[r-t] An alternative approach to method extension

[Philip Saddleton]

The recent amendments to the Decisions on Methods and Extension were 
limited in scope, mainly because the MC wanted a proposal that could be 
put at this year's CC meeting (next year is the first of a new 
triennium, and with the election of committees on the agenda is not a 
good time to introduce complex motions). There were other areas that we 
considered, but decided were too complex to deal with at the time, 
including hunt bell paths that are not currently covered, and increasing 
the lead length of principles. I hope that we will return to these in 
due course.

Prior to our discussions the year before last, I produced a paper that 
attempted to provide a general framework for all extensions by an even 
number of stages. I discussed some of my ideas with Rod Pipe after the 
HJ Dinner, after which he sent me his extensions of Thursday Maximus, 
and it was as a result of this that he got upset when our proposals 
didn't go far enough to allow these.

I have since had a few more thoughts, and now think that a quite simple 
formula can encapsulate all currently permitted extensions and many 
more. The idea behind all extensions is that every place made in the 
extension has its counterpart in the parent, and vice versa, although in 
some cases this relationship is many to one. The Decisions stipulate how 
these places are related at different stages.

Consider Cambridge:

Minor:
12-36-14-12-36-14-56
AB CD EB FG EB FG EH

Major:
12-38-14-1258-36-14-58-16-78
AB CD EB FGCD EB FG EB FG EH

Rather than using conventional place notation, specify each place as an 
ordered pair (change,place). Taking the lead end as change 0, the places 
are:

Minor
A (0,1)
B (0,2)(4,4)(8,6)
C (2,3)
D (2,6)
E (4,1)(8,3)(12,5)
F (6,1)(10,1)
G (6,2)(10,4)
H (12,6)

Major
A (0,1)
B (0,2)(4,4)(8,6)(12,8)
C (2,3)(6,5)
D (2,8)(6,8)
E (4,1)(8,3)(12,5)(16,7)
F (6,1)(10,1)(14,1)
G (6,2)(10,4)(14,6)
H (16,8)

In each case the coordinates on the same row are in arithmetic 
progression.
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.

Clearly the only permitted values are linear functions of n, and the 
differences are fixed. If the extension is to exist at an indefinite 
number of stages then the lengths of the series cannot grow any more 
quickly than that of the lead, and the places cannot grow more quickly 
than the number of bells, so these impose natural restrictions. For 
external places to remain external, strictly speaking the external 
places in B and E belong to separate rows. The requirements of (G)B 
should be retained. Need there be any others?

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