[r-t] Extent of Cambridge-over minor methods
Richard Smith
richard at ex-parrot.com
Thu Jun 23 17:28:13 UTC 2005
Philip Saddleton wrote:
> Can the same courses be used with Carlisle backwork to give an extent in
> nine methods (i.e. excluding Northumberland and Whitley)? If not, why not?
No.
If it were possible, then it would certainly be much harder
than for Cambridge methods, as the Cambridge-over methods
without sixth-place variants (York and Durham) are the same
ones the are difficult to get into a composition on this
plan. This is good because only two plain leads -- one for
each method -- are needed. For Carlisle-over methods, the
opposite is true -- the methods without sixth-place variants
(Carlisle, Chester and Munden) are those which are easy to
get into the composition. This means the other methods all
have to be squeezed into a small number of leads.
The key to the original composition is the slightly unusual
slice bewteen Beverley, Durham and Bourne. One way of
looking at this is with the idea of an "imaginary method".
(This is the concept that Michael Foulds introduces in
"Intermediate Spliced TB Minor" to explain grid splices.
Basically, an imaginary method is an ordinary method but
with the half-lead change replaced by some jump change to
join together two otherwise-unconnected lead ends.)
Taking a lead of Beverley and changing the half-lead change
so that it produces a J lead head, gives you an imaginary
method, "J-Beverley", that three-lead splices with Bourne
with 3,5 fixed. (Note that 3rd and 5ths place bells do not
do the same work in both methods, but when the 10th and 12th
rows are swapped to account for one method being right-place
and the other being wrong-place, they do do the same work.)
J-Bv Bo
...153 + ...153 + 8
...513 + ...513 + 9
...531 + \ / ..5.31 - 10
..5.13 - X ..5.13 - 11
..5.31 - / \ ...531 + 12
..3.51 - ...351 + 13
..3.15 - ..3.15 - 14
...351 + ..3.51 - 15
...315 + ...315 + 16
...135 + ...135 + 17
This is all well and good, but how do you get three leads of
J-Beverley into an extent without any other jump changes?
Let's start with an extent of Beverley and take the six
leads with a pair of bells in 3,5. (There are six rather
than three because 3,5 are not a crossing pair in Beverley
as they are in J-Beverley.) Six of the twelve half-lead can
joined together to produce J-Beverley leaving another six
half-leads.
Beverley
1.3.5. \
.1.3.5 |
1...35 |
.1..53 |
..1.35 |
...153 | join to give J-Beverley and thence Bourne
..1.35 |
...153 |
...513 |
...531 |
..5.13 |
..5.31 /
------
...531 \
...513 |
..5.31 |
.5..13 |
5..1.3 |
.51.3. | join to give Durham
.5.1.3 |
5.1.3. |
51...3 |
15..3. |
51.3.. |
153... /
The six remaining half-leads, which have the fixed bells in
2,3, can be joined with a first place half-lead which turns
2,3 into a crossing pair. This gives three leads of Durham
(or York as they are three-lead splices on this pair).
As Beverley, Durham and Bourne have different lead heads,
the effect of making a single replacement like this is to
split some of the courses into two round blocks.
For example, if 5,6 are the fixed bells, then the three
tenors together courses split as follows.
123456 Bv 142635 Bv
156342 Du 135264 Bo
164523 Bv ------
------ 142635
123456
This gives nine separate round blocks spanning the extent,
and four Q-sets can readily join these together (though not
in a three-part arrangement). There is also scope to
course splice additional Durham into the composition, or to
use a six-lead splice to add Cambridge. Even if this is
done, however, the composition still doesn't have
particularly good method balance.
Applying the splice more than once is the obvious way to get
better method balance. This, however, requires care.
Suppose I choose 2,3 as fixed bells for one instance of the
splice, and 2,4 as fixed bells for the other. What does
this mean the lead 142635 should be? 2,4 are in 2,3 which
suggests it should be Durham, but 2,3 are in 3,5 suggesting
Bourne. This is a problem.
The solution is to choose disjoint pairs, e.g. 2,6 and 3,5.
Having two pairs means one can be used to introduce York and
one for Durham. And, by not involving the fourth in either
splices, it is free to be used to get Cambridge via a
six-lead splice.
Putting all this together produces eight round blocks which
need four Q-sets to join to give an extent. (In this case,
the four Q-sets are those in which the fifth is unaffected.)
720 Spliced Surprise Minor (6m)
123456 Bo 154326 Cm 145362 Bv
164523 Cm - 142635 Su 162534 Yo
- 142356 Su 135264 Du - 162345 Yo
156234 Su 156342 Bv - 162453 Yo
- 163425 Su - 164235 Cm 125634 Bo
125346 Su 135426 Du - 134625 Cm
146532 Su 152364 Bo 125463 Bv
- 163254 Bo - 164352 Cm 163542 Bv
142563 Bo 152436 Bv - 134256 Cm
135642 Du - 123645 Bv 156423 Bo
------ ------ ------
154326 145362 - 123456
Contains no 65s at back and a plain lead of each method.
Methods: 144 each of Beverley (Bv), Bourne (Bo), Cambridge
(Cm) and Surfleet (Su); and 72 each of Durham (Du) and
York (Yo).
Twelve bobs in an extent is obviously undesireable, and, by
using a mixture of sixth and seconds place methods, we can
escape from the requirement for the bobs to be in Q-sets.
It also means the four sixth-place variants can be added to
the composition.
720 Spliced Surprise Minor (10m)
123456 Bo 125634 Hu 145623 Bv
164523 Bo 134562 He 123564 Bv
135264 Du 146325 Pr 164352 Bv
156342 Cm 162453 Su 152436 Bv
142635 Yo 153246 Du - 123645 He
- 142356 Yo - 153462 Du 134256 Su
125463 Bk 136524 Cm 156423 Cm
156234 Pr 124653 Yo - 162345 Bo
163542 Bk 145236 Bo - 145362 Su
134625 Hu - 136245 Cm 162534 Su
------ ------ ------
- 125634 145623 - 123456
Contains no 65s at back and a plain lead of each method.
Methods: 48 each of Berwick (Bk), Hexham (He), Hull (Hu)
and Primrose (Pr); 72 each of Durham (Du) and York (Yo);
and 96 each of Beverley (Bv), Bourne (Bo), Cambridge (Cm)
and Surfleet (Su).
Repeating this with the Carlisle-over methods, we can easily
get the six seconds-place methods into an extent.
720 Spliced Surprise Minor (6m)
123456 Cl 135426 Cl 145362 Nw
135264 Ch 152364 Sa 123645 Sa
156342 Ch - 143526 Cl - 156234 Ch
164523 Ch - 143265 Sa 163542 Ch
142635 Ch - 152436 Cl 134625 Ak
- 142356 Cl 123564 Nw - 125634 Nw
- 142563 Mu 145623 Sa 146325 Mu
126435 Mu 164352 Sa 162453 Mu
163254 Mu 136245 Ak - 162534 Sa
135642 Mu - 145236 Cl 156423 Ak
------ ------ ------
- 135426 - 145362 - 123456
Contains a plain lead of each method.
Methods: 144 each of Carlisle (Cl), Chester (Ch), Munden
(Mu) and Sandiacre (Sa); and 72 each of Alnwick (Ak) and
Newcastle (Nw).
Unfortunately, I've been unable to produce an arragement
without 65s at back.
With the Cambridge-over methods, I just took the same plan
and inserted some sixth-place lead ends. However, with the
Carlisle-over methods, 3/5ths of the leads are in methods
that do not have corresponding sixths-place variants. This
severely restricts how you insert the sixth-place leads
ends.
My composition in eight methods (no Carlisle) got around
this by only starting with one instance of the J-Beverley -
Bourne splice, and course splicing three whole courses of
Newcastle into the composition. Unfortunately this course
splice is incompatible with the six-lead splice needed to
get Carlisle into the composition. This, has the
unfortunate effect of skewing the method balance, leaving
too much Newcastle and Morpeth, and very little of some of
the other methods. (Even by doing this, it is still
slightly non-trivial to get a composition as Morpeth and
Newcastle do not Parker splice.)
720 Spliced Surprise Minor (8m)
123456 Wo - 125346 Mo - 142563 Mo
164523 Ak 146532 Mo 163254 Mo
135264 Mu 132654 Mo - 154263 Mo
156342 Mu - 154632 Mu 163425 Mo
- 156423 Sa - 154326 Mo - 125463 Nw
- 134562 Ch 126435 Mo 134625 Nw
146325 Ch 135642 Mo 156234 Nw
162453 Ct - 142635 Mu - 134256 Ak
153246 Sa - 142356 Nw 162534 Ch
125634 Mu 163542 Nw 123645 Ch
------ ------ ------
- 125346 - 142563 - 123456
Contains a plain lead of each method.
Methods: 240 of Morpeth (Mo); 120 each of Munden (Mu) and
Newcastle (Nw); 96 of Chester (Ch); 48 each of Alnwick
(Ak) and Alnwick (Ak); and 24 each of Canterbury (Ct) and
Wooler (Wo).
I believe that John Leary has a similar composition to this,
too, though I've never seen it.
I actually came across this when doing an exhaustive search
for five-part compositions of minor about 18 months ago.
The following composition turned up and, when I worked out
how it worked, I produced the 8m composition above.
720 Spliced Surprise Minor (4m, atw)
123456 Wo
- 164235 Wo
152364 Nw
143652 Ch
135426 Mo
126543 Wo
------
- 135264
Four times repeated.
(In the event that anyone is interested in five-parts, there
is an exhaustive list of these, using just methods from the
41, here: http://www.ex-parrot.com/~richard/minor/5part )
Compositions based on this plan are possible with the
Cambridge-over methods, but it is not possible to join them
into a five-part structure.
This is actually subtly different to the earlier
compositions, as there are different quantities of Wooler
(Bourne for Cambridge-over) and Newcastle / Morpeth
(Durham). This is because the unused spare half-leads of
Chester (Beverley) are joined up differently.
An exhaustive search of mutually-true leads with these three
underworks suggests that these are the only possible ways of
splicing these methods:-
* 3 Bo + 3 Du + 24 Bv (one application of the splice)
(This has scope for up to three six-lead splices with
Cm or up to three course splices with Du.)
* 6 Bo + 6 Du + 18 Bv (two applications of the splice)
(This has scope for either a six-lead splices with
Cm or up to two course splices with Du.)
* 15 Bo + 10 Du + 5 Bv (five-part variant of the splice)
Clearly if we want both the London and Wells frontworks, and
if we want to include the Cambridge frontwork, this
restricts the underlying plan to the one I used in the
composition with just seconds-place methods.
The only remaining question is can it be joined together to
include plain leads of all nine methods? This is now within
easy reach of an exhaustive search. And running the search,
I find that I cannot have both Carlisle and a plain lead of
Wooler with this plan. So no, it's not possible.
Richard
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