[r-t] Decisions / Algorithms for generating the extent

[Richard Smith]

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