Spliced treble-dodging minor - 4

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Richard Smith richard at ex-parrot.com Sat Oct 16 02:10:49 BST 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 preceding parts.

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'.

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 relabelled] > | > (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 dropping 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 relabelling, 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 development 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 labelling 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.