[r-t] Extents in half leads - was Colin Wyld's extent of Yorkshire Major

[Richard Smith]

Mike Ovenden wrote:

> Excellent post; I look forward to the next instalment.

Thanks.  I've nearly got around to writing a second
instalment.  But for now, a couple of comments on the points
you raise.


[...]

> Suppose a half lead head x appears in an extent (it could actually be a lead
> head or lead end).  Then x.f1 and x.inv(f1) are precluded.  But if x.f1
> can't occur, what about its fifth row x.f1.a?  That MUST occur as the
> seventh row of the half lead x.f1.a.inv(b) = x.f1^2.  The same thing is true
> of any FHLH fi:  if x occurs as a half lead head, x.f cannot occur, and
> x.f^2 must occur.

It's perhaps worth being completely explicit that the reason
that this logic works is that there are precisely two places
in a half-lead in which any given row can occur.  For
example, the row 321456 must either be row #5 or row #7: it
cannot possibly be present as row #11.

This means that, unfortunately, a similar argument cannot be
applied to general variable-treble extents as there are no
restrictions on where a row can occur.  (For variable-treble
extents based around Hudson's group, which does provide the
necessary restriction, the argument can still be used.)

> Now construct two sets, L and R, based on that observation.  L will be what
> must be present; R will be what can't be.  Start L off by including rounds.
> Calculate some elements of R by transposing each element of L (initially
> just rounds) by each FHLH in turn.  Now calculate some new elements of L by
> transposing each element of R by each FHLH in turn.  And so on, stopping
> when no new elements are found.

In some ways this is very similar to the procedure I outline
in a post on 20th Sept 2004 on the use of singles in
treble-dodging minor.

  http://bellringers.net/pipermail/ringing-theory_bellringers.net/2004-September/000156.html

If we want to express your procedure a bit more formally, we
could describe L and R in terms of the set of FHLHs which
I've written this as F(M) where M is the set of rows in half
a lead.

  F(M) = { a b^-1 : a in M, b in M }.

We can then write

  L = < (F(M) \ {I})^2 >  and
  R = (F(M) \ {I}) < (F(M) \ {I})^2 >,

where I is rounds, the product of two sets is defined

  AB = { a b : a in A, b in B },

and the notation <X> means the group generated by X.

(As can be seen from the fact that all of these equations
use (F(M) \ {I}), the set of FHLHs excluding rounds, rather
than just F(M), this is an occasion where a definition of
falseness that excludes the trivial falseness (a half-lead
being false against itself) would be better.)

So how is this similar to looking at singles in treble-
dodging minor?  With half-leads, we know that the half-leads
with HLHs, L, contain exactly the same rows as those with
HLHs, R, this means we can write LM = RM.  This is exactly
what we were looking for with touches of treble-dodging
minor involving singles, except that there M is the whole
lead, not just half of it.  (In my Sept 20th email, I used
L' to denote what I've called R, here, and R for something
different, but otherwise the notation is the same.)

[...]

> In the case of Yorkshire S Major, L is a representation of A6.  I suppose
> there are likely to be Major methods where L is considerably smaller that
> maybe afford the composer more latitude?

Indeed there are.  Running through a slightly-dated
collection of 4580 rung surprise major, I found 754 methods
for which L is not equal to R and where the union of L and
R is smaller than the extent.  (Although this sounds
surprisingly many, a lot of these are Derwent variants from
the 1983 peal of 497 spliced.)

One obvious set of possibilities for L are the alternating
groups, and alternating groups on 3, 4, 5 and 6 bells all
appear.  (Examples are Derwent, Portsmouth, Cornwall and
Yorkshire respectively.)  However other, more interesting,
possibilities also exist.  One such method is Go Surprise

  &-56-16-58-1238-34-1238-1256-38,12

where the L = C_2 x A_5.  This is particularly interesting
because C_2 x A_5 is a mixed parity group (it is [7.10] from
Brian Price's "Composition of Peals in Parts").  This means
that, like Mike Ovenden's example with Delight Minor, this
doesn't have a traditional parity structure, but an extent
might well still be possible.

Another curious example is Vinales (a not-unpleasant
Bristol-over method).  This time L is an even-parity group
of order 20, [7.12].

Richard