[r-t] Singles in treble-dodging minor

[Richard Smith]

A few weeks ago, someone (probably Phil) asked me about
extents of treble dodging minor using singles -- do any
exist?  And how can you find them?  I've given the subject
some thought and have found a way of identifying which
methods have such extents.  This email has become fairly
mathematical, though I hope not unjustifiably so.



If a TDMM can be rung both bobs-only and in a composition
involving singles, then there must exist a set of in-course
lead heads and ends, L, that is only present in the
bobs-only composition, and which is replaced by a set of
out-of-course lead heads and ends, L', in the composition
with singles.

If we restrict this discussion to palindromic methods, and
use M to denote the rows in half a lead of the method, then
LM is the set of rows appearing in leads that are only
present in the bobs-only composition, and L'M is the set of
rows appearing in leads only present in the composition
including singles.  Clearly, as both compositions are
true extents, LM = L'M.

(What exactly do I mean by the product LM?  It is just the
set containing each member of L multiplied by each member of
M.  A mathematician would probably write this as

   LM = { lm : l in S, m in M },

or similar.)



At this point it is useful to introduce the idea of a
falseness table, F(M), of a method, M.  We can define this
to be the set of lead heads that are false against the lead
starting with rounds.  Mathematically, this can be written

  F(M) = { f : fM intersection M != {} }

where M is, again, the set of rows in half a lead of the
method.  (This is to be read: the set of rows, f, such that
the half-lead starting f and the half-lead starting with
rounds contain at least one row in common (i.e. are false
against each other).)

This notation is all well and good, but it hasn't got us any
closer to a way of constructing the falseness table.  It can
be proved that an equivalent expression for F(M) is

  F(M) = { a b^-1 : a in M, b in M }

(This can be read as follows: take two rows, a and b, from
the first half-lead of the method and multiply a by the
inverse of b to get an element of the falseness table.)
This expression is much more useful as it provides an
algorithm for generating the falseness table.



Back to the problem in hand.  How do we find a pair of sets,
L and L', one in-course and one out-of-course, that satisfy
LM = L'M?  Let's start by considering some element, l, of L.
By removing this lead from the composition, we get a set of
rows, lM that are now missing.  Now consider one of these
rows, lm (for some m in M): we want this occur in some
out-of-course lead, l', which means we want lm = l'm' (for
some other m' in M).  As l' must have the treble in first
place and is out-of-course, m' can be found uniquely -- it
has the treble in the same position as m, but the oposite
parity.

For notational convenience, I am going to use H to
denote the set of all in-course lead heads and Hx to denote
the set of out-of-course leads heads.  This provides a way
of expressing l',

  l' = (lm M^-1) intersection Hx.

Repeating this for each row, m, of the method gives an
expression for the set of out-of-course leads that must be
added to the composition to regain the missing rows caused
by removing lead l:

    l F(M) intersection Hx
  = l R

where lR = F(M) intersection Hx.

In general, this will add too many rows -- the set lRM of
added rows will be larger than the set lM of removed rows.

Consider some (out-of-course) lead, k in lR.  We now want to
find the set of in-course leads that need to be removed to
avoid falseness.  By a similar argument to the above, we
find this is

   (k F(M)) intersection H
 = k R.

Repeating for all the new out-of-course leads, we get l R^2.
So what have we established?  That if l is in L (the set of
removed rows), then so are each of the rows l R^2.
Applying this recursively, we find that by removing lead l,
we need to remove all of the leads, l <R^2>, where <R^2> is
the group generated by R^2.

In practice, unless we want half-lead singles, we want to
replace whole leads of the method rather than just
half-leads.  This means including the lead-end row, e, as
a generator of the group.



Much of the time, the group <R^2,e> will just be H -- the
group of in-course lead-heads.  This is simply saying that
if you replace all the leads with out-of-course leads, you
still have a true extent.  Put another way, if you start the
extent from an out-of-course row instead of rounds, it will
still be true.

However, some of the time, <R^2,e> will be a subgroup of H,
and it is these cases that are of interest.  Of the 147
"regular" TDMMs, 19 methods are of interest.  Modulo lead
splices, they are: Old Oxford, Neasden, Westminster,
Bamborough, Bacup, Morning Star, Pontefract and Wath.  In
each case, the group is the 12 in-course lead heads and ends
with the pivot bell fixed.  (E.g. Bamborough has 5th place
bell as pivot bell, so the group is the 12 in-course rows
of the form 1xxx5x.)

What does this mean about a composition?  Let's use
Bamborough as an example.  For each pivot bell, we can
choose whether we want to rows to rung in-course or
out-of-course.  As an example, let's ring the leads when the
6 pivots out-of-course, and the rest in-course.  This
produces a two-course block:

  120 Bamborough S Minor

    123456
    142635
    164523
    156342
  s 132564
  s 154326
    135642
    163254
    126435
  s 145263
    ------
  s 123456

Unfortunately, it turns out to be impossible to join the
three blocks together using ordinary bobs.  It turns out
that, the same is true for all of the 2nds place methods in
this list.

The 6ths place methods have a different problem.  They all
have N lead end order which means the 5th is always the
pivot.  With a standard 1456 single, this poses a problem as
it affects the pivot bell.  If, instead, we use a
non-standard 1236 single, we can get an extent.

  720 Bacup S Minor

    123456
    164523
  s 153264
  s 124653
    136524
    145236
    162345
  s 135462
  s 142635
  - 156423
    ------
    134256

  Twice repeated

  s = 1236, - = 14

This is not particularly elegant, and an alternative way of
joining it up is by using a mixture of 2nds and 6ths place
leads (i.e. splicing Bamborough with Bacup), for example,
with this slightly curious extent:

  720 Spliced S Minor (2m)

    123456 Bm
    142635 Bc
    156342 Bm
  s 132564 Bm
  s 154326 Bm
  - 163542 Bm
  - 125634 Bm
    162453 Bm
    146325 Bm
  s 135462 Bm
    ------
  s 142356

  Twice repeated.



What about methods outside of the 147 "regular" methods?
The vast majority of methods have the same problems as
Bamborough / Bacup.  However, there are some methods which
don't have this problem.  Stockwith Delight (-5-6-2-3-2-5,
lh6) is such a method.  Here, the group <R^2,e> has just two
elements -- rounds and e.  This means that there are no
restrictions on which leads can be rung out-of-course.

  720 Stockwith D Minor

    123456
    134625
    146532
    165243
  s 152634
    126453
  s 164235
    142563
  s 125436
  s 154263
    ------
    142356

  Twice repeated



So far, all of these compositions have been very much
curiosities: the singles are unnecessary, and the standard
bobs-only extent is at least as elegant.  Where this
technique really becomes useful is in spliced when the
singled-in leads might be of different methods.  I think
that should probably be the subject of another email,
though.

Richard