[r-t] "Magic" blocks of minor

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