[r-t] Jump methods

[Richard Smith]

A few weeks ago, Ben Willetts wrote:

> Leigh:
> > It's recently occured to me that I don't know
> > a way to notate methods with jump changes
>
> Currently the only jump change methods that have been rung have had only one
> bell jumping at a time.  These could be notated with arrows.  For instance,
> the change from 123456 to 234561 could be notated 1->6.  All the bells that
> the jumping bell passes over have to move one place in the other direction.

I gave a little thought to this a year or two ago when we
first came up with Mersey Ferry.  My aim was to produce a
notation that was identical to ordinary place notation in
the absence of jumps, that allowed (shortish) jumps to the
easily expressed in the same framework, and that would allow
an arbitrarily complicated jump change to the expressed.

My starting point was the standard mathematical notation for
expressing permutations in terms of cycles.  Thus the
permutation

  123456
  312546

can be expressed in cycles as

  (132)(45)(6)

(From a ringing perspective, I often feel the 3-cycle here
is the wrong way round -- the treble moves to seconds place,
not thirds place.)

Starting from these cycles:

  (i)   remove all 2-cycles on adjacent bells;
  (ii)  remove the brackets surrounding 1-cycles; and
  (iii) if nothing remains insert a x or -.

In this example we end up with (132)6.  Had we had an
ordinary change, we would have ended up with the standard
place notation.

Using this notation, we can provide a place notation
for Mersey Ferry:

  123456
  231465    (123)4
  324615    (345)6
  236451    x
  326145    3(465)
  312654    1(243)
  ------
  316245    x

  Mersey Ferry:  (123)4.(345)6.x.3(465).1(243).x

I guess we might want to make it particularly easy to
express jumps where one bell moves up (down) by n places and
the intervening n bells move down (up) by one place.  If so,
taking something like Ben's idea and converting (123456) to
(1->6) and (165432) (which is the same as (654321)) to
(6->1) might achieve this.

Using this extension, the place notation for Mersey Ferry
becomes

  (1->3)4.(3->5)6.x.3(6->4).1(4->2).x

Richard