[r-t] 147 TDMM
Richard Smith
richard at ex-parrot.com
Sat Oct 16 01:10:49 UTC 2010
Despite impressions, I haven't yet got bored of this.
I've now analysed all of the composition plans that can be
explained solely in terms of simple splices -- that is,
course, three- and six-lead splices. This was described
in the following series of emails:
http://ex-parrot.com/~richard/r-t/2010/09/003650.html
http://ex-parrot.com/~richard/r-t/2010/09/003660.html
http://ex-parrot.com/~richard/r-t/2010/10/003675.html
That explained 2280 of the 4614 plans. The remaining 2334
plans are listed here:
http://ex-parrot.com/~richard/minor/147/compound-plans.txt
I've noted in another thread that these can be partitioned
into 492 clusters of plans where each cluster contains plans
related by simple plans. These plans are interesting
because this is where any potential new compositions will be
found. So let's start to look at these. This email covers
grid splices, the related (though often overlooked)
triple-pivot grid splice, and a new (if rather useless)
generalisation which I've termed the hidden triple-pivot
grid splice.
GRID SPLICES
Grid splices are the best-understood splice that is not a
simple splice. A grid splice involves three methods with
different lead-ends in the ratio 2:2:1, and the choice of
method depends solely on the position of the observation
bell for the splice. (That is sufficient for a definition.)
The most rung example of a grid splice must be the one
between Cambridge, Ipswich and Bourne which have H, K and J
lead ends, respectively:
123456 Ip
142635 Bo
156342 Ip
- 123564 Cm
164352 Cm
- 145236 Ip
124653 Bo
136524 Ip
- 145362 Cm
162534 Cm
---------
134256
Twice repeated
By using a Parker splice to get both 2nds and 6ths place
lead ends, A.G. Driver produced a three-part arrangement
with six methods -- the so-called 'Cambridge six'.
(Incidentally, I was slightly surprised to see no mention of
Driver's work composing spliced minor in his obituary in
the RW a fortnight ago [RW 2010, p1000].)
Grid splices were mentioned in passing in the 'OTHER EXTENTS
WITH FOUR METHODS' section of my third email looking at
simple splice plans, when I said
> X --(3)-- G --(3)-- Y [diagram relabeled]
> |
> (6)
> |
> Z
>
> This arrangement of splices is the one that makes a grid
> splice work, except that for a regular grid splice, G is
> an irregular method and entirely removed.
When all four methods are regular methods, they all must
have the same lead-end order, and so under the definition of
a grid splice that I've adopted, an arrangement of X, Y and
Z doesn't count as a grid splice.
However, if method G is irregular then X, Y and Z must each
have different lead-end orders if they are regular. This
means that, because we're only searching for regular
methods, we will see three-method plans for X, Y and Z in
the list of compound (i.e. non-simple) plans. More
generally, G might have other undesirable properties such as
multiple consecutive blows in one place or jump changes.
ENUMERATING GRID SPLICES
This raises an interesting question. To enumerate simple
splices, we simply look at each pair of methods in turn and
ask what splice they have. Even for a fairly long list of
methods, that's quite efficient. But how do we efficiently
enumerate grid splices?
One possibility is to take a list of methods that include
irregular methods, and look at the simple splices between
all pairs of methods. Then whenever we have a set of of
four methods G,X,Y,Z with different lead end orders that
share splices as marked shown in the diagram above, we know
that X,Y,Z have a grid splice. The problem with this is
that the list of methods needs to be very long so as to
include G even when it contains jump changes or some other
undesirable property.
Another possibility is to look at all choices of three
method, X,Y,Z, put them into a grid splice and see if it's
true. The strategy I've used is a variation on this. I've
taken my code for searching for extents of the 147 and
modified it to search for plans with some part-end group.
The relevant part-end group is the 12 in-course rows of the
form 1....6 -- these are the course heads and course ends of
the composition, and by having both, we're taking into
account the palindromic nature of the grid splice.
This loses some of the search's efficiency as it means
droppping rotational pruning; it also complicates the
inter-method falseness handling. (Mathematically, one way
of thinking about the latter is as a consequence of the fact
that, unlike Cayley graphs, Schreier graphs are not vertex
transitive.)
That search turns up 53 grid splices listed below:
X Y Z course #plans
--------------------------------------
Cc Lo Ke/Ce (S) HKJKH 2
Mp So Ke/Ce (S) HKJKH 2
Li/Pv Pn Sg (S) HKJKH 5
Fo Sa/Te Ti/Tr (S) HKJKH 10
Cm Ip Bo/Ne (S) HKJKH 2
C1 Mp So/Pn (Q) GHKHG 2
Cu/Cl Nb Sa (Q) GHKHG 5
Di/Ws Es Po (Q) GHKHG 5
Dn Yo Cm/Su/Bs (R) JGHGJ 3
Wl/Bo Ey Cj (R) JGHGJ 5
Bp Bu Cm/Su/Bs (R) JGHGJ 3
Dk Di/Ms Be (R) JGHGJ 5
Ip Bo/Ki Ey (P) KJGJK 5
Rc Bp Bu (P) KJGJK 1
El/Ol Bo/Ki Bu (P) KJGJK 36
Te Tr Ms (P) KJGJK 1
Qu/Tr Kt Po (U) MONOM 5
Dk Ox Po (U) MONOM 1
No El/Ol Be (W) ONLNO 5
No Ip Es (W) ONLNO 1
Do Fr Ey (-) OHGLO 1
Do Cj Bu (-) OHGLO 1
C1 Pn Kt (-) GNOKG 1
Di/Ms Rc Ox (-) GNOKG 5
No Ms Ki/Ma (-) OOJGG 2 } Cannot be made
No Di/Ws Bo/Ne (-) OOJGG 10 } to join up
In each grid splice, there are 12 leads of each of the
methods listed in the first two columns (headed X and Y),
and 6 of the method in the last (Z) column. Where several
grid splices just differ by a simple splice (i.e. a three-
or six-lead splice), they're listed on the same line above.
With two (or more) methods in the Z column, it's not
possible to get more than one of them in the composition,
because they share a six-lead splice.
But when there are two methods in the X or Y columns, the
methods share a three-lead splice and both methods can be
present in composition. There are four 3-lead splice slots
for X or Y (with the observation bell each each other bell
as the fixed bells). This gives rise to five plans (up to
rotation and reflection) depending on whether 0, 1, 2, 3
or 4 of the slots are used.
In one case, both columns X and Y have two methods. We can
label the four X splice slots a, b, c and d, and there are
four corresponding Y splice slots with the same fixed bells.
There are two X methods: lets call them X1 and X2. If we
have no X1 or no X2 then, we have five different ways of
applying the Y splice (solely depending on the ratio of the
two Y1 methods). With one splice slot used to get X1
(say slot a), we have eight ways of choosing Y:
0, a, b, a+b, b+c, a+b+c, b+c+d, a+b+c+d
With equal amounts of X1 and X2 (say by having X1 at a and
b) we have nine ways of choosing Y:
0, a, c, a+b, a+c, c+d, a+b+c, a+c+d, a+b+c+d
However we need to think about chirality. This is relevant
in one case -- when a+b are X1 and a+c are Y1:
Y1
a ------ c
| |
X1 | | X2
| |
b ------ d
Y2
If we relabel (say) a and b, we also need to relabel c and
d. That's an even parity relabeling, so we've got two
versions of that plan. That a total of gives 5+8+10+8+5 =
36 plans.
The table above shows the number of plans for each grid
splice. Adding them all up gives 124 plans. With a few
moments thought, we can see that it's not possible to build
on a grid splice by adding further methods.
COMPOSITE COURSES
[This section is a digression from the analysis of the
extent plans found in the search.]
The fourth column of the table of grid splices shows, in
parentheses, the lead end order of the grid method -- the
method G in the diagram at the top of this email. Of the
eight irregular lead-end codes, only six are represented
above. S and V are just lead-end variants of each other;
the lead-end code that's really missing is T. With a larger
selection of methods to play with, it's possible to get grid
splices were the grid method is T-group method; however, it
turns out that there are no suitable methods in the 147.
The other thing in the fourth column is the composite course
-- that is, the sequence of lead-end codes that make up the
course. There are eight of these corresponding to the eight
possible irregular lead-end codes.
Base Composite Parker Base Composite Parker
---------------------- ----------------------
S HKJKH NLJKH V NLMLN NLJKL
HLJNK HLJNN
HKMLK HKMLN
NKMHH NKMHL
P KJGJK NJGMK T LMOML HMOJL
NJNJG HMHMO
GMKMK OJLJL
Q GHKHG NLHGG W ONLNO OOKKL
GGLLK HKNOO
R JGHGJ GMLJG U MONOM OJKMO
The left hand set of columns corresponds to grid splices
with seconds place lead ends; the right hand to sixths place
ones. Only the S / V line corresponds to the same methods,
because that is the only irregular lead end that produces a
five-lead method with both 2nds and 6ths place lead ends.
It's easy enough to see that, with 2nds place lead ends,
there ought to be 24 ways of ordering the lead heads in the
course. (The first, rounds is fixed, the remaining four can
be in any order.) 20 of these are the composite courses
shown in the second column above (five rotations of each of
the four courses); the remaining four are single method
courses (GGGGG, HHHHH, JJJJJ and KKKKK). Similarly for 6ths
place lead ends.
The 'Parker' column corresponds to courses with mixed lead
heads. A Parker course is not a round block. Just as it
would go false, a bob is called to bring up the course
head 156423 -- the 4th there is observation, and if the 4th
is at the back, a 12 l.e. is rung, and if the 4th is at the
front, a 16 l.e. is rung. As with the 2nds and 6ths place
courses, the lead end/head pairs can crop up in 24 different
orders. 16 are in the table above, and are derived from the
composite courses above.
What are other eight? Six are miscellaneous courses that do
not correspond to a 2nds or 6ths place course because they
have both G and O group methods:
OHGLO GGMOO OOJGG GNJLO OHMKG GNOKG
The remaining two are derived from single-method courses,
instead of composite courses. These are are standard Parker
splices for, say, Cambridge/Primrose and Ipswich/Norfolk.
HLLHH NKKNN
TRIPLE-PIVOT GRID SPLICES
This neglected splice is closely related to the grid splice.
I believe Michael Foulds mentions it in passing in the
fourth of his excellent series of books, but I've lent my
copies to someone and so can't check. (If whoever has them
is reading this, can I have them back?)
When I generated the list of grid splices, I asked my
computer to generate a list of all plans with a
twelve-element part-end group (A_4). This is effectively
looking for palindromic courses that can be rung in each of
the six courses to give the extent.
Obviously this produced all the single-method plans, as well
as all the three-lead splices (showing up with methods in
the ratio 3:2, with the splice applied solely based on the
position of a single observation) and all the six-lead
splices (with methods in the ratio 4:1). Grid splices
turned up with a method ratio of 2:2:1; plans with both
three- and six-lead splices and plans with two three-lead
splices also had a 2:2:1 ratio, looking much like grid
splices, except that all the methods had the same lead end
group. That was all I anticipated finding.
In fact I found a further 51 palindromic course plans, many
of which are triple-pivot grid splices.
In grid splices, the grid method has a lead end that swap
two pairs of bells. For example, the grid method to the
Cm/Ip/Bo grid splice is King Edward which has lead end
156423, swapping 2-5 and 3-6; 4 is the pivot bell. 2-5 are
then used as fixed bells for the three-lead splice with Cm,
3-6 for the three-lead splice with Ip, and 4 for the
six-lead splice with Bo. That means the half lead change in
the grid method must be in the 2,2,1,1 equivalence class --
by which I mean it has two pairs of bells swapping and two
fixed bells.
But what if the half-lead change in the grid method is in
the 3,1,1,1 equivalence class? Clearly that's not possible
for an ordinary change, but there's no reason why the grid
method shouldn't have a jump change at the half lead. For
example, Norwich with the following underwork:
234165 +
243615 +
423651 +
246315 -
426351 - } jump
642351 - } change
462315 -
643251 +
463215 +
436125 +
It's fairly straightforward to see that this 'method' has a
six-lead splice with Bedford (with 3 fixed), and also with
Old Oxford (with 5 fixed). (The 3 just rings pivot bell
Bedford, and the 5 pivot bell Old Oxford.) Perhaps less
clearly, it also has a three-lead splice with with Marple
with 3 and 5 fixed. That's because we can relabel 3 and 5
at the half-lead and make a corresponding relabelling to 2,
4 or 6 to preserve parity
This allows us to take an extent of this 'method', do a
six-lead splice with Be when the observation is 3rds place
bell, do another six-lead splice with Ol when the
observation is 5ths place bell, and do three-lead splices
with Ma whenever the observation is not 3rds or 6ths place
bell. The reason for the name (triple-pivot grid splice) is
that the observation bell rings the pivot bell in all three
methods.
Joining the parts up can be a little delicate because each
course fragments into two bits meaning lots of bobs are
required, but it's often possible. For example,
123456 Ta
- 156423 Ol
- 134562 Ta
125634 Ta
- 134625 Ol
163542 Ta
- 142563 Be
163254 Ta
- 154263 Ta
132654 Be
---------
- 125463
Twice repeated;
no 65s at back
In total, there are 37 triple-pivot grid splices using
methods from the 147:
X Y Z Additional splices
-----------------------------------------------------
Av/Ca So/Pn Ke/Ce [3-lead: Ca/Gl] *
C3/C2 So/Pn Cc/Pv/Mp/By/Bh/Bw
Ma/Ta Bm/Ol Be [6-lead: Ma/Ki]
Bs Bu Ta [6-lead: Bs/Cm/Su]
[* = these plans cannot be joined up]
The method(s) in the X column are the three-lead splice of
which there are 18 leads -- the six splice-slots involving
the non-observation bells (a,b), (a,c), (a,d), (b,c), (b,d)
and (c,d). This means that when there are two methods (X1
and X2) in the X column, we can incorporate both.
Clearly there's one way of having just X1, and one way of
having one slot of X2. Two X2 slots: either they overlap or
they don't: (a,b), (a,c) or (a,b), (c,d). And there are
three ways of chosing three slots:
a --- b a --- b a --- b
/ \ / / \
/ \ / / \
c --- d c d c d
The first of these is a chiral pair. That gives twelve
plans when there are two methods in the X column. So there
are 144 = 12 * 2 * 6 plans in the cluster containing C2/C3
as X.
But with 18 leads of an X method, if X has a six-lead
splice, it's possible to incorporate six leads of that
method too -- for example Cm or Su splices with Bs. These
are indicated in the table above, and generate one more
plan for each choice of Y and Z. That means the Ma/Ta
cluster contains 26 = (12+1)*2 plans, and the Bs cluster
contains just 3.
It's also possible to include a three-lead splice into X.
As there's only one set of methods where this applies, we
may as well be concrete about it. The six X slots can each
be either Av or Ca, and if we have enough Ca we can splice
Gl in using Ca's other 3-lead splice. If bell e is the
observation for the triple-pivot grid splice, the six slots
for Ca or Av are (a,b), (a,c), (a,d), (b,c), (b,d) and
(c,d).
If all of these are Ca (i.e. we have no Av), then we have
four Ca-Gl splice slots: (a,e), (b,e), (c,e) and (d,e).
(Bell e must be involved in the Gl splice, because otherwise
some of the Gl leads will fall in the Y or Z methods.) This
gives four extra plans depending on whether 1, 2, 3 or 4 of
these slots are used. It's worth noting that if all four
slots are used, the only leads of Ca remaining are when
the observation is pivot bell.
If one slot is Av -- say (a,b) -- then either a or b must
also be involved in the Gl splice leaving Gl two slots:
(a,e) and (b,e). That gives another two plans. With two
overlapping slots of Av, (a,b) and (a,c), then there's just
one Gl slot: (a,e). The same is true when there's three
mutually overlapping Av slots: (a,b), (a,c), (a,d).
In all, that gives a total of 8 extra plans involving Gl.
So the number of plans in the Av/Ca cluster is (12+8)*2*2 =
80.
Adding these all up, we have 80+144+26+3 = 253 plans.
HIDDEN TRIPLE-PIVOT GRID SPLICES
There's one final developement to the triple-pivot splice
that warrants discussion. The triple-pivot splice works by
having an imaginary method G which has two different
six-lead splices (methods Y and Z, with fixed bells a and b,
respectively) and a three-lead splice (with X when a,b are
fixed). Sometimes X has a six-lead splice with another
method, W, allowing the pivot leads of X to be removed. And
sometimes X has a different three-lead splice with a method,
V, which allows some or all of the non-pivot leads of X to
be removed.
W Y
\ /
\ /
X ------ G
/ \
/ \
V Z
In principle, with suitable methods, this means we might be
able to remove all of X, just leaving V, W, Y and Z. There
are no suitable choices of V,W,X,Y,Z amongst the 147 to make
this possible, but why does X need to be one of the 147?
We've already accepted that the grid method, G, can be
outside of the 147 (e.g. by having jump changes) -- the same
can be true of X.
Any examples of this will have been found by my search for
grid splices, and the only such plans are given below:
V W Y Z
-----------------------
Av Mu/Cl/Gl Te Ti/Tr
This accounts for 6 = 3*2 further plans. It's fairly clear
that we cannot apply any further simple splices to this to
add additional methods.
Sadly none of these plans can be joined up to give a working
extent.
The labeling of W, Y and Z is somewhat arbitrary. Despite
the diagram above, it's not the case that W has a different
status to Y and Z in the splice by virtue of the fact that W
splices with X, and Y and Z with G. As X and G are really
just arbitrary sets of rows, we can recombine them
differently to get two other methods X' and G' such that
it's Y that splices with X' and W and Z with G'.
I wonder whether further investigation of this kind of use
of imaginary methods might yield a general theory of
splicing that would allow us to understand splicing of three
or more methods as well as we currently understand the
splicing of two methods.
SUMMARY
My initial intention in this email was to look just at grid
splices. To create an exhaustive list of grid splices I ran
a plan search using the twelve in-course 1....6 rows as the
part-end group. This found all plans where each course was
the same and also palindromic. As well as finding grid
splices, this turned up some triple-pivot grid splices
(about which I had forgotten), and a generalisation of this.
We've now enumerated all plans related to these by simple
(i.e. course, three- or six-lead) splices.
To update the running count of plans, this shows:
Single method plans . . . . . . . . . . 75 \
Course splices . . . . . . . . . . . . . 108 | See first
Six-lead splices . . . . . . . . . . . . 176 | email
Three-lead splices . . . . . . . . . . . 798 /
Multiple course splices . . . . . . . . 36 \ See second
Multiple six-lead splices . . . . . . . 286 | email
Multiple three-lead splices . . . . . . 412 /
Combined course & three-lead splices . . 198 \ See third
Combined six- & three-lead splices . . . 163 | email
Other simple extents with four methods . 28 /
Grid splices . . . . . . . . . . . . . . 124 \ See this
Triple-pivot grid splices . . . . . . . 253 | email
Hidden triple-pivot grid splices . . . . 6 /
---------------------------------------------
TOTAL . . . . . . . . . . . . . . . . . 2663
Only 1951 plans left to explain, and these promise to be
particularly interesting as we've now basically exhausted
the standard splicing recipes. Stay tuned.
RAS
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