[r-t] Spliced Plain Major
Richard Smith
richard at ex-parrot.com
Thu Mar 5 14:47:46 GMT 2020
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
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