# [r-t] An alternative approach to method extension

Mon Aug 30 11:21:22 UTC 2004

```Philip Saddleton <pabs at cantab.net> wrote at 14:04:26 on Sat, 14 Aug 2004
>The general formula for 2n bells is then
>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
(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)

Brocket TP
(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)

Thursday
(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