[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:


That explained 2280 of the 4614 plans.  The remaining 2334 
plans are listed here:


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 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

   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.


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 

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:

      a ------ c
      |        |
   X1 |        | X2
      |        |
      b ------ d

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.


[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:


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.



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 

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 

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 = 

Adding these all up, we have 80+144+26+3 = 253 plans.


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 

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.


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.


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