[r-t] Extent of caters

[Wyld Family e-mail]

I can confirm Eddie Martin's belief that the extent of Grandsire Caters can 
not be broken down into a set of complete plain courses.

It must, as has already been shown, be possible to break any extent down 
into a set of complete B blocks.  If an extent could also be broken down 
into a set of plain courses that would mean that a bob called at any lead of 
any plain course would have to generate a lead from within that set.  Since 
the set of all plain courses contains every change twice the identification 
of a set of plain courses containing the extent means that the remaining 
courses are also a set containing the extent.  Because a bob called in any 
course in the first set generates a lead within that first set the two sets 
would be separate and it would not be possible to get from one to the other 
using bobs alone.  If the course with rounds at backstroke is in the first 
set the course with rounds at handstroke must be in the second.  If the two 
sets were separate it would be impossible to compose a touch of Grandsire 
Caters, that comes round at hand, using bobs alone.  Since such touches are 
known and have been published it is clear that the initial proposition, that 
a set of plain couses containing the extent can exist, is wrong.

Colin

----- Original Message ----- 
From: "edward martin" <edward.w.martin at gmail.com>
To: <ringing-theory at bellringers.net>
Sent: Thursday, August 27, 2009 9:30 AM
Subject: Re: [r-t] Extent of caters


2009/8/13 Ted Steele <ted.steele at tesco.net>:

>
> What, if any are the special problems in composing the extent of Grandsire
> caters? I assume that because of the different nature of the rows, being 
> all
> quadruple changes the problems will be different to triples and the
> possibility of a bobs only extent does not exist; but can the extent be
> obtained with just two singles?

In both Triples & Caters the extent can be set out in so many bobbed
blocks. these are symmetric in structure such that any bell occupies
every possible positional relationship with the treble. The composer's
problem is how to link up these bobbed blocks by omits. In Triples
every row with treble at lead is related to 4 other rows with treble
at lead (6 in caters) such that to include every row, if 3rds occurs
when treble is in 1-2 (as from rounds) then the other 4 rows must each
have 3rds- conversely, if one has 7ths then all must have 7ths. (This
also applies to caters which has q-sets of 7 members).
The first apparent difference between triples & caters is that until a
single is called, in triples all hand strokes are negative & all
backstrokes are positive, which means that initially all the bobbed
blocks can be set out from positive lead heads, the blocks can only
flow in the one direction. But in caters until a single, all rows are
positive thus the lead block is truly reversible & could appear as
from either a lead head or lead end,
I say apparent difference because in actual fact there is a marked
similarity. Because we start Grandsire with rounds as a LH followed by
3rds we have created the q-set which decrees that wherever the other
members occur (125374968 etc) they also must have 3rds when treble is
in 1-2 - ie 125374968 could either occur as a LH or as a LE if bobbed
from 179826543 which suggests a difference between composing caters.
However, having established that 123456789 is a 3rds LH, we have also
established the direction of the flow of that particular bobbed block.
Without singles, it can only be joined to other bobbed blocks by omits
(9th place LE) which in turn establishes the direction of the flow of
each bobbed block that contains this plain q-set. Eventually every
bobbed block will be established as only being able to flow in one
direction and the argument used by Colin Wylde in his recent posting
holds true. As with triples, the q-sets have an odd number of members,
thus plaining an otherwise bobbed q-set will add an even number of
blocks to an odd. Odd + even cannot = even, so we cannot get the
extent of either pure triples or of pure caters using only plain &
bobbed leads.

I do not believe that the extent of Grandsire caters can be set out in
so many mutually exclusive plain courses as it can be in so many
bobbed blocks.

To come round two changes short of the in-course extent would require
the comp to fall into two isolated blocks. In one would be rounds in
the other would be each of the q-set members to include rounds viz
143628597, 163849275 etc. If having rung the one block instead of
running round with say a plain lead, we call a bob, this would  shunt
us to the other block where we would meet and pass each member of the
plain q-set until the second block is complete. We are now left high &
dry with the rows 132547698 123456789 being missing, with no way of
including them

Cheers
Eddie Martin

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