[r-t] A new category of splice?

[Richard Smith]

Philip Earis wrote:

> I do like this half-lead spliced concept.  Has it been 
> documented before?  Are there rung methods which have this 
> feature?

I don't know whether it's been documented before, but there 
are plenty of Surprise Minor examples out there.  Obviously 
a half-lead splice is a special case of a lead splice, in 
just the same way that a two-lead splice is a special case 
of a six-lead splice, and so many examples of half-lead 
splices are known as lead splices and the fact that they are 
also half-lead splices is not so well known.

Probably the best example of this is Allendale and 
Fryerning.

   &-34-4-2-3.2-2.3,2  (Allendale)
   &-34-4-2-23-2-3,2  (Fryerning)

The following rows appear in Allendale when the treble is 
dodging in 5-6:

   234165   8
   324615   9
   326451  10
   234615  11
   236451  12
   326541  13

If we swap rows 9 and 11, we get Fryerning.

Obviously you can do the same above the treble with the 
5-5.4 and -5-45 overworks.  If you want to avoid more than 
two consecutive blows in one place, these are the only two 
types of half-lead splice in treble dodging minor.

More generally, though, there are lots of odd splices if you 
allow half-lead changes of method.

The lead heads and lead ends of the unit affected by the 
splice must form a group.  For a minor method, the group 
must be a subgroup of S_5; and for a treble dodging minor 
method, a subgroup of A_5.  The usual requirement to only 
change method at the lead head coupled with palindromic 
symmetry means the group must contain an element conjugate 
to the half-lead change, which typically means an even 
parity element of order two.  An examination of the table of 
groups in Brian Price's 'Composition of Peals in Parts' 
shows that the only such groups for TDMMs are the ones of 
order 12, 10, 6, 4 and 2.  These correspond to the six-lead 
splice, the course splice (and five-lead splice), the 
three- and two-lead splices, and the lead splice.

But allowing half-leads to be changed independently gives us 
a lot more types of splices, and also makes it much more 
likely that two more methods do splice.  However, the more 
complex half-lead splices are much harder to exploit than 
the full lead ones.  Take Cambridge and London, for example. 
These have a two-half-lead splice which in principle might 
allow a composition fo half-lead spliced Cambridge and 
London, though I can't immediately see how to exploit it in 
a composition.

RAS