and J really are the shining Helixoid 
splice-sisters, or if other similar or even better examples exist.

I hope that this explanation of my recent endeavours has been of 
interest, and I'd be fascinated to hear if anyone can devise an 
explanation for the "plain hunt" conundrum described above.

Part of the fun of this particular composing exercise has been the 
opportunity it has given me to play with a (relatively) new programming 
language, Scala, and to discover how superbly well-suited it appears to 
be to the task of computer composition. It was quite a revelation. I'll 
write a bit about that in my next email.

MBD