[r-t] Gangnam, etc.

[Robin Woolley]

Richard Smith said :"I would be extremely interested to see a proof that 
there is no true multi-extent composition of Gangnan with an odd number 
of extents. Are you sure it's true?"

I can't come up with a proof - but I can explain why I think it's likely 
to be true.

1. For 'normal' methods, extents are available which involve ringing 
some parts of some courses backwards - but this is OK since this is what 
a symmetric method means - it's the same starting from the end of the 
lead backwards. An example of this is the lead the 6th lead of the 
standard 720 of PB Minor. This is from the course '2453' but backwards. 
(Symmetry here  = palindromic).

2. The falseness in asymmetric methods (non-palindromic) means that 
there are groups of leads (and the set including 23456 is exactly what 
we mean by a group) which have to be rung in one way only. In 
Bishopthorpe, the group is {23456, 43256} and it turns out that this 
group, along with the lead-end order of the method (Group K - 'F' in new 
money) yields extents. (The lead-end order is an important part of the 
equation).

3. For those asym. methods which cannot have an extent, Bailey's 1440 in 
the diary works. I worked out, when lying in bed one afternoon, that if 
we had a 1440 in which each row occurs at both hand- and back-stroke, it 
should be true. The reason is as follows. Consider the method in its 
half-leads. This type of comp. has each half-lead rung as the first half 
of the lead and each half-lead rung backwards as the 2nd half. This is 
permissible as it is easy to show that any row in a *half-lead* can be 
in only one half-lead. Therefore, in the 1440, we have every row rung in 
the top half of a lead somewhere, and in the bottom half of some 
half-lead. All that remained was to search for a suitable comp. This was 
the only one in the on-line collection and works because it contains 
just two singles - 720 rows apart. (It is possible to construct those 
with more than two singles but care needs to be taken as to the actual 
positioning so every lead end group has a slightly different version.)

4. How about the 5040? Since *I expect* the method under construction to 
have 'required lead heads' it must have every treble lead-end row to 
appear as a lead-head also, and 7 is not divisble by 2.

These, then, are the basic 'bones' of my thoughts on the subject.

Best wishes
Robin

PS -  the following is the footnote to a 5760 by Don Morrison - "The 
unreduced callings contain every row four times each at handstroke and 
at backstroke, with half the 65s appearing at backstroke. To avoid 
falseness care must be exercised in choosing which singled-in courses to 
omit when reducing the length to 5,040."