[r-t] "Magic" blocks of minor
Richard Smith
richard at ex-parrot.com
Thu Aug 12 01:43:48 UTC 2004
Phil Earis has suggested that I should post something to the
new list about some of the compositions I have produced
using an idea by Philip Saddleton. Philip's original
composition is given below (rotated to remove the 65s at
back.)
720 Half-lead Spliced Surprise Minor (20m) Comp PABS
123456 df ef 123564 dj cj 152346 bj aj
143625 eh ah 134256 ci ei 164352 ah bh
163254 ai ei 145263 eg cg 156243 bj cj
143256 eh dh 135264 ci bi 134562 cg ag
125634 dj bj 156432 bh eh 165324 ah dh
156324 bi ei 145326 eg ag 124536 df bf
146235 ef bf 165432 aj dj 154236 bh dh
153426 bi ai 123645 dg ag 126435 dg eg
162453 aj cj 165243 ai ci 146523 ef cf
134625 cf df 132654 cf bf 136524 cg dg
------ ------ ------
123564 152346 123456
The method symbols refer to the over and underworks, so "dj"
is overwork "d" with underwork "j". None of the changes of
method at the half leads never affects the underwork and the
changes of method at the lead end never affects the
overwork. The under and overworks are as follows. All have
5ths place half-leads and 2nds place lead ends.
Overworks Underworks
a = George Orwell f = S2 (aka Rhyl)
b = Westminster g = Chelsea
c = London h = Kelso
d = Carlisle i = Cambridge
e = Cambridge j = Beverley
Some of the methods created from these over and underworks
have short courses -- for example a lead of "ci" brings up
the lead head 134256. The logical way of describing this
that the Central Council would be happy with, is as a lead
of Cunecastre ("ci" with a 3rds place half-lead) and then
saying that there is a half-lead call to get the 5ths place
half-lead. This is a shame as it detracts from the elegance
of the composition.
Philip's composition is a logical extension of the idea of a
6-lead splice as between, say, Cambridge and Beverley.
Each of these underworks 6-lead splice with each of the
others (as will any other underwork with 5ths at the
half-lead and -12- when the treble is in 3-4). The same is
true for the overworks.
As there is not time for the bell making 2nds at the lead
end to get to the back in time to make 5ths at the half
lead, the overwork assigned to a particular pivot bell will
never be rung with the underwork assigned to that bell.
This is why there are only 20 methods rather than 25. By
judiciously pairing over and underworks, the composition can
be made to join up into a 720 without needing calls.
It turns out to be quite simple to extend this idea to get
180 methods in a 2160 with a three-part structure -- i.e.
exactly half a lead of each method. In total there are 18
over and underworks with 5ths at the half-lead and with -12-
when the treble dodges in 3-4. These are given on Phil's
web page <http://ringing.8bit.co.uk/minor_grids.pdf> and
divide into three classes according to which pairs of bells
swap at the half-lead.
class S: S1-6, Mendip, Chelsea (8)
class D: D1-4, Seddlescombe, Kelso (6)
class B: Beverley, Surfleet, Cambridge, Burslem (4)
(These are underworks: the same applies for overworks. The
numbers in paretheses are the number of different underworks
in each class.)
To get a three-part structure, each extent needs two class S
frontworks, two D frontworks, and one B. This rules out
Philip's original composition, so I started with the
following variant.
720 Half-lead Spliced Treble Dodging Minor (20m)
123456 Ae 263451 Ee Overworks
163542 Eb 546231 Ab
124653 Ae 234651 Be A = Westminster
135264 Ba 435621 Ca B = Dover
142356 Cb 654231 Ab C = Oxford
125463 Ad 423561 Cd D = George Orwell
146325 Cb 524631 Eb E = Cambridge
162345 Ea 564321 Da
152634 Dd 354261 Bd
136245 Ba 536421 Da
152346 Db 465231 Ab Underworks
126543 Ac 253641 Dc
154362 Db 625431 Cb a = S2
143526 Ce 423651 Ae b = Mendip
123645 Ac 265341 Ec c = D2
162453 Ea 365421 Ba d = Kelso
132465 Bc 362541 Ec e = Cambridge
163425 Eb 426531 Db
153642 Dd 452361 Cd
146253 Ca 645321 Ea
165243 Ee 643251 Ce
145236 Ca 543621 Da
152463 Dc 563241 Ec
165432 Ee 632451 Be
135642 Bd 435261 Cd
146532 Ce 436251 Be
135426 Bc 325641 Ac
123564 Ad 524361 Dd
156432 Dc 532641 Bc
134625 Bd 234561 Ad
-------------------
123456
Repeat this twice replacing each above/below work with
a different one of the same type. For example,
First extent Second extent Third extent
A = Westminster F = Stotfold K = Averham
B = Dover G = London L = Collingham
C = Oxford H = Kent M = Maltby
D = George Orwell I = Bunwell N = Leckhampton
E = Cambridge J = Carlisle O = Hills
a = S2 f = S4 k = S6
b = Mendip g = Chelsea l = S1
c = D2 h = D4 m = D1
d = Kelso i = Seddlescombe n = D3
e = Cambridge j = Beverley o = Burslem
This gives 2160 in 60 methods with 36 changes of each
method.
Note that the six occurances of, say, overwork B occur with
each possible pair of underworks from a,c,d,e: a/c, a/d,
a/e, c/d, c/e and d/e. (B never occurs with b, likewise A
with a, C with c, D with e and E with d. D and E are
interchanged simply so that D and d can represent the same
piece of work used as an over and underwork respectively.)
These six occurances of B can be paired so that each pair
contains all four methods Ba, Bc, Bd and Be. I.e. a/c with
d/e, a/d with c/e, and a/e with c/d. One of these pairs is
left alone. In the other two pairs the B overwork is
swapped with the overworks in the other two extents: one in
the order B -> G -> L, the other B -> L -> G. This will
give half-leads of twelve different methods with B over the
treble: Ba, Bc, Bd, Be, Bf, Bh, Bi, Bj, Bk, Bm, Bn and Bo.
Repeat with the other five overworks. (The case of A is
slightly more subtle and the technique described in the
preceeding paragraph only produces nine methods. This is
because A is the overwork used to join the extents. Cutting
each extent in half at another A overwork and reassembling
in a different order solves the problem.)
This finally gives a 2160 in 180 methods and has no changes
of overwork at the lead end or underwork at the half lead.
2160 Half-lead Spliced Treble Dodging Minor (180m)
123456 Ae 263451 Je 123465 Ko 623451 Oo 123465 Fj 432651 Ej
164253 Jb 546231 Kb 163524 Ol 625431 Fl 163542 Eg 456231 Ag
124635 Ke 234651 Ge 124635 Fo 324651 Lo 124653 Aj 642351 Bj
135246 Ga 435621 Ha 136452 Lk 345621 Mk 135264 Bf 345621 Cf
143256 Hb 654231 Ab 143265 Ml 426531 Kl 142356 Cg 564231 Fg
125463 Ad 423561 Md 125436 Kn 423561 Cn 125436 Fi 532461 Hi
143652 Mb 524631 Ob 146325 Cl 465231 El 143625 Hg 254631 Jg
162354 Oa 564321 Ia 162345 Ek 654321 Nk 164532 Jf 654321 Df
154362 Id 354261 Ld 152634 Nn 354261 Bn 152634 Di 245361 Gi
134562 La 536421 Na 136245 Bk 356421 Dk 136254 Gf 356421 If
152346 Nb 465231 Fb 152346 Dl 524631 Al 156432 Ig 645231 Kg
126534 Fc 253641 Ic 126543 Am 526341 Nm 126534 Kh 256341 Dh
152634 Ib 625431 Cb 154362 Nl 546231 Hl 154362 Dg 265431 Mg
143526 Ce 423651 Fe 145326 Ho 243651 Ao 145362 Mj 634251 Kj
123654 Fh 263541 Eh 123645 Ac 265341 Jc 123654 Km 623541 Om
162453 Ef 635421 Lf 165342 Ja 365421 Ba 162435 Ok 635421 Gk
136524 Lh 365241 Oh 132465 Bc 362541 Ec 132456 Gm 635241 Jm
163452 Og 246531 Ng 163425 Eb 426531 Db 162543 Jl 654231 Il
153642 Ni 325461 Mi 153642 Dd 452361 Cd 152463 In 452361 Hn
142635 Mf 465321 Of 146253 Ca 645321 Ea 142653 Hk 465321 Jk
165234 Oj 236451 Hj 165243 Ee 643251 Me 164325 Jo 463251 Co
142536 Hf 453621 Nf 142563 Ma 543621 Da 145236 Ck 453621 Ik
152463 Nh 562341 Jh 152463 Dc 563241 Oc 153642 Im 652341 Em
163245 Jj 426351 Lj 165423 Oe 632451 Be 165432 Eo 362451 Go
134256 Li 253461 Ci 135642 Bd 435261 Hd 135624 Gn 435261 Mn
146532 Cj 264351 Gj 145632 He 436251 Le 145623 Mo 346251 Bo
135462 Gh 326541 Ah 132654 Lc 325641 Kc 135426 Bm 236541 Fm
123564 Ai 342561 Ii 123546 Kd 524361 Nd 123546 Fn 524361 Dn
152346 Ih 536241 Bh 156432 Nc 532641 Gc 156432 Dm 356241 Lm
134625 Bi 543261 Ki 134652 Gd 234561 Fd 132546 Ln 234561 An
------------------- ------------------- -------------------
123465 123465 123456
Repeating all this, but with different pairings of overworks
with underworks allows a 4320 in 225 methods to be produced.
This is every combination of the 15 overworks and 15
underworks selected, but now contains many methods twice.
The three missing underworks are all lead splices with
methods present in the extent and this can be used to
increase the number of methods to around 270, but to all 324
(= 18*18) methods, I had to add a seventh extent making a
round 5040 in 324 methods.
This idea can be applied to higher stages as well. Ander
Holroyd and I produced a 5040 of 70 Plain Triples based on
this idea, but instead of the 6-lead splices based on the
group A_4 that were used in Minor, we used 24-lead splices
based on the group S_4 x C_2. The practical effect is that
instead of one bell making seconds at the lead end, there
are two bells making thirds as at a bobbed lead of
Grandsire.
I also used the idea to produce a 40,320 of spliced surprise
major but it is a bit inelegant as there is a double change
every half lead and lead end. Nevertheless it is possible
pair over and underworks such that it all joins up without
calls (and without all the methods being Derwent variants).
Richard
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