[r-t] Gangnam, etc.
Robin Woolley
robin at robinw.org.uk
Wed May 14 17:37:48 UTC 2014
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."
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