[r-t] Singles in treble-dodging minor

Philip Saddleton pabs at cantab.net
Tue Sep 21 18:53:02 UTC 2004

Richard Smith <richard at ex-parrot.com> wrote at 18:18:51 on Mon, 20 Sep 
>If a TDMM can be rung both bobs-only and in a composition
>involving singles, then there must exist a set of in-course
>lead heads and ends, L, that is only present in the
>bobs-only composition, and which is replaced by a set of
>out-of-course lead heads and ends, L', in the composition
>with singles.

An alternative way of looking at it is to construct a dual "method" with 
the same rows as the original, but with each pair of rows in a half-lead 
with the treble in a given place interchanged. The nature of the rows 
when starting from an out of course lead head will match those of the 
original starting from an in-course lead head.

Westminster   becomes
   124356-     123456+
   213465+     214365-
   123456+     124356-
   214365-     213465+
   241356-     421356+
   423165+     243165-
   421356+     241356-
   243165-     423165+
   423615-     243615+
   246351+     426351-
   243615+     423615-
   426351-     246351+

Now, if the method can be rung in an extent containing singles, this is 
equivalent to finding a splice with in-course leads of its dual (ignore 
the fact that it contains jump changes). Then it is easily seen how 
conventional splices can be used:

6-lead splices are precisely those where the pivot bell is in the same 
place in each pair of rows that are swapped, i.e. it must dodge at the 
same time of the treble.

3-lead splices are those when a pair of bells crossing at the half-lead 
both dodge at the same time of the treble (either together, or one with 
the treble, of necessity) - adjacent places may be introduced where this 
pair would cross.

Or, turning it on its head, a similar analysis to Richard's can be used 
to determine whether two methods can be spliced in-course. Each row can 
be rung in one of two ways, so picking a lead of one method rules out 
certain leads of the other, which in turn implies that rows contained in 
the ruled-out leads must be rung to the first method etc. Eventually we 
finish up with a subgroup of les/lhs that must all be the same method - 
if this is not the entire group then the extent may be partitioned.


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