[r-t] 40320 Spliced Major (3m)
Alexander Holroyd
holroyd at math.ubc.ca
Mon Jul 2 22:01:48 UTC 2012
Never mind, I see Philip sent it.
On Mon, 2 Jul 2012, Alexander Holroyd wrote:
> This sounds fascinating, Mark. Are you going to show us the composition?
> Ander
>
> 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
>> (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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