[r-t] Multi-cyclic composition

[Jonathan Agg]

For a while I've been interested in cyclic compositions which mix up the
parts, by changing the bells which are pivot bells mid-part. Examples
include this one by DJP, which changes twice, once using a link method, and
again using mega-tittums: https://bb.ringingworld.co.uk/view.php?id=387809.
His particles compositions do this too as 12-parts.
A fairly natural conclusion to this idea is a composition where the cyclic
part changes every lead! I set myself the challenge of finding a
composition that I thought might be ringable, both in terms of learning,
and also importantly enjoyable enough to stand a chance of persuading
enough others to try and ring it in the future.

I started out by looking at the existing link methods, and labelling them,
firstly by the standard number of leads of Plain-bob and second by their
cyclic jump. For example, Slinky was labelled by (+5, +4) as its lead end
(14523ET90786) is the fifth leadend of Plain Bob starting from
167890ET2345, which is "rounds + 4". Other examples include Cyclone,
designed to move from 1ET907856342 to 1796E820ET53, which was labelled:
(-4, +4). Like other good link methods, these are musical, palindromic and
well-structured.

Restricting the problem to only using palindromic methods and only ringing
the 121 leadheads found in the classic cyclic 6 composition, the first step
was to find a plan which solved the "11 queens" version of the "8 queens"
problem, making sure each part included each plain bob lead end, and each
cyclic part once. I tried using the existing methods, but thought there
aren't currently enough.

>From the 121 leadheads, I found the possible leadends by 'undoing' sensible
leadend changes from each of them. I then looked for all the possible
methods from a leadhead to a leadend which were possible with palindromic
symmetry, i.e. requiring pairs swapping, or pivot bells. Interestingly, but
after investigation predictably, some of these options corresponded to
multiple relevant methods if rung with different leadends (e.g. a method
from leadhead 1234567890ET to leadend 12TE09876543 can be rung with a 2nds
lead end to give 12ET90785634 and also a 12ths place lead end to give
1T20E8967453).

This raised my hope of not having to invent 11 methods, with clear
associated benefits in reducing the method learning! However, it will still
need care to make sure any leads which reuse a method were conducive to
music in all of them, probably by requiring the pivot bells to be sensible.

These candidate methods can be used as edges in a graph with the leadheads
as nodes, and this can then be searched. I tried to find frameworks with 6
methods, in the hope the framework would itself be palindromic, but didn't
find any. The smallest number of methods used I found was 8, with lots of
candidate frameworks. Very fortunately, it felt like the gods were smiling,
and I stumbled across frameworks like the one below where the pivot bells
in the leads with repeated methods corresponded to the 2 and the tenor for
the cyclic part for each of the leadheads. The only issue was the fairly
comedic middle method, with 9 pivot bells.

This lists the lead head and the lead ends:

11,1 5,11 1,10 4,9 2,8 3,2 8,3 9,4 7,5 10,6 6,7 11,8	(8 methods)
1234567890ET
1E098765432T
10E89674523T (A 1E098765432T,+T, pivot:T)
1TE325476980
1T3E52749608 (B 1T3E09876542,+2, pivot:3)
18694725E30T
1684927E503T (C 1T09876543E2,+T, pivot:E)
1T0395E72846
1T30597E8264 (D 1T0E59876342,+2, pivot:5)
108T624E3957
1806T423E597 (E 1492E0T83657,+T, pivot:8)
1806T324E597
108T63425E79 (F 1234587690ET,+6, pivot:lots)
19E7254638T0
197E5264830T (G 1T0E89756342,+2, pivot:7)
1T30584627E9
13T504826E79 (D 1T0E59876342,+T, pivot:5)
19E728406T53
197E8204T635 (H 1T0E87659342,+2, pivot:9)
15736T4028E9
175634T20E89 (B 1T3E09876542,+T, pivot:3)
19E02T436587
190ET2345678 (C 1T09876543E2,+2, pivot:E)


There are thousands more, so there may well be a nicer one hiding in there
somewhere, particularly if 11 methods are allowed. This lists the ones
using up to 9 methods:
https://drive.google.com/drive/folders/134is8T8uoWd54U8xWtjXJ__f5Oi8p4hd

Coming up with methods was a challenge given the opposite pairs are
unfamiliar, and there's also no possibility of rollups at the half lead. I
had various goes at this, and eventually came up with these ones which I'm
broadly happy with, though can certainly be refined further:
https://complib.org/composition/76082

My favourite one is the first method, which gets to megatittums over the
half lead with lots of plain hunting, with the tenor being pivot, then 2,E,
3,0, 4,9, 5,8, 6,7 being opposites. Normally a method which is a cyclic
shunt of 1 is less popular as it's not as disruptive music-wise as shifting
by a larger amount, but in this composition this concern goes away!

The issue with the middle method can be reduced by ringing a silly
Bastow-style method with places which fairly naturally has the desired
effect in a very short space of time.

If anyone has ideas for how to improve any of these methods, all feedback
is gratefully received.

Overall, I'm pretty happy with what I've managed to come up, especially in
structure, though some of the methods are a bit too fiddly in places.
Learning these unfamiliar methods would be tricky, but I think possible.
Perhaps in another year's time...

Jonathan
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