[r-t] An alternative approach to method extension

[Philip Saddleton]

Philip Saddleton <pabs at cantab.net> wrote at 14:04:26 on Sat, 14 Aug 2004
>The general formula for 2n bells is then
>(lead=8n)
>A (0,1)
>B (0,2)+(4,2)[n]
>C (2,3)+(4,2)[n-2]
>D (2,2n)+(4,0)[n-2]
>E (4,1)+(4,2)[n]
>F (6,1)+(4,0)[n-1]
>G (6,2)+(4,2)[n-1]
>H (4n,2n)
>where +() gives the difference between successive elements of the 
>series, and [] the number of elements.
>
>To allow for asymmetric methods we ought to repeat rows B-G with change 
>i replaced with -i (changes are equivalent modulo the lead length). 
>Note that the half-lead and lead-end places in B and E appear in two 
>series. This is necessary for extensions that allow these to be repeated.
>
>Any extension complying with the existing Decision (G)B-D can be 
>expressed in these terms. It remains to determine what restrictions 
>should be placed on the formula.

We could attempt to reproduce the effect of the existing decisions - 
this would be quite easy where the lead length (and hence the number of 
places) is fixed, harder where it expands - but I prefer to keep things 
as general as possible.

Allowing each place to extend completely independently probably gives 
too much freedom. Places in the same change must be well-behaved for an 
indefinite extension, but in order to get some similarity of blue line 
it is necessary to consider at least adjacent changes. We cannot require 
that places in the same or adjacent changes remain so, or separate 
extension above and below a hunt bell is not possible: however, where 
they do remain adjacent, we should not allow, say, 14.36 to extend to 
14.58, since changing the order of the places has a significant effect 
on the character of the line. Perhaps the simplest way of achieving this 
is the following:

The direction of the blue line in the changes either side of each place 
must remain fixed in any extension.

Note that such a restriction has no effect for right-place methods, but 
would disallow some existing extensions.

Examples of extensions that are not currently permitted:

Bristol S
(stage=2n,lead=8n)
(0,1)
(4,1)+(4,2)[n-1]
(9,1)+(4,0)[n-3]
(9,4)+(4,2)[n-3]
(4n-5,1)
(4n-5,2n-4)
(4n-2,1)
(4n-2,2n-4)
(4n,1)
[+reflection about change 0, rotation about change 2n]

Brocket TP
(stage=2n,lead=8n)
(0,1)
(0,2)
(2,1)+(2,0)[n]
(2,2n)+(2,0)[n]
(2n,2)
(4n-2,1)+(-2,2)[n]
(4n-2,4)+(-2,2)[n-1]
(4n,1)
(4n,4)
[+reflection about change 0]

Thursday
(stage=8n+4,lead=16n+2)
(2,3)+(2,2)[2n]
(4n+2,1)+(4,0)[n+1]
(4n+2,4)+(2,2)[2n]
(4n+4,3)+(4,0)[n]
[+reflection about change 1, mirror symmetry]

Are there any unsatisfactory possibilities that would require further 
restrictions to prevent?

-- 
Regards
Philip
http://www.saddleton.freeuk.com/