[r-t] Spliced Plain Major

[Richard Smith]

Andrew Johnson wrote:

> I've been thinking about plain major recently, thought not
> quite along Richard's lines.
>
> First I listed all the plain major methods with
> right place
> conventional symmetry
> at most 2 places per change
> (at most 2 blows per place - implied by right place)
> no 78s below treble
> plain bob half leads

I think you mean no 78s above the treble.

This list of fifteen methods reminds me of a conversation 
started by Ander on the 'Ringing Methods' Facebook group 
back in Nov 2018.  Ander wrote:

   I've always liked Double Court Minor. Here is a a family
   of closely related plain methods (I'm not going to use the
   E word) that I think deserve attention. They all follow
   the same rule - treble bob until you meet the treble, then
   do court places or miss a dodge. This always gives regular
   lead ends and full coursing music above and below the
   treble. On stages 6,10,14,... there are two double
   methods, depending on whether you dodge or hunt at the
   lead end/half lead. On stages 8,12,16,... the same choice
   gives two methods that are reverses of each other. [...]

   Minor
   &-4-3-1,1 Double Court
   &-1-1-5,2 Double Bob

   Major
   &-4-3-1-7,1 Highbury
   &-1-6-5-1,2 Edmonton Bob

   Royal
   &-4-3-8-7-1,1 Double Norwich
   &-1-6-5-1-9,2 Double Isleworth Bob

I replied saying:

   These are part of a slightly broader family of methods,
   which are formed from Plain Hunt by adding pairs of court
   places around the treble, and/or seconds over the treble,
   and/or places immediately below the treble's lie. None of
   these pieces of work disrupts the coursing order, so you
   are guaranteed regular lead heads (albeit possibly
   differential) and full coursing music above and below the
   treble. On six bells, this gives 2^3-1 methods: Plain
   Bob†, Reverse Bob†, Double Bob, Double Court, Hereward*,
   London Bob* and Double Oxford*. On eight bells, this gives
   2^4-1 methods, all named: Plain Bob†, Reverse Bob†, Double
   Bob†, Shipway Court†, Winchester*†, Highbury,
   Marlborough*, Lavenham Court†, Edmonton, Maidstone*†,
   Chesterfield*, Double Norwich*, Hereward*, Pershore* and
   Double Oxford*. You're then saying that for aesthetic
   reasons, you don't want multiple dodges, which eliminates
   those listed here with an asterisk, or those with missed
   dodges except around the treble, which are those listed
   here with a dagger.

The reason you have 2^m-1 methods, where m is the number of 
dodging positions (i.e. half the number of bells), is 
because you have an independent choice of whether to add 
court places at each internal treble position, 3-4, 5-6, 
etc., and also seconds at the lead-end and (n-1)ths at the 
half-lead.  The only combination that doesn't work is Plain 
Hunt, hence the -1; though I guess you could consider that a 
degenerate case of a differential hunter.

What I hadn't realised is that these could be found with a 
simple methsearch search – specifically that the requirement 
for regular half-leads was equivalent to this on eight 
bells.  It ceases to be the case on twelve bells because of 
this method (and its half-lead and lead-end variants):

   &-8-8-58-5-5-1,1

> Note the absence of a m lead head code method.

Indeed.  If we think of things in terms of lead head numbers 
(e.g. +1 for Plain Bob, +2 for Double Bob, etc), then Plain 
Hunt is a 0, and we each piece of work (meaning court 
places, seconds over the treble, of (n-1)ths under it), we 
add 1 – bearing they palindromic symmetry requirement means 
that court places appear in both halves of the lead if at 
all.  The maximum number of pieces of work we can add is 
n-2: m-2 pairs of court places, and two half-lead / lead-end 
places. 2(m-2)+2 = n-2.  An 'm' lead-end group would require 
n-1 pieces of work.

Drop the requirment for palindromicity and you also find 
some once-popular methods like Yorkshire Court.  And the 
numbers of methods with each lead head are the binomial 
coefficients, with Plain Hunt and Double Oxford being the 
two extremes.

> Removing the plain bob half leads adds 17 more:
>
> X.16.X.36.X.18.X.58.X.18.X.36.X.16.X.12 a Longney
> X.18.X.16.X.18.X.38.X.18.X.16.X.18.X.12 b Richmond
> X.14.X.16.X.18.X.18.X.18.X.16.X.14.X.12 c Single Oxford
> X.18.X.18.X.58.X.58.X.58.X.18.X.18.X.12 d Ashbourne College
> X.16.X.16.X.58.X.38.X.58.X.16.X.16.X.12 d Gonville
> X.14.X.36.X.18.X.38.X.18.X.36.X.14.X.12 d Loughborough
> X.14.X.38.X.58.X.58.X.58.X.38.X.14.X.12 f Painswick College
> X.16.X.38.X.38.X.58.X.38.X.38.X.16.X.12 f Pulford
> X.18.X.38.X.38.X.38.X.38.X.38.X.18.X.12 f St Clement's College
> X.18.X.16.X.18.X.38.X.18.X.16.X.18.X.18 g New London Court
> X.14.X.16.X.18.X.18.X.18.X.16.X.14.X.18 h Single Norwich Court
> X.14.X.36.X.18.X.38.X.18.X.36.X.14.X.18 j Cambridge Court
> X.18.X.18.X.58.X.58.X.58.X.18.X.18.X.18 j Spalding College
> X.16.X.16.X.58.X.38.X.58.X.16.X.16.X.18 j Trinity Court
> X.16.X.38.X.38.X.58.X.38.X.38.X.16.X.18 l Avalon Court
> X.18.X.38.X.38.X.38.X.38.X.38.X.18.X.18 l Crayford College
> X.14.X.38.X.58.X.58.X.58.X.38.X.14.X.18 l Kidlington College

I'm not sure any of these grab me as particularly exciting. 
But maybe there's space for one of them, if Edmonton (or 
Highbury) were dropped.

> but still no m lead head code method,

This doesn't surprise me, though I'm struggling to give a 
convincing explanation.

> so search for more:
>
> Allowing wrong places, but requiring PB half leads gives:
>
> X.14.58.36.14.58.X.18.X.58.14.36.58.14.X.18 m Double Coslany Court
> 36.18.56.38.12.58.34.18.34.58.12.38.56.18.36.18 m Untitled
> 56.14.X.36.12.58.36.18.36.58.12.36.X.14.56.18 m Untitled

One of those the last two would keep us on our toes!  I'm 
keen to keep Coslany.

> Allowing more than two places per row, with PB HL gives:
> X.1458.X.1236.X.3458.X.18.X.3458.X.1236.X.1458.X.18 m Untitled
> X.1458.X.16.X.123458.X.38.X.123458.X.16.X.1458.X.18 m Untitled
> X.1458.X.16.X.3458.X.38.X.3458.X.16.X.1458.X.18 m Untitled
> X.1458.X.36.X.123458.X.18.X.123458.X.36.X.1458.X.18 m Untitled
> X.18.X.1236.X.38.X.18.X.38.X.1236.X.18.X.18 m Untitled
> X.18.X.36.X.1238.X.18.X.1238.X.36.X.18.X.18 m Untitled

Again, not sure anything here really grabs me.  However, for 
an 'm' group method with a more distinct line, how about 
this:

   &-45-1.34.1.3456.1,1

> Just for a completeness factor it would be nice to see
> the 8 12-lead-end methods from the first set spliced, or
> perhaps all 15 (with 16 bob for the 18 lead-ends).

Ander made a similar comment on Facebook in 2018, but I'm 
not convinced by this.  I don't think there's enough variety 
for the composition to be very fulfilling.

RAS