[r-t] An alternative approach to method extension
Philip Saddleton
pabs at cantab.net
Sat Aug 14 13:04:26 UTC 2004
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/
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