[r-t] 40320 Spliced Major (3m)
Mark Davies
mark at snowtiger.net
Mon Jul 2 21:20:33 UTC 2012
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
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