[r-t] Summary falseness of royal methods

Philip Saddleton pabs at cantab.net
Sat Feb 3 16:53:52 UTC 2007

I think of falseness groups as permutations of the coursing order, e.g.

B(In): reverse five adjacent bells 53246 -> 64235
B(Out): swap an adjacent pair: 35246, 52346, 53426, 53264

U1: rotate three adjacent bells: 32546, 52436, 53462
(and the inverse) 25346, 54326, 53624

Looking at it this way, the anomalies arise in Major and Royal, as a 
result of set of bells of the same size as the number of fixed bells 
keeping the same relationship as in the plain course or its reverse. 
E.g. we can think of U1 as being a bob: as well as W, M, H in Major, a 
Before keeps the tenors together, giving two more elements in group U:
65324 and its inverse 32465. On higher numbers these are not equivalent 
to a bob, and form a separate group, U2.

Considering B(Out), if we reverse the tenors, we get coursing order 
7853246, which is 8764235 reversed, equivalent to B(In). Similar 
relationships exist between the other in and out of course groups.

Coming to Royal, the only non-trivial permutations keeping four working 
bells coursing either forwards or backwards are
B(In): reverse five adjacent bells
D(In): reverse four adjacent bells
U2: move one bell over four or its inverse.

But reversing the order of five bells is equivalent to reversing all 
nine then four, and B(In) and D(In) are equivalent.

For Maximus and above we would need to keep at least six working bells 
coursing, which is clearly impossible.

Of course, if we are not restricted to tenors-together falseness there 
are many more groups as the number of bells increases.

Don Morrison said  on 03/02/2007 15:24:
> When applying the well known false course head "group" letters to
> royal methods some changes are needed, since the partitioning into
> sets of related false course heads was designed for major. Well known
> is that the in and out of course chunks that are always present
> together in major need not be together in royal; and that some of the
> in course chunks divide into two chunks which no longer have to appear
> together.
> Is it also true that some of the chunks that in major can appear
> separately, in royal will always appear together? For example, is it
> true that in royal the in course B (24365) and D (32546, 46253) chunks
> will always appear together? If so, what other bits, potentially
> separate in major, coalesce in royal?
> I guess another way to ask this, is in royal there really is a
> different partitioning into false course head groups than in major.
> Closely related, but different. What is it?
> And what happens at higher stages? It appears the in course B and D
> are no longer coalesced there.
> All this is, of course, assuming plain bob lead heads, usual symmetry,
> and bells 7 and higher fixed.


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