[r-t] Monday afternoon puzzle

[Richard Smith]

Philip Earis wrote:

> OK folks, time for a quick starter for ten:
>
> - In what way is the seemingly unremarkable method Biddlesden Surprise
> Major (&-5-4.5-5.36-4-5.36.2.36.7,2) similar to Cambridge Surprise
> Minor?

Phil does seem to want to explain his somewhat cryptic
question, and, unsurprisingly, no one has managed to
guess the similarity.

Cambridge Minor is a regular method and has a regular
half-lead variant, Ipswich.  There's nothing particularly
unusual in that, except that Cambridge does not have regular
half-leads -- that is, back rounds does not appear at a
half-lead in the plain course of Cambridge.

Most half-lead variants (e.g., on six bells, Durham and
Beverley) have regular half-leads.  Changing an (N-1)ths
place half-lead to a 1sts place half-lead cycles the bells
one step through the coursing order, meaning that if one
method has a regular lead end, the other will too.  (Or it
will come round after one lead, or be a regular short course
method.)

So how does Cambridge work?  Instead of having all the
coursing pairs coursing at the half-lead, it has none of
them coursing:

  Cambridge  Ipswich

  624513     624513
  265431     265431
  256413     256413
  524631     524631
  256431     542361
  524613     453216
  542631     435261
  456213     342516

At the half-lead, the bells are coursing in the order 25436.
(This is easier to see in Ipswich than in Cambridge.)  The
important thing to notice is that the pairs coursing at the
half-lead are pairs separated by one in the lead-end
coursing order.  By changing the half-lead, the bells move
one position through the half-lead coursing order which is
equivalent to moving two positions through the lead-end
coursing order.

All pairs of six-bell half-lead variants must have either
the regular coursing order or this involution[*] of it at
the half-lead.

(I'm not convinced this is the correct use of the term
involution, but Brian Price uses it extensively in "The
Composition of Peals in Parts" to mean a pair of coursing
orders (or N-cycle permutations) where adjacent pairs in one
coursing order are M bells apart in the other, and
where M+1 and N are coprime.)


Phil's example, Biddlesden S Major, also has this property,
and is one of very few major methods to have it.

  Biddlesden      [Unnamed]

  45237618        45237618
  54273681        54273681
  54726318        54726318
  45762381        45762381
  54673281        47526831
  45637218        74562813
  45362781        74658231
  54326718        47685213

The half-lead coursing order is 3654728 and the bells in
every adjacent pair in this coursing order are two apart in
the lead-end coursing order.  (On eight bells, suitable
coursing orders can be found by taking bells one or two
apart in the regular coursing order.)

Unsurprisingly there are relatively few rung methods with
this property -- no doubt partly because separating coursing
pairs in this manner is very bad from a musical perspective.
However it is probably also partly because it is quite
difficult to move the bells through each to separate all of
the coursing pairs in the available space.

The difficulty in separating the coursing pairs is even more
apparent on higher numbers.  There are no rung surprise
royal methods that have this property, and it is quite
difficult to produce one.  Starting with Cambridge and
inserting pairs of adjacent places is often a good start,
but ends up producing truly hideous methods, for instance,

  &9.3-4-25-36-47-58-6-7.56.8.23456.1,2
  &9.3-4-25-36-47-58-6-7.56.8.23456.9,2

After a bit more thought, I came up with the following
creation, which at least has an interesting line and grid.

  &7.36.5.4-56-3.4-34.5.36-36.7.6-36.1,2
  &7.36.5.4-56-3.4-34.5.36-36.7.6-36.9,2

It's just a shame it has almost no musical merit.  Counting
its 4-runs gives 22 -- quite an achievement given that the
regular lead heads and ends alone give 13!

Richard