[r-t] Decisions / Algorithms for generating the extent
Richard Smith
richard at ex-parrot.com
Wed Jun 21 14:42:38 UTC 2006
Philip Earis wrote:
> Reading the online version of Tintinnalogia, it's very
> interesting to see how the 'algorithm' of plain changes is
> introduced as an iterative way of generating the extent on
> n bells.
>
> What other, more dynamic, ways would people use to do the same thing?
Another way is to use Plain Changes on (n-r) bells as the
coursing order for a singles-only touch of an n-bell method
with r fixed bells which get into each position relative to
each other in the plain course.
A very simple example is a touch of Original Minimus with 34
for x singles. We can only have one fixed bell (as no pair
of bells get into each position relative to each other in
the plain course); let's choose the treble. The initial
coursing order is 3124, but as this is only meaningful up to
rotation, we can rotate this to put the fixed bell (the
treble) at end of the coursing order and then ignore it
giving 243.
Now lets write out Plain Changes on these three bells.
243 swap 3 & 4
234 swap 2 & 3
324 swap 2 & 4
342 swap 3 & 4
432 swap 2 & 3
423 swap 2 & 4
---
243
This defines the bells that need to be swapped in the
coursing order via singles and allows us to write out the
touch.
1234 1342 1423
s 2134 s 3142 s 4123
2314 3412 4213
3241 4321 2431
3421 4231 2341
4312 2413 3214
4132 2143 3124
s 1432 s 1243 s 1324
---- ---- ----
1342 1423 1234
In this case, the touch of Original Minimus is more familiar
as Single Court Minimus.
(It's worth noting that this has unnecessarily many Q-sets
of singles. In fact, the touch can be joined with just two
Q-sets, instead of three, giving a palindromic one-part:
1234 1243 1432
s 2134 s 2143 4123
2314 2413 4213
3241 4231 2431
3421 4321 2341
4312 3412 3214
4132 3142 3124
1423 s 1342 s 1324
---- ---- ----
1243 1432 1234
This touch seems quite reminiscent of Grandsire Doubles,
with the fourth -- the pivot bell in the palindromic
symmetry -- serving the role of observation.)
This works on higher stages too. In Plain Bob Minor the
treble and tenor get into all positions relative to each
other, so we can keep both bells fixed and ring use Plain
Changes on four to get a three-part composition. There's a
slight complication, however: ringing a 1234 single instead
of a 12 plain swaps two non-adjacent bells in the (usual)
coursing order. Fortunately, we can just use an unorthodox
definition of coursing order that takes alternate bells from
the usual coursing order, therby putting them together. So
instead of 53246, we start with 34526. To get 5 fixed at
the part-end, we want 5 as the hunt for the plain changes:
3452 swap 2 & 5
3425 swap 3 & 4 (by choosing a "far" extreme)
4325 swap 2 & 5
4352 swap 3 & 5
4532 swap 4 & 5
5432 swap 2 & 3 (again, by choosing a "far" extreme)
5423 swap 4 & 5
4523 swap 2 & 5
----
4253 x 3
The position of the swap in this unorthodox coursing order
determines the calling position (just as it does with a
traditional coursing order): swaping the first pair is a
home, the middle pair, seconds, and the last pair, wrong.
23456 W 2 H
--------------
54326 s s
25436 s s s
34526 s s
34256 s
--------------
Twice repeated
(As all coursing orders are visited, including those with
the tenors coursing backwards, this inevitably gives a touch
with 65s at back.)
The algorithm for generating an extent on n by applying
plain changes on n-1 to the coursing order of plain hunt on
n has the elegant property that only three different changes
are needed. By contrast, plain changes needs n-1 different
changes, as does the Original Singles, Bob Minimus, Old
Doubles, ... sequence.
RAS
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