[r-t] A new category of splice?

[Dale Winter]

Hi Richard.

is it correct that you're finding a way to get the half full firkin from michaelhouse to GSM? Do you need a hand with this at all?

dale


On Jul 2, 2012, at 2:20 PM, Richard Smith wrote:

> 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
> 
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