[r-t] Singles in treble-dodging minor
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
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.
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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