Spliced treble-dodging minor - 1 - Revision history

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2010-11-06T18:36:09Z

Formatting improvements

GACJ at 18:15, 6 November 2010

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GACJ: moved Spliced treble-dodging minor plans 1 to Spliced treble-dodging minor - 1

2010-10-28T21:31:39Z

moved Spliced treble-dodging minor plans 1 to Spliced treble-dodging minor - 1

Holroyd: Created page with 'Richard Smith richard at ex-parrot.com Tue Sep 28 17:21:54 BST 2010 I'm going to start by cataloguing those plans that can be explained simply in terms of well-understood splic…'

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Created page with 'Richard Smith richard at ex-parrot.com Tue Sep 28 17:21:54 BST 2010 I'm going to start by cataloguing those plans that can be explained simply in terms of well-understood splic…'

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Richard Smith richard at ex-parrot.com
Tue Sep 28 17:21:54 BST 2010

I'm going to start by cataloguing those plans that can be
explained simply in terms of well-understood splices,
probably in two separate emails. This will then leave the
shorter list of plans that deserve further study.

This email covers all plans with one or two methods. That
means there's nothing new in this email as splicing two
methods (at least with a fixed treble) is well understood.

SINGLE METHOD PLANS

As we've got 75 methods (modulo lead splices and lead-end
variants), 75 of the 4614 plans contain just a single
method.

The fact that the plans only include a single method doesn't
mean that they cannot produce extents of spliced -- for
example, we can easily produce an 8-method extent of spliced
using Old Oxford's lead-splices and lead-end variants.
Similarly an extent of Beverley, Surfleet, Berwick and
Hexham is derived from one of these single method plans.

SIMPLE SPLICES

Let's call a splice 'simple' if it can involve just two
methods. So for example the three-lead splice between York
and Durham is a simple splice -- sure, we can continue by
combining, say, course of Beverley into the touch, but this
is optional -- the touch works with just York and Durham and
so the three-lead splice is 'simple'.

On the other hand, the grid splice with Ipswich, Bourne and
Cambridge is not simple as all three methods are an integral
part of the splice -- we cannot get an extent of just
Ipswich and Bourne, for example.

For the remainder of this email, I shall refer to methods
using their standard two-letter abbreviations. These are
give on John Warboy's website:

http://website.lineone.net/~jswcomps/comp06.htm#TD

It's well understood how to generate a complete list of
simple splices. I'm not sure an explanation of this has
ever been covered explicitly on this list, though it has
been mentioned in passing. But I'm not going to break this
discussion to explain how to do it -- though I might write
another email on it.

Splices are usually described in terms of the minimum number
of leads of the method that can be inserted. For TDMMs,
this number can be 1, 2, 3, 5 or 6. In practice 2-lead
splices are rare and do not occur amonst any of the standard
147. The most common form a of a 5-lead splice is the
course-splice where the five leads to be replaced form a
course. There are no non-course 5-lead splices using
methods from the 147. Let's take these types of splices one
by one.

LEAD SPLICES

The following lead splices exist amongst the standard 147.

(i) Using the D1, D2, D3 & D4 underworks

[Ci, Ks, Ls, Sd] / [Cf, Dk, Ny, Oc] J/M
[Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi] K/N
[Cd, Ce, Sw, Va] J

(ii) Using the S1, S2, S3 & S4 underworks

[Ba, Cs, Fg, Sk] / [Bg, Kn, Rs, Wl] J/M
[Bt, Le, Md, Pv] H

(iii) Using the Westminster & Allandale underworks

[Ad, Ws] G
[Co, Li] H
[Ck, Wt] / [Dt, Po] K/N

(iv) Using the Beverley & Surfleet underworks

[Bk, He] / [Bv, Su] H/L
[Ed, Kh] / [By, Pm] H/L
[Ch, Mu] G

A note on notation. [Bk, He] / [Bv, Su] means that Bk and
He are lead splices and that Bv and Su are their 2nds place
lead-end variants and which also form a pair of lead
splices. Whether you consider He and Bv to be lead splices
is simply a matter of definition and of no great relevance
here. The letters in the last column are the lead-end
orders.

Because lead splices were excluded when reducing the list of
methods to 75, they do not appear in the list of plans.

Unfortunately it is not possible to include both J and M
variants in an extent (without also including other lead-end
orders). This means that the Old Oxford group is the only
one of these that can give an eight method extent. This
plan is responsible for 55% of the 5.86 x 10^21 extents.
This is because there are 4^30 lead splices and 2796
possible callings (allowing 2nds, 4ths and 6ths lead ends).
Multiplying these together gives 3.2 x 10^21 extents.

COURSE SPLICES

The following is a table of all course splices using methods
from the 147. This table was calculated from first
principles (and is much the same as the one in Michael
Foulds' books on spliced TDMM) rather than extracted from
the results of the search.

Br, [Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi], Ma o
Ab, Ro / Lf, Ne
[Ci, Ks, Ls, Sd], Ox / [Cf, Dk, Ny, Oc], Ms o #
Nf, Pr / Cm, Ip o
[Bk, He] / [Bv, Su], Du o

[Ba, Cs, Fg, Sk], Do / [Bg, Kn, Rs, Wl], Ey o #
[Ck, Wt] / [Ad, Ws], [Dt, Po] +
Wh / Cl, Nb o
Mo / [Ch, Mu], Nw o
C1, Mp o

C3, Pn
[Bt, Le, Md, Pv], Cx o
Av, [Cd, Ce, Sw, Va] o
Cu, Lo +

Notation. As with lead splices, a slash separates 6th
place and 2nds place methods. Where a group of methods are
enclosed in square brackets, they are lead splices. An o
denotes that the course splice is just a half-lead variant,
often with a set of lead splices. A # notes that the line
contains three separate course splices, e.g. Ox, Ms and the
eight lead splices are three sets of course splices. A +
notes that multiple backworks are present.

It's worth calculating the number of plans that can be
accounted for solely in terms of course splices. This is
worthwhile because the easiest way of checking that there's
nothing interesting hidden amongst the list of
seemingly-ordinary plans is by checking that the search
found the predicted number.

With six courses, we would expect 2^6 = 64 plans. However,
our list of 4614 plans exclude rotations and reflections,
and many of the 64 plans will just be rotations of each
other.

If we apply the splice zero times, then we have a single
method plan (already considered above). All ways of
applying it once are equivalent -- we can always rotate /
reflect the plan so that the splice is applied to the 123456
l.h.

What about two applications? Put succinctly, are all
choices of two courses equivalent? We know that from the
plain course we can reach any other course using just one
bob -- therefore all pairs of (distinct) courses are related
by cycling three coursing bells and are thus equivalent.

By symmetry, four, five and six applications of the splice
will be the same as two, one and zero respectively. This
just leaves the case of three applications of the course
splice. Are all choices of three courses equivalent? No.
For example, we know that a block of three bobs can join the
three tenors-together courses, but the same is not true of
the three split-tenors courses.

We know that any two courses must share two coursing pairs.
Three distinct courses cannot all share two coursing pairs
because there are only 10 pairs in total and 3*(5-2)+2 > 10.
So they must either all share a single coursing pair (as the
tenors together courses do, which is what allows them to be
joined by a Q-set of bobs on this pair) or none (as the
split tenors courses do).

How many of each type of choice of three courses are there?
Once we've selected two courses, there are four remaining.
Two of the unselected courses each share a (different)
single coursing pair with the two courses, and therefore the
other two do not share any coursing pair with both the
already chosen courses. One way of looking at this is that
courses A,B,C can be joined with a block of three homes,
A,D,B with a block of three before, but A,B,E and A,B,F
cannot be joined in any order using a block of three calls.
So of the 20 ways of selecting three courses, 10 share a
single coursing pair, and 10 do not. Once rotations and
reflectins have been factored out, this just leaves two ways
of selecting three courses.

So we have 1+1+2+1+1 = 6 plans for course splices (excluding
those that none of one or other method). The table above
has 18 course splices (noting that the two lines marked with
a # each contain three pairs of course splices). This means
that 108 = 18 * 6 out of the 4614 plans can be explained
just in terms of a single course splices, perhaps applied
multiple times.

SIX-LEAD SPLICES

The following is a table of all 6-lead splices using methods
from the 147.

Do, No 2
Bl, Wk 2
[Bk, He], Pr, Wa / Bs, [Bv, Su], Cm 3 [3]
[Ed, Kh], Os, Wf
/ Bh, [Bt, Le, Md, Pv], Bw, [By, Pm], Cc, Mp 3 [6]
Ml / [Co, Li], Fo 3

[Ba, Cs, Fg, Sk], [Ci, Ks, Ls, Sd], Pe, Ri, Wv
/ [Bg, Kn, Rs, Wl], Bp, [Cf, Dk, Ny, Oc], Cn, Dn 4 [5]
Br, Lv / Ki, Ma 4
Ab, Hu / Bo, Ne 4
Km, Sh / Ti, Tr 4
Ct, Cy / Ak, Cz 4

Lu, Mo / Nw, Ww 4
[Cd, Ce, Sw, Va], Ke 4
Bc, [Cw, Ns, Sl, Wr] / Bm, [Cb, Ng, Ol, Wi] 5
Pn, So 5
Bn, Lo 5

Cx, We 5
[Ch, Mu], Cl, Gl 6 [3]

Notation. As above, a slash separates lead end variants,
and lead splices are enclosed in square brakcets. The
number in the right-hand column is the fixed (pivot) bell
for the splice. Where a number is given in square brackets
at the end of the line, this is number of groups of mutually
six-lead splicing methods on the line.

Counting the plans that these are responsible for is
trivial. Because the splice uses all six rows where a given
bell pivots, up to rotation, there is exactly one way of
applying the splice once, one way of applying it twice, one
way of applying three times, and one way of applying if four
times. (Zero or five applications results in a single
method extent, already considered above.)

The table above has 17 lines, but four rows list multiple
six-lead splices. With n six-lead splice clusters (i.e. a
row marked [n]), there are n(n-1)/2 separate pairs of
six-lead splicers. This gives 17-4 + 2*(3*2/2) + 5*4/2 +
6*5/2 = 44 six-lead splices.

This means that 176 = 44 * 4 out of the 4614 plans can be
explained just in terms of a single 6-lead splices, perhaps
applied multiple times.

THREE-LEAD SPLICES

[Ad, Ws], Di 2&3 +
Du, Yo 2&3 *
Ca, Gl 2&3 +
Cl, Cu 2&3 +
Cr, [Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi], El 2&4 +

[Ck, Wt], Wo / [Dt, Po], Sa 2&4 +
Nm, Pn 2&4
Lo, We 2&4 *
Ro, St / Lf, Wm 2&5 *
[Bt, Le, Md, Pv], [Co, Li] 2&5 +

[Ba, Cs, Fg, Sk], Hu / [Bg, Kn, Rs, Wl], Bo 2&6 +
Km, Sn / Qu, Tr 2&6
Ct, Mo / Ak, Nw 2&6 *
Br, Hm / Ma, Ta 3&5 *
Hu, Lv / Bo, Ki 3&5

Cy, Lu / Cz, Ww 3&5 *
C2, C3 3&5 *
Ev, Wo / Sa, Te 3&6
Bn, Cx 3&6 *
Di, Ms 4&5

Av, Ca 4&5 *

(Notation. As with lead splices, a slash separates 6th
place and 2nds place methods. Where a group of methods are
enclosed in square brackets, they are lead splices. The
numbers in the right-hand column are the fixed place bells
for the splice. A * notes that the splice works like
London and Wells by swapping 34.16.34 for 14.36.46 at the
half-lead. A + notes that multiple backworks are present.)

With 30 leads in the extent, we can apply the 3-lead splice
any number of times from 0 to 10. Another way of looking at
this is that there are ten ways of choosing a pair of bells
from the five working bells (10 = 5*4/2). This means that
there are 2^10 different plans for each extent. However,
our list of 4614 plans exclude rotations and reflections,
and many of the 1024 = 2^10 will just be rotations of each
other which complicates things a bit.

If we apply the splice zero times, then we have a single
method plan (already considered above). All ways of
applying it once are equivalent -- we can always rotate /
reflect the plan so that the splice is applied to the 123456
l.h. With two applications, either the two applications
share a fixed bell (e.g. 2&3 and 2&4) or they do not (e.g.
2&3 and 4&5). Up to rotation and reflection, that's the
only choice left. We can show these diagramatically with
letters A-E indicating the five working bells and a
representing each application of the splice by joining the
two fixed bells.

(1.1) A --- B C D E

(2.1) A --- B --- C D E

(2.2) A --- B C --- D E

With three applications, we apparently have four
possibilities.

(3.1) A --- B --- C D --- E

(3.2) A --- B --- C --- D E

(3.3) A --- B --- C E
|
|
D

(3.4) A --- B D E
\ /
\ /
C

However, this isn't what the search found. For example, it
found five plans (up to rotation and reflection) containing
21 leads of London and 6 of Wells -- (3.2) appeared twice.

The reason is to do with parity. Because the plan only uses
in-course l.h.s and l.e.s we can only rotate or reflect the
plan by an even permutation. In (3.1), A and C are
equivalent as are D and E. When rotating (3.1), if we find
we need an odd permutation, we simply swap the labels on A
and C and use an even permutation.

But with (3.2) we can't do that. Yes, A and D are
equivalent as are B and C. But we cannot indepdently swap
labels on one pair of these -- if we swap the labels on A
and D we also need to swap the labels on B and C for the
graph to remain unaltered. This means we cannot simply
relabel so that an odd permutation 'rotation' converts into
an even permutation. The result is that there are two
versions of (3.2) which we might term a right-handed and a
left-handed version.

What of four applications of the splice?

(4.1) A --- B --- C --- D --- E [has l+r versions]

(4.2) A --- B --- C --- D
|
|
E

(4.3) A --- B --- D E
\ /
\ /
C

(4.4) A --- B D --- E
\ /
\ /
C

(4.5) A --- B E
| |
| |
C --- D

(4.6) A
|
|
B --- C --- D
|
|
E

And finally, for five applications:

(5.1) A --- B --- C [l+r variants]
\ /
\ /
D --- E

(5.2) A --- B --- C
| |
| |
D --- E

(5.3) A --- B --- C --- D
\ /
\ /
E

(5.4) A --- B --- C --- D [l+r variants]
\ /
\ /
E

(5.5) A
|
|
B --- C --- D
\ /
\ /
E

(5.6) A --- B E
\ / \
\ / \
C --- D

Six or more applications of the splice are, by symmetry, the
same as four or fewer. This gives the total number of plans
for a 3-lead splice as: 1+2+5+7+8+7+5+2+1 = 38. There are
21 3-leads splices in the table above, so that means that
3-lead splices are responsible for 798 = 21 * 38 of the 4614
plans.

SUMMARY

Scanning through the results of the search, I find 1157
plans with one or two methods. If I add the numbers above,
I get:

Single method plans . . . . . . . . . . 75
Course splices . . . . . . . . . . . . . 108
Six-lead splices . . . . . . . . . . . . 176
Three-lead splices . . . . . . . . . . . 798
---------------------------------------------
TOTAL . . . . . . . . . . . . . . . . . 1157

This isn't surprising. As I noted at the beginning of the
email, the theory of spliced with just two methods is well
understood and we wouldn't expect to find anything new.
However, this has been a productive exercise on two counts.
First, it increases my confidence that the search results
are correct as it agrees with the already well-tested theory
on splicing two methods. Secondly, it has allowed me to
work out techniques for counting extents -- for example,
identifying the potential problem with chirality (handness)
of certain three-lead splices.

Of course, with 3459 plans left to study, there's still
plenty to do!

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		+	[[Spliced treble-dodging minor - 2\|Plans 2]]
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		+	[[Spliced treble-dodging minor - 3\|Plans 3]]
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		+	[[Spliced treble-dodging minor - 4\|Plans 4]]
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		+	[[Spliced treble-dodging minor - 5\|Plans 5]]
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		+	[[Spliced treble-dodging minor - 6\|Plans 6]]
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	Richard Smith richard at ex-parrot.com		Richard Smith richard at ex-parrot.com
	Tue Sep 28 17:21:54 BST 2010		Tue Sep 28 17:21:54 BST 2010

@@ Line 25: / Line 25: @@
 methods (at least with a fixed treble) is well understood.
+====SINGLE METHOD PLANS====
 As we've got 75 methods (modulo lead splices and lead-end
@@ Line 39: / Line 38: @@
 Hexham is derived from one of these single method plans.
+====SIMPLE SPLICES====
 Let's call a splice 'simple' if it can involve just two
@@ Line 77: / Line 75: @@
 by one.
+====LEAD SPLICES====
 The following lead splices exist amongst the standard 147.
+<pre>
     (i) Using the D1, D2, D3 & D4 underworks
@@ Line 93: / Line 90: @@
     [Bt, Le, Md, Pv]                               H
-    (iii) Using the Westminster & Allandale underworks
+    (iii) Using the Westminster & Allendale underworks
     [Ad, Ws]                                       G
@@ Line 104: / Line 101: @@
     [Ed, Kh] / [By, Pm]                            H/L
     [Ch, Mu]                                       G
+</pre>
 A note on notation.  [Bk, He] / [Bv, Su] means that Bk and
 He are lead splices and that Bv and Su are their 2nds place
@@ Line 125: / Line 122: @@
 Multiplying these together gives 3.2 x 10^21 extents.
+====COURSE SPLICES====
 The following is a table of all course splices using methods
@@ Line 133: / Line 129: @@
 Foulds' books on spliced TDMM) rather than extracted from
 the results of the search.
+<pre>
     Br, [Cw, Ns, Sl, Wr] / [Cb, Ng, Ol, Wi], Ma        o
     Ab, Ro / Lf, Ne
@@ Line 150: / Line 146: @@
     Av, [Cd, Ce, Sw, Va]                               o
     Cu, Lo                                              +
+</pre>
 Notation.  As with lead splices, a slash separates 6th
 place and 2nds place methods.  Where a group of methods are
@@ Line 227: / Line 223: @@
 The following is a table of all 6-lead splices using methods
 from the 147.
+<pre>
     Do, No                                               2
     Bl, Wk                                               2
@@ Line 250: / Line 246: @@
     Cx, We                                               5
     [Ch, Mu], Cl, Gl                                     6 [3]
+</pre>
 Notation.  As above, a slash separates lead end variants,
 and lead splices are enclosed in square brakcets.  The
@@ Line 276: / Line 272: @@
 applied multiple times.
+====THREE-LEAD SPLICES====
-THREE-LEAD SPLICES
+<pre>
     [Ad, Ws], Di                                   2&3  +
     Du, Yo                                         2&3 *
@@ Line 304: / Line 299: @@
     Av, Ca                                         4&5 *
+</pre>
 (Notation.  As with lead splices, a slash separates 6th
 place and 2nds place methods.  Where a group of methods are
@@ Line 312: / Line 307: @@
 London and Wells by swapping 34.16.34 for 14.36.46 at the
 half-lead.  A + notes that multiple backworks are present.)
 With 30 leads in the extent, we can apply the 3-lead splice
@@ Line 334: / Line 328: @@
 representing each application of the splice by joining the
 two fixed bells.
+<pre>
     (1.1)  A --- B     C     D     E
@@ Line 340: / Line 334: @@
     (2.2)  A --- B     C --- D     E
+</pre>
 With three applications, we apparently have four
 possibilities.
+<pre>
     (3.1)  A --- B --- C     D --- E
@@ Line 357: / Line 351: @@
              \ /
+              C
+</pre>
 However, this isn't what the search found.  For example, it
 found five plans (up to rotation and reflection) containing
@@ Line 380: / Line 374: @@
 What of four applications of the splice?
+<pre>
     (4.1)  A --- B --- C --- D --- E      [has l+r versions]
@@ Line 410: / Line 404: @@
+                 |
+                 E
+</pre>
 And finally, for five applications:
+<pre>
     (5.1)  A --- B --- C       [l+r variants]
             \         /
@@ Line 445: / Line 439: @@
              \ /   \
               C --- D
+</pre>
 Six or more applications of the splice are, by symmetry, the
 same as four or fewer.  This gives the total number of plans
@@ Line 453: / Line 447: @@
 plans.
+====SUMMARY====
 Scanning through the results of the search, I find 1157

← Older revision	Revision as of 21:31, 28 October 2010
(No difference)