by B. L. Burrows
In his article on falseness in irregular methods (RW p.455 1979), Roger Baldwin described a method of establishing a correspondence between regular methods and certain irregular methods. At first sight it would appear that the procedure breaks down for some lead-heads such as 6574382. If we follow the prescription and define the coursing order by pairs of bells in 7-8 we obtain 8745362. It can easily be seen that in this order the bells affected by a conventional bob (6, 5, 7 at the first lead-end) are not consecutive in the coursing order and consequently the equivalence of the effect of the bobs on this order and the regular order 8753246 cannot be established. Furthermore in this irregular system only bobs at Home leave both 8 and 7 unaffected so that it would be useless to look for a composition for a regular method which would correspond to a composition in this system where 8 and 7 are unaffected. If however we redefine the coursing order by the pairs of bells in 3-2 we obtain 8432756 in which bells affected by the bobs are consecutive and the bobs would have an equivalent effect on this coursing order to bobs in a regular method on the regular order 7532468.
Suppose now that the irregular method in question has one false course head 2754638 which corresponds to coursing order 8572346. This coursing order has the same relationship to 8432756 as the coursing order 7642358 has to the regular order 7532468. This relationship for regular methods is "B" falseness, that is it corresponds to a false course head (f.c.h.) 2436578. Consequently a composition in a regular method which is true for "B" falseness can be used for the irregular method (fixing 6-8 in the irregular method instead of 8-7) provided that the order of the leads can be matched. In the correspondence established between the regular order and 8432756 we have
|8 corresponds to 6|
7 corresponds to 8
5 corresponds to 4 etc.
Thus the correspondence between the irregular lead-heads and the regular lead-heads is:
|2345678 corresponds to 3527486 (W)|
7236854 corresponds to 2345678 (H)
5728463 corresponds to 4263857 (M) etc.
What we therefore require is a composition for a regular method with first lead-head 6482735 which is true for "B" falseness, since then the first lead-head of the irregular method 6574382 corresponds in the sense that the bells which would be affected by the bob occupy the equivalent position in the coursing order. Care is needed if the regular composition relies on incidence of falseness but the composition can be used directly if the leads of incidence correspond. (For example if in the regular method we have "B" falseness H.V.W., W.V.H. the corresponding falseness in the irregular method needs to be M.V.H., H.V.M. to apply the composition automatically).
The algebraic approach introduced by Roger in his article works with the irregular coursing order q taken as 8432756 and the regular coursing order p taken as 7532468 provided that the formula P = KRK-1 is changed to P = KRK-1H where H is a factor introduced to take into account that 2345678 in the regular method is not necessarily equivalent to 2345678 in the irregular method. In the above system H = 3527486 since when R = 2345678 P = H. Similarly the false lead-head of the regular method F is related to the false lead-head of the irregular method G by F = KGK-1H. (The f.c. heads for the regular and irregular methods f, g respectively are still related by f = KgK-1 however K being defined in Roger's article as pq-1). It is also possible, of course, to use coursing orders, together with the position of an observation bell, without any reference to course heads. Perhaps this demonstrates the equivalence of the falseness in a better way. For example "D" falseness corresponds to the regular false coursing orders 8756423 and 8742356 (compared to 8753246) and the irregular orders 6845723 and 6872345 (compared to 6843275).
This analysis shows that there are alternative methods of defining coursing orders. For a regular method instead of taking the order of the bells in 7-8 (or 3-2 which is equivalent for regular methods) we can use the order in 3-4 viz. 8526734, or in 5-6 viz. 8365472. If one takes the latter and fixes 2-8 we can then derive the false coursing orders false with 36547. For Superlative the groups are K and R with false coursing orders 56374, 63745, 53476, 67354, 43675, 56734. This shows that courses with orders 7xxxx or xxxx7 are 24 true courses. (The out of course orders in K are irrelevant for this analysis). To obtain a composition one is now faced with the problem that consecutive bells in the coursing order are not those affected by conventional bobs. We must therefore use unconventional bobs. If a bob is defined with place notation 56 (instead of 36) when the treble dodges 7-8 up and an (in course) single is defined as 125678 at the half-lead the changes in the coursing order 36547 when the bob is called at the tenor position 5ths and the single at W are 53647 and 74563 respectively. Thus with bobs at 5ths, M, 3rds and singles W it is fairly easy to join the 24 courses. In this way we obtain 12 courses of the form 87xx2xx and 12 of the form 8xx72xx using conventional coursing orders. There are no 8-7's at backstroke and 8/2 are never together at the back. I will not give an example of one of the many ways of joining these courses lest it be thought that I am committing the, by now, unoriginal sin of publishing compositions by the back door!
The Ringing World No. 3616, August 15, 1980, page 718