by Roger Baldwin
In his review of Eton S. Major, Marcus Sherwood asks for details of the techniques I use to relate falseness in irregular methods to the F-groups associated with regular methods. It is possible to transform not only all f.c.e's of irregular methods, but also the lead-ends and bobs, so that they correspond exactly with an equivalent system of f.c.e's, lead ends and bobs for regular methods.
The clue is the coursing order of the method. The effect of bobs on coursing orders in most irregular lead end systems is exactly the same as in regular methods, i.e. it affects the first three, middle three or last three bells of the coursing order. It follows that there is an exact correspondence between calls in irregular methods and those in regular methods; compositions for regular methods can be transposed by simply renaming the calling position and re-calculating the course-ends. Care must be taken over internal falseness (f.c.e's) however.
For example, Pritchard's well-known peal of Plain Bob Major can be transposed for the lead-end 3624785 as follows:
23456 W B 5ths H 23645 - X 3 23564 - X 3 25463 - 2 32465 - 3
The coursing order associated with 3624785 is 63245 (write down all pairs of bells in 7-8 at the lead-end). A bob at 5ths produces the coursing order 63452, and is thus equivalent to a M in Plain Bob.
Similar, but rather more difficult arguments can be applied to f.c.e's. Suppose we transpose compositions as above and wish to establish the f.c.e's for regular methods corresponding to those for the irregular method for which we are seeking a composition. Clearly since the relationship between coursing orders is preserved by bobs, we must seek a similar property for f.c.e's. The question is, which regular f.c.e. has the same effect on coursing orders as the irregular f.c.e.?
An example will illustrate the procedure. Consider the lead end 3462857, associated with the coursing order 52346. Suppose the method has f.c.e. 64352. Now the coursing order of 64352 is
64352 × 52346 = 56432.
We notice that the plain course coursing order (52346) must be transposed by 26543 to give the coursing order of the f.c.e (56432), that is
56432 = 52346 × 26543.
Thus the effect of the f.c.e. 64352 is to transpose the coursing order by 26543.
Now consider regular methods. The coursing order of the plain course is 53246, and if we transpose by 26543 we have
53246 × 26543 = 56423.
But 56423 is the coursing order of the course 46253. Hence the effect of f.c.e. 46253 in regular methods is to transpose the coursing order by 26543.
It follows that the f.c.e. 64352 for the irregular method corresponds to the f.c.e. 46253 for regular methods. This procedure can be applied to all the other f.c.e's of a method, for any lead end system (except those where bobs are not possible).
Thus there is an exact correspondence between the falseness, lead ends and bobs of irregular methods and regular methods, and composition problems for irregular methods can be transformed into a search for a composition for regular methods.
I personally find this procedure a little cumbersome, so I have generalised the above results and translated them into a procedure which can be applied to course ends directly. The basic theorem is best expressed algebraically; anyone acquainted with elementary algebra should have no difficulty in understanding it. The trick is to write transposition in exactly the same notation as multiplication in conventional algebra: "ab" means "a transposed by b".
Consider an irregular method, and let its lead-end be R, coursing order q. Let p denote the regular coursing order (53246). Let q-1 be the inverse of q, such that
qq-1 = 23456.
Then (a) The regular lead-end P, corresponding to R, the irregular lead-end, is
P = kRk-1
where k = pq-1
(b) The f.c.e., f, of a regular F-group, corresponding to an f.c.e., g, of the irregular method is f = kgk-1
(c) Composition true to regular methods with lead-end P and f.c.e. f (preserving the incidence of falseness) will be true when transposed for the irregular method.
I will conclude with an example. Consider Cardington S. Major (rung recently). The lead end is 3462857, with 8ths place l.e., 4ths place bob, coursing order 52346.
In terms of the above theorem,
R = 3462857.
q = 52346, q-1 = 34526.
p = 53246.
k = pq-1 = 32456.
k-1 = 32456.
The f.c.e's of Cardington are 32546, 64352, 25634, 56423.
| We have P | = kRk-1 |
| = 32456 × 3462857 × 32456 | |
| = 4263857 and |
32546 corresponds to 32456 × 32546 × 32456 = 32546
64352 corresponds to 46253
25634 corresponds to 53624
56423 corresponds to 65432.
Hence the f.c.e's of Cardington S. Major corresponding to the f.c.e's 32546, 46253, 53624, 65432 for a regular method (i.e. the F-groups d and k), and that regular method must have the lead-end 4263857.
I find it convenient to transform the falseness table direct, as follows:
Table of Table of falseness
falseness for equivalent
for Cardington regular method
I I
2345678 2345678
64352-I 46253-I
25634-II 53624-II
32546-VII 32546-VII
56423-VII 65432-VII
II II
3462857 4263857
64352-I 46253-I
25634-I 53624-I
IV IV
8756243 7856342
- -
VII VII
5237486 3527486
32546-I 32546-I
56423-I 65432-I
We see that the equivalent regular method is an mx method with falseness b and d occurring in the M, W and H leads, so we must choose a composition joining the Middleton courses. There are several available; we rang one by J. R. Mayne.
The Ringing World No. 3553, June 1, 1979, page 455