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

[Richard Smith]

Philip Earis wrote:

> 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?

Well, there's another one in Tintinnalogia, though
Stedman/Duckworth isn't explicit that it works on arbitary
stages.

The starting point of the series is the extent on three
bells, Original Singles.

  123   3
  213   1
  231   3
  321   1
  312   3
  132   1
  ---
  123

All bells hunt throughout and this is enough to get the
whole extent.  If you try this on four bells, though, you
only get a third of the extent:

  1234  x
  2143  14
  2413  x
  4231  14
  4321  x
  3412  14
  3142  x
  1324  14
  ----
  1234

However, we can take this as one lead of a three lead extent
by making lead heads and lead ends form the extent on the
back three bells:

  1234
  1324
  ----
  1342
  1432
  ----
  1423
  1243
  ----
  1234

This is Plain Bob Minimus.  Now let's repeat the operation.
Take plain hunt on five as the lead, and make the lead heads
and ends produce the extent on the back four.  This gives an
extent of Plain Bob Doubles using 123 singles with the
second as observation.  It is called Old Doubles in
Tintinnalogia.

This can be continued indefinitely, with each stage
introducing a new type of call.  On six bells this gives an
touch of PB6 whose calls are 1236 and 1234 singles; this is
described in the "Trebles and Doubles on six Bells" section
of Tintinnalogia.  It seems pretty clear from the last
paragraph of that chapter (and elsewhere) that this
principle was understood back in 1671:

| In this Peal is contained the Six-score changes of Old
| Doubles on five bells, and also the Twenty-four changes
| Doubles and Singles on four bells; the half hunt, the
| quarter hunt, and three extream bells, makes the Six-score
| changes of Old Doubles in a perfect course; and the
| quarter hunt, and three extream bells, do make the
| Twenty-four changes Doubles and Singles. Every time the
| whole hunt leads, there are made two changes of the
| Six-score; and every time the whole hunt leads, and the
| half hunt lies next to it, there are two changes mades in
| the Twenty-four.

In general, you can take any n bell extent and use it as the
lead heads and ends of an n+1 bell (reversing alternate
changes under the treble).  The extent called "Grandsire
Bob" on six bells (which is what we would now call Plain Bob
Minor) is formed from an extent of one of the extents of
Grandsire Doubles (or Reverse Grandsire) in this manner.

Richard