[r-t] Triples differentials (2)

[Philip Earis]

The only previously rung triples differential is Upham, a 3-4 helixoid differential, with bells 1,2 and 3 ringing in all 7*6*5 = 210 possible relative orders in their cycle (giving 210/3 = 70 change leads, equating to a plain course of 840 changes).  The method is therefore designed so that the extent can be simply rung in six courses.



Upham is by no means unique - there are many possible alternative "helixoid" 3-4 triples differentials that have similar extent-generating possibilities.



Indeed, in November last year Richard Smith found 14369 such methods consisting of only triple changes and with leadhead notation 147 (it's not possible to get any pure triples methods) and palindromic symmetry. Allowing 147 changes in the lead as well gives 361440 methods, which range from the very static to very dynamic.



Perhaps Richard could (when the file attachment problem is sorted!) upload the file to here?



I was especially interested in splicing together some of these differential methods to produce an extent.  A simple splice for this purpose is the course splice, and happily there are a number of possibilities. Here are 12 mutually-course-splice triples differentials that Richard found:



> &5.1.3.5.1.7.5.1.3.5.7.5.3.1.7.1.5.3.1.3.1.3.5.3.5.3.4.5.1.7.3.5.7.3.4,1

> &5.1.3.5.1.7.5.1.3.5.7.5.3.1.7.1.5.3.1.3.1.3.5.3.5.3.4.5.1.7.3.5.7.3.5,4

> &5.1.3.5.1.7.5.1.3.5.7.5.3.1.7.1.5.3.1.3.1.3.5.3.5.3.4.5.1.7.3.5.7.3.7,4

> &5.1.3.5.1.7.5.1.3.5.7.5.3.1.7.1.5.3.4.3.4.3.5.3.5.3.4.5.1.7.3.5.7.3.4,1

> &5.1.3.5.1.7.5.1.3.5.7.5.3.1.7.1.5.3.4.3.4.3.5.3.5.3.4.5.1.7.3.5.7.3.7,4

> &5.1.3.5.1.7.5.1.5.1.3.1.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.4,1

> &5.1.3.5.1.7.5.1.5.1.3.1.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.4,3

> &5.1.3.5.1.7.5.1.5.1.3.1.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.5,4

> &5.1.3.5.1.7.5.1.5.1.3.1.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.7,4

> &5.1.3.5.1.7.5.1.5.4.3.4.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.4,1

> &5.1.3.5.1.7.5.1.5.4.3.4.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.4,3

> &5.1.3.5.1.7.5.1.5.4.3.4.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.7,4



Some of these are just trivial half lead / leadend variants, but others are not, and indeed "feel" quite different.  As an illustrative example of a very simple spliced extent see this arrangement of 5040 triples in 3 methods (expressed in Sirilic code):



==
7 bells

peal=2((11p1,b1),(11p2,b2),(11p3,s3))

conflict=

p1=meth1,+4
b1=meth1,+6

p2=meth2,+4
b2=meth2,+6

p3=meth3,+4
s3=meth3,+1236

meth1=&5.1.3.5.1.7.5.1.3.5.7.5.3.1.7.1.5.3.1.3.1.3.5.3.5.3.4.5.1.7.3.5.7.3.7
meth2=&5.1.3.5.1.7.5.1.5.4.3.4.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.7
meth3=&5.1.3.5.1.7.5.1.5.1.3.1.3.1.5.1.7.1.3.5.7.5.1.3.5.3.4.5.1.7.3.5.7.3.5
==



I am especially interested in course splices - is there a "general theory" of necessary conditions to constitute a course splice?  I'm thinking of an almost algorithmic list - ie the ways of substituting notations, jumping between parts of a lead / course etc that will generate a course splice method.



This has all resurfaced and become very topical, because I've been in recent contact with MBD about interesting possibilities for an extent of spliced major.  Mark has made some fantastic discoveries and progress, about which more soon (hopefully). But the question of splicing is key here, and indeed what I think is a new type of splice - both incredibly simple in its manifestation but really quite clever - has been discovered.  Again, hopefully more soon.



This is what is called a teaser... :)













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