[r-t] 40320 Spliced Major (3m)
holroyd at math.ubc.ca
Mon Jul 2 21:56:57 UTC 2012
This sounds fascinating, Mark. Are you going to show us the composition?
On Mon, 2 Jul 2012, Mark Davies wrote:
> 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
> 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:
> 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:
> 21435687 56
> 21435768 58
> 21435678 5678
> 12435687 1256
> 12435768 1258
> 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.
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