[r-t] Variable hunt treble-dodging minor
Mike Ovenden
mike.ovenden at homecall.co.uk
Mon Dec 12 21:29:14 UTC 2005
> We rang one of these, a rather nice London-over delight method, at
> the beginning of 2004, and called it Hudson Delight Minor (3-3.4-2-1.4-4.5
> lh2). The footnote said "This method gives all the work for all bells when
> rung with variable treble. Believed to be the first treble dodging method
> with this property to be rung"
> (http://www.campanophile.co.uk/show.html?Code=16251)
>
Performances of Jonathan Deane's composition for Disley and for Minster were
reported in RW for 1 Nov 1991 (p1051). The band were less restrained in
their footnote for the first of them: "Containing the first true 720 of a
variable hunt treble-dodging Minor method ever rung in the universe."
> Now fast-forward to this evening. I saw for the first time a copy of
> Jonathan Deane's 1993 book "Minor Curiosities". There's lots of very
> interesting stuff here
Do tell. Maybe not all at once though; I can only take so much
excitement ;-0
> The
> composition given for Disley (which is obviously atw) uses 14 bobs and 36
> singles, and is:
>
> 123456
> ------
> S 312456
> 325164
> - 325641
> - 325416
> 351264
> S 135264
> ------
> 5-part
>
> So my question is, how do these extents work?
I don't know (I'm not actually sure what the question is really asking),
but the following factors in the method construction surely contribute:
- The methods are formed from sections of four rows based on subgroups of
the group G generated by {x,12,34} - Price [6.34] (of order 8)
(but it's not obvious under what circumstances relevant bits can be joined
together)
- When the hunt bell is in 34, the same pairs appear together in 12, 34,
and 56 in either half of the lead, so these two sections form a coset of G
(which simplifies things a bit)
There are other methods with similar properties that don't yield an extent
(at least not with any reasonable choice of calls),
eg Morning Star TB and Shelford D. It's no great surprise to find that
they can be used in spliced extents.
Disley D Minster D Morning Shelford D
Star TB
123456 123456 123456 123456
x 214365 x 214365 x 214365 x 214365
34 124356 34 124356 34 124356 34 124356
x 213465 x 213465 x 213465 x 213465
16 231645 16 231645 16 231645 16 231645
x 326154 x 326154 x 326154 x 326154
12 321645 56 231654 12 321645 12 321645
x 236154 x 326145 x 236154 x 236154
36 326514 36 236415 16 263514 1236 236514
x 235641 x 324651 x 625341 x 325641
34 325614 34 234615 34 265314 34 235614
x 236541 x 326451 x 623541 x 326541
36 326451 36 236541 16 632451 36 236451
x 234615 x 325614 x 364215 x 324615
34 324651 34 235641 34 634251 34 234651
x 236415 x 326514 x 362415 x 326415
36 326145 36 236154 16 326145 1236 326145
x 231654 x 321645 x 231654 x 231654
12 236145 56 236145 12 236145 12 236145
x 321654 x 321654 x 321654 x 321654
16 312564 16 312564 16 312564 16 312564
x 135246 x 135246 x 135246 x 135246
34 315264 34 315264 34 315264 34 315264
x 132546 x 132546 x 132546 x 132546
12 135264 12 135264 12 135264 12 135264
720 Spliced Minor (variable hunt)
=================================
(s = 36)
123456 Morning Star
s 312456 Disley
325164 Morning Star
- 325641 Disley
- 325416 Morning Star
351264 Disley
s 135264 x5
720 Spliced Minor (variable hunt)
=================================
(s = 36)
123456 Disley
s 312456 Disley
325164 Shelford
- 325641 Shelford
- 325416 Shelford
351264 Disley
s 135264 x5
Composition in Parts
====================
Apart from being a 5-part, the composition is palindromic, so it's
effectively
a 10-part with half the parts rung backwards, and matching rows are cosets
of
the dihedral group on 5 - Price [5.04]. (The symmetry points occur at half
leads.)
Brian Price says a lot of useful stuff about this in "The Composition of
Peals in Parts"
http://www.ringing.info/bdp/peals-in-parts/parts-0.html etc
He considers the use of parts where the part heads form a group. We can
take that a bit further by considering the case where the basic building
blocks (sections) from which the extent is constructed are cosets of some
other group.
t p2.t p3.t ... pi.t ... pm.t
t.g2 p2.t.g2 p3.t.g2 ... pi.t.g2 ... pm.t.g2
t.g3 p2.t.g3 p3.t.g3 ... pi.t.g3 ... pm.t.g3
... ... ... ... ... ... ...
t.gj p2.t.gj p3.t.gj ... pi.t.gj ... pm.t.gj
... ... ... ... ... ... ...
t.gn p2.t.gn p3.t.gn ... pi.t.gn ... pm.t.gn
The array above represents some corresponding section in each part of an
m-part composition. The sections (written in columns) each comprise perms
making a coset of some group G = {rounds, g2, g3, ..., gn} of order n. The
part heads form a group P = {rounds, p2, p3, ..., pm} of order m.
Reading across the array we find cosets of P. Reading down we find cosets
of G. (Different sorts of cosets though. One set are left cosets, the
others are right cosets. Don't ask me which is which.) As noted by Price,
moving down the array we have transpositions, seen as post-multiplication by
elements of G, and moving across we have transfigures, seen as
pre-multiplication by elements of P.
An array of this form, which could succinctly be described by the notation
P.t.G, has particularly useful properties, such as
(A) If there are no elements of P having the same cycle structure as any
element of G, then the array has no internal falseness.
(B) If the array P.t.G has any perm in common with the array P.s.G, then
their entire contents are identical (but shuffled around a bit if s /= t).
(C) The *extent* can be partitioned into a set of mutually true arrays
P.t1.G, P.t2.G, ... if and only if there are no elements of P having the
same cycle structure as any element of G.
--------------------------------------------------------------------------
As a concrete example, let
P = {123456, 135264, 156342, 164523, 142635}
G = {123456, 214365, 124356, 213465}
G is consistent with sections having place notation x34x or 34x34 or x1256x
or 1256x1256.
P is (isomorphic to) [5.05] in the Price catalogue, with signature
5,1 N=4
G is [6.36], with signature
2,2,2 N=1
2,2,1,1 N=1
2,1,1,1,1 N=1
So, no common cycle structure, and an array based on these groups is not
internally false, eg
123456 135264 156342 164523 142635
214365 312546 513624 615432 416253
124356 132564 153642 165423 146235
213465 315246 516324 614532 412653
This suggests for example that a 5-part extent of Forward Minor is a
reasonable thing to look for. (And it happens to be achievable in
practice.)
--------------------------------------------------------------------------
Now consider the part heads in the variable hunt extent of Disley
P = {123456, 135264, 156342, 164523, 142635,
132546, 153624, 165432, 146253, 124365}
G = {123456, 214365, 124356, 213465}
P is (isomorphic to) the dihedral group on 5 [5.04], with signature
5,1 N=4
2,2,1,1 N=5
G is the same as before.
There is some common cycle structure, both groups containing elements with 2
2-cycles. There may be internal falseness in P.t.G, but it depends on t.
The following array is true:
123456 135264 156342 164523 142635 132546 124365 146253 165432 153624
214365 312546 513624 615432 416253 315264 213456 412635 614523 516342
124356 132564 153642 165423 146235 135246 123465 142653 164532 156324
213465 315246 516324 614532 412653 312564 214356 416235 615423 513642
On the other hand, this one isn't
231645 351426 561234 641352 421563 321654 241536 461325 651243 531462
326154 534162 652143 463125 245136 236145 425163 643152 562134 354126
236145 354126 562134 643152 425163 326154 245136 463125 652143 534162
321654 531462 651243 461325 241536 231645 421563 641352 561234 351426
And in case you're wondering, neither is
326154 ... nor indeed any array containing these 4 perms,
231645 ... but we knew that from Property (B), yes?
321654 ...
236145 ...
(So there's no point looking for a 10-part (or palindromic 5-part) extent of
Forward Minor based on P above because a number of rows (e.g. 231645) cannot
appear without making it go false. Indeed there's no point looking for
*any* 10-part (or palindromic 5-part) extent of Forward Minor, because
[5.04] is the only available group of order 10 and it has cycle structure
in common with G.)
In the Disley composition, blocks that can't appear in the form of x34x
sections CAN (and do) appear as x12x sections.
There is no overlap between the blocks that cannot appear as x34x
sections and those that cannot appear as x12x sections.
Mike
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