Palindromes in Parts

Consider a particular case of a palindrome in parts, that of a 3-part palindrome with part-ends A1, A2, A3 forming group [3.02]:

               |                         |
    P1   4 6 2 5 8 3 7             4 5 3 6 7 2 8   Q1
               ^                         |
               |            W1           V
    A1   2 3 4 5 6 7 8  <--------  3 2 4 6 5 8 7   Z1
               ^                         |
               |            W2           V
    A2   3 4 2 5 6 7 8  <--------  4 3 2 6 5 8 7   Z2
               ^                         |
               |            W3           V
    A3   4 2 3 5 6 7 8  <--------  2 4 3 6 5 8 7   Z3

As transpositions, the three changes W1, W2, W3 are the same, all being at Bobs; but as transfigures they are different, and it is as transfigures thay they must be considered:

                W1  (2x3, 4, 5x6, 7x8)
                W2  (2, 3x4, 5x6, 7x8)
                W3  (2x4, 3, 5x6, 7x8)

There will be three more apices at the other ends of the palindrome. (A1, A2, A3) will constitute a row-type under the group [3.02], and so will (Z1, Z2, Z3). We may consider these two row-types to be images in the multiple axis of the palindrome.

P1 is the row 4625837 which appears somewhere as a leadhead in the branch of the palindrome starting with A1, and is to be regarded as representative of any row in that branch. Q1 is its image in the loop ending Z1-A1. What will be the corresponding rows in the other loops?

       P1   4 6 2 5 8 3 7         Q1   4 5 3 6 7 2 8
       P2   2 6 3 5 8 4 7         Q2   2 5 4 6 7 3 8
       P3   3 6 4 5 8 2 7         Q3   3 5 2 6 7 4 8

These six rows form the group [7.40] of order 6, and its rows are transpositions of the group of rows (A1, A2, A3, Z1, Z2, Z3).

A crucial fact is this: since (A1, A2, A3, Z1, Z2, Z3) form a group, they marshall the extent of the rows being considered, into discrete sets of 6 such as (P1, P2, P3, Q1, Q2, Q3). Thus although the composition is a 3-part one, the palindromic structure is equivalent to a 6-part, three of the parts being rung in reverse, and this 6-part structure has as its basis a group of order 6.

Is this group structure of a palindrome generally true? No, it is not (refer to the section Sufficient but Not Necessary later in this paper), but it is seen in the great majority of observed palindromes and, like the technique of part composition in Price 1989, it forms the basis of an extremely useful way of producing peals. It is desirable for the group of transfigures of the part-ends, plus the set of transfigures at the apices, to form an outer group of double the order. If this is so, then all rows, and in particular all leads of a treble-bob major method, will fall into discrete image sets, and proof may be conducted in terms of "image-pair types", i.e. types of image-pairs of leads (see Price 1989 for the definition of "types"). From the pragmatic viewpoint, the need is for a routine to discover peals, and the above theory provides us with one.

The method of proof by image-pairs of leads really is a matter of significance. Take the case of Bristol Surprise Major, using as "universal set" the 2,520 lead-heads which are in-course (positive parity). Every lead is false with 6 other leads in the set. If now we construct a one-part palindrome, most leads pair off into image-pairs (a few mid-lead apex leads remaining unpaired) and the falsity of these image-pairs is the falsity of the leads which they contain. But are there 12 image-pairs false with a given image-pair? No; there is again a maximum of 6 false. And this economy is true for a larger number of parts, owing to group structure. The palindrome scores over non-palindromic composition as it effectively doubles the number of parts being assembled without any increase in the extent of the proof scale.

This goes a long way to explain the observed fact that, when peals of treble-bob are composed by an exhaustive tree search on a trial-and-error basis, many of them turn out to be palindromes, particularly so when long lengths are sought; in fact it has been suggested that all maximum lengths are palindromes. The palindrome asserts itself more particularly in the field of treble-bob, because of the falsity problem, but it is known in peals of Stedman Triples, where the apices are in mid-six. However, Stedman has the limitation that all the rows must appear in a peal, hence one cannot select or reject at will from possible sixes with mid-six apices, one has to use a comprehensive selection of them.

 

Possible Palindromic Systems on up to 7 Working Bells

In the following list of palindromic systems, the nomenclature of groups used is that outlined in Price 1996; further to this, rounds itself, as the group of a 1-part peal, is styled as [0.01] by an extension of the same nomenclature (mathematically speaking, this is the Identity Group). Up to seven working bells are considered, the ones employing more than five necessarily parting the tenors.

The first group quoted is the part-end group of transfigures, given in the first column under the group name; the second group includes the first column, together with the apical transfigures which are given in the second column under its name, so that the order of the second group is twice that of the first.

The first group is a normal subgroup of the second and of half its order, though in the case of the one-part palindromes the relationship is trivial. Subgroups of half the order of a group are always normal subgroups, a well-known fact in group theory, hence in looking for possible outer groups containing a given part-end group, one need only look for the "normal supergroups" of twice the order of the part-end group, listed in Price 1996.

The process of searching for possible palindromic systems is therefore:

  1. Consider a particular group for part-heads.
  2. Look up the normal supergroups (a terminology defined in Price 1996 of the particular group, selecting only those of twice the order.
  3. Examine the transfigures of the outer group which do not occur in the inner group, i.e. the possible apical transfigures. There must be at least one apical transfigure of a useful kind, of which there are two:

                *-kind  (2,2,2,1)       $-kind  (2,2,1,1,1)            
    

    The symbols * and $ are used in the ensuing list of palindromic systems in this way. The comparison of the two groups may be carried out quickly by using the tables in Price 1996, where the columns corresponding to * and $ may be highlighted.

  4. If such a minimum of apical transfigures exists, a data file may be made up from the groups for use in a computer program to compile arrays for a tree search.

The last process 4 may not be straightforward, as if one group is given as a subgroup of the other, the sets of rows given in Price 1996 may not be directly related in this way; the transposing or transfiguring of one may be necessary. The dihedral groups are particularly difficult to handle. There is no real difference between the two forms of the part-end group:

         2 3 4 5 6 7 8     and      2 3 4 5 6 7 8
         3 2 5 4 6 7 8              2 4 3 6 5 7 8

Both are manifestations of the group [4.07] (two pairs of working bells swapping in tandem) and differ only by a transposition. Either could be used to initiate a computer search, as the computer will generate all possible variations of the types involved, and only the particular labelling of the types will be different; hence no loss of generality is involved, as type labelling is an arbitrary process. In the list below, the variation chosen is that which might be useful for an actual peal of treble-bob major.

The rows of the groups are annotated thus: Rows of the parthead group on the left which indicate the possibility of P/B Q-sets of disposable calls (3-part shift) are marked "b"; and of P/S or B/S disposable calls (2-part shift) are marked "s". With the set of apical transfigures making up the overall group, those of practical use in a major treble-bob method are transfigures of kind (2,2,2,1) marked "*" and of kind (2,2,1,1,1) marked "$".

In the listings the following notations are observed:

@ denotes an apex of the palindrome. If placed within a leadhead or on either side of it, the apex is a mid-lead one.

b denotes a transfigure of a leadhead-type of kind (3,1,1,1,1) which will give rise to P/B disposable calls in the resulting blocks.

s denotes similarly a transfigure of kind (2,1,1,1,1,1) giving rise to P/S and B/S disposable calls.

* apical transfigure of useful kind (2,2,2,1)

$ apical transfigure of useful kind (2,2,1,1,1)

(TT) after a system heading denotes that the tenors may be kept together. For this to be true, the part-head group must involve not more than 5 working bells, and both apical transfigures used must allow 7x8 to swap.

The links used in the peals below may be direct or asymmetric. A direct link is a change in one call in a palindrome which does not disturb the sequence of the remaining calls (a Plain counts as a call); such direct links are of B for P (B/P), S/P or S/B, and can only occur when the parthead group contains the requisite kinds of transfigures. As noted above, these are labelled against the parthead group as 'b' or 's'.

An asymmetric link is a linkage between points in different blocks which are not in the same position in the palindrome. The two positions are labelled '%' and '&'. Such links are in practice 2-fold links, S/P or S/B for several reasons: 3-fold asymmetric links B/P are much rarer and more difficult to find; and if they exist, the Q-set parity law may defeat the required linkage into one round block, whereas a 2-fold link is more amenable. Asymmetric links make the conductor's task more difficult, and a palindromic system requiring them may be considered inferior.

The listings are intended to be used in conjunction with the description of groups given in Price 1996, which are essential to the understanding of the palindromic systems; repetition here is thought to be unnecessary.

In many cases, examples are given of peals of Plain Bob Major, because of the difficulties of falseness in Treble-Bob or Surprise methods. Either falseness appeared to make such peals impossible, or else the arrays involved were beyond the capacity of the computer available, or of its software. One-Part Palindromic Systems


[0.01] order 1p  [4.07] order 2p    (TT)

 2 3 4 5 6 7 8    2 3 4 6 5 8 7 $

The apical transfigure available requires a single at each apex. Six short-course peals of Cambridge Major were given in Price 1989, of which one is given below. Superlative enables peals with full courses to be found on this system. Although the overall group [4.07] is a positive one, the two sides of the palindrome are rung as mutually reverse, hence leads are out-of-course with their image leads.

5,120 Cambridge S M     5,184 Superlative S M

 B   H   2 3 4 5 6     B  M  W  H   2 3 4 5 6
 -----------------     ----------------------
 x   -   2 3 5 6 4              -   4 2 3 5 6
 5   -   5 2 3 6 4              -   3 4 2 5 6
 x   S   2 5 6 4 3        S  S      5 4 2 6 3
 x   -   2 5 4 3 6        -  -  -   3 6 2 4 5
 3   S   6 3 4 2 5        -     S   2 5 6 4 3
 x   -   6 3 2 5 4           S  S   4 6 5 2 3
 3   -   5 4 2 6 3        S  -  -   5 2 3 6 4
 2   S   6 4 3 2 5        S  -  -   3 6 4 2 5
 2   -   4 2 5 3 6        -     S   4 5 6 2 3
 x   S@  2 4 3 6 5           S  S   2 6 5 4 3
 ... and               x            6 4 2 3 5
  reflected to ...        S  S  S@  3 2 4 5 6
 5   -   3 2 6 4 5     .. and reflected to ..
 x   S@  2 3 4 5 6              S@  2 3 4 5 6
 -----------------     ----------------------


[0.01] order 1p [6.37] order 2m (TT) 2 3 4 5 6 7 8 3 2 5 4 6 8 7 *

The apical transfigure (2,2,2,1) enables peals with bobs only on this system. Examples in Price 1989. When attempts were made to find a one-part palindromic peal of Kent Treble Bob Major with all 144 CRUs, the shortest length was 4,800! 5,248 is the next palindromic size (Graham Scott has a non-palindromic 5,152). Where there is an apex labelled 2@ it is in mid-lead between two Home bobs. In the 4,800 the 4th is the place-maker at each apex, and the pairs 2x3, 5x6, 7x8 swap. It was found not possible to do the same for Oxford TBM as this method has its CRUs in relatively diverse courses.

     4,800  Kent Treble Bob Major  5,248

M  B  W  H   2 3 4 5 6    M  B  W  H   2 3 4 5 6
----------------------    ----------------------
2     2      3 5 2 6 4    2     2      3 5 2 6 4
2        2   5 3 4 6 2    2  -  2      6 2 4 5 3
2        1   5 2 3 6 4    2        2   2 6 3 5 4
2  -  1  2@  2 4 3 6 5    1  -  2   @  5 2 3 6 4
1  -  2  1   3 6 2 4 5    2  -  1  2   2 4 3 6 5
      2  2   4 2 6 3 5          2      4 6 3 2 5
      2      2 3 6 4 5    2  -  2      2 3 5 6 4
2     2      3 4 2 5 6    2     2      3 6 2 4 5
2  -  2      5 2 6 4 3    2  -         6 4 5 2 3
2        2   2 5 3 4 6    1     2  1   3 4 2 5 6
1  -  2   @  4 2 3 5 6          2  1   2 4 5 3 6
2  -  1  2   2 6 3 5 4          1  2@  2 5 3 4 6
      2      6 5 3 2 4    1        1   6 3 5 4 2
2  -  2      2 3 4 5 6    2        1   6 2 3 4 5
----------------------    2     1      4 5 6 2 3
                             -  2      2 3 4 5 6
Both blocks: 144 CRUs     ----------------------


Two-Part Palindromic Systems


[2.01] order 2m   [4.06] order 4m    (TT)

 2 3 4 5 6 7 8     2 3 4 5 6 8 7
 3 2 4 5 6 7 8 s   3 2 4 5 6 8 7 $

As there is only one usable apex transfigure, two separate 1-part blocks must result, to be linked by a P/S or B/S Q-set. The 5,120 Superlative SM below consists of two separate identical palindromes. Curiously, at every plain Home 5 & 6 are dodging in 3-4 and a pair of singles in any of these positions will effect linkage.

         5,120 Superlative Surprise Major

B   M   W   H   2 3 4 5 6   B   M   W   H   2 3 4 6 5
-------------------------   -------------------------
        S   S   5 4 3 2 6           S   S   6 4 3 2 5
x           -   5 4 2 6 3   x           -   6 4 2 5 3
x               4 6 5 3 2   x               4 5 6 3 2
    S           2 6 5 3 4       S           2 5 6 3 4
    S   S       3 6 5 4 2       S   S       3 5 6 4 2
x           S   6 3 4 2 5   x           S   5 3 4 2 6
        S   S@  2 4 3 6 5           S   S@  2 4 3 5 6
    S       S   5 3 4 6 2       S       S   6 3 4 5 2
x               3 6 5 2 4   x               3 5 6 2 4
    S   S       2 6 5 4 3       S   S       2 5 6 4 3
        S       4 6 5 2 3           S       4 5 6 2 3
x           -   4 6 2 3 5   x           -   4 5 2 3 6
x           S   6 4 3 5 2   x           S   5 4 3 6 2
    S       S@  2 3 4 5 6       S       S@  2 3 4 6 5
-------------------------   -------------------------

N.B. As in many of the examples, the working bells in the peal do not correspond exactly to the group [2.01] as given. In the group listing 2x3 swap, and the one apical transfigure $ has 2x3 and 7x8 swapping; but in the peal 5x6 are the swapping bells between parts, and at the apices 5x6 and 7x8 swap.


[2.01] order 2m [6.36] order 4m (TT) 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ 3 2 4 5 6 7 8 s 3 2 4 6 5 8 7 *

Both kinds of apical transfigure must be used for a 2-part block, * and $, from reasons of parity. In the 5,056 Superlative SM below, there is an apex at mid-lead between the central pair of Singles at M and W; and another at the final Single Home. The peal is an exact 2-part, and in this case follows exactly the transfigures as given in the above listing of the groups concerned, with for the mid-lead apex and $ for the final one.

5,056 Superlative S. Major

B   M   W   H    2 3 4 5 6
--------------------------
            -    4 2 3 5 6
    S   S        5 2 3 6 4
x                2 6 5 4 3
        -   -    5 4 2 6 3
x           S    4 5 6 3 2
    S            2 5 6 3 4
    S @ S        3 5 6 4 2
        S   S    4 6 5 3 2
x           -    4 6 3 2 5
    -            3 6 5 2 4
x                6 2 3 4 5
    S   S   -    3 4 2 5 6
            S@   3 2 4 5 6
--------------------------


[4.07] order 2p [4.04] order 4p 2 3 4 5 6 7 8 4 5 2 3 6 7 8 $ 3 2 5 4 6 7 8 5 4 3 2 6 7 8 $

This system inevitably parts the tenors objectionably, and is much inferior to the next one.


[4.07] order 2p [6.27] order 4m (TT) 2 3 4 5 6 7 8 4 5 2 3 6 8 7 * 3 2 5 4 6 7 8 5 4 3 2 6 8 7 *

A Rotating-Sets palindrome. One 2-part block is possible as there are two different apical transfigures of kind *. In the 5,120 Superlative SM below the four working bells 3456 swap in pairs at the apices (refer to the section later on Rotation-Sets Palindromes); at the first of the @x Before apices pairs 3x5, 4x6 swap and at the second one 3x6, 4x5 swap giving an overall swapping of 3x4, 5x6 at the partend 24365. In Price 1989 this category is termed a "Pairs of pairs" palindrome. On probability assumptions the result of a tree search are equally likely to be one 2-part block or two 1-part blocks.

5,120 Superlative S. Major

 B   M   W   H   2 3 4 5 6
 -------------------------
 x           -   2 3 5 6 4
     -   S   S   6 4 3 5 2
@x           S   4 6 5 2 3
     S   -   -   5 2 3 6 4
 x               2 6 5 4 3
         -       4 2 5 6 3
     S       S   3 5 2 6 4
 x           -   3 5 6 4 2
             -   6 3 5 4 2
@x           -   6 3 4 2 5
             -   4 6 3 2 5
 x           S   6 4 2 5 3
         S       5 4 2 6 3
     -           2 4 3 6 5
 -------------------------


[4.07] order 2p [6.35] order 4p (TT) 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ 2 4 3 6 5 7 8 2 4 3 5 6 8 7 $

Both apical transfigures are of kind (2,2,1,1,1) hence 2-part peals might be achieved with Singles at both apices. The peal of Pudsey below has this; at the first apex the bells 2x4 (and 7x8) swap, at the second bells 3x5. The overall swaps of the part are both these. Also in the category are the short-course peals of 5,120 Cambridge SM given in Price 1989.

5,120 Pudsey Surprise Major

B   M   W   H   2 3 4 5 6
-------------------------
x               3 5 2 6 4
x           -   3 5 6 4 2
        -       4 3 6 5 2
        S   S@  5 6 3 4 2
    S           2 6 3 4 5
    -       -   5 3 6 4 2
x               3 4 5 2 6
x               4 2 3 6 5
x               2 6 4 5 3
    S           3 6 4 5 2
    -       S@  4 2 6 5 3
        -       5 4 6 2 3
        S       2 4 6 5 3
x               4 5 2 3 6
-------------------------


[4.07] order 2p [6.36] order 4m (TT) 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 2 5 4 6 7 8 3 2 5 4 6 8 7 *

Only one apical transfigure is usable, hence two 1-part blocks are inevitable, requiring asymmetric linkages. Examples of this with Cambridge Major are given in Price 1989: 5,120 with 74 CRUs (the apparent maximum); 5,056; 4,992 in two blocks giving a peal with a coda ending at the treble's snap after Wrong.


[6.37] order 2m [6.27] order 4m 2 3 4 5 6 7 8 4 5 2 3 6 7 8 $ 3 2 5 4 7 6 8 5 4 3 2 7 6 8 *

One apex of each of kind * and $, giving hope of one 2-part block. Unfortunately the arrays involved proved too large to handle, even for plain methods.


[6.37] order 2m [6.36] order 4m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 3 2 5 4 7 6 8 3 2 5 4 6 7 8 $

Hope only of two 1-part blocks, to be linked asymmetrically. But again, the arrays involved proved too large to handle, even for plain methods.

 


 

Three-Part Palindromic Systems


[3.02] order 3p   [5.07] order 6p    (TT)

 2 3 4 5 6 7 8     2 4 3 5 6 8 7 $
 3 4 2 5 6 7 8 b   3 2 4 5 6 8 7 $
 4 2 3 5 6 7 8 b   4 3 2 5 6 8 7 $

The presence of alternative apical transfigures of kind $ makes it possible to construct an exact 3-part peal, as the 5,184 of Bristol below. At the first S@ apex the pair 4x5 swap, while at the second 6x4 swap giving overall a 3-part rotation of 456. Note that, unlike in the more usual 3-part Cyclic-Dihedral system [3.02]-[7.40], the fixed bells 2, 3 do not swap at the apices.

5,184 Bristol S. Major

M    W    H    2 3 4 5 6
------------------------
-         S    4 6 3 5 2
     S    S    5 3 6 4 2 
-         S@   6 2 3 4 5
     -    S    4 3 6 2 5
S         S    5 6 3 4 2
     -    S@   2 3 5 6 4
------------------------
3-part    (Singles 1234)


[3.02] order 3p [7.39] order 6p (TT) 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ 3 4 2 5 6 7 8 b 3 4 2 6 5 8 7 4 2 3 5 6 7 8 b 4 2 3 6 5 8 7

One usable apical transfigure, making three 1-part blocks feasible with singles at both apices. P/B Q-sets might be found for block linkage. The palindrome of Peterborough SM below extends to a peal using a P/B Q-set by Bob for plain at either of the two locations marked % (not both) making a 3-part peal. The more usual methods did not cooperate! The fixed bells 5, 6 swap at the apices, but not the rotating bells.

  5,184 Peterborough S.M.

M   B   W   H    2 3 4 5 6
--------------------------
        S        5 3 4 2 6
S       -   -    4 2 6 3 5
-   x    %  -    6 2 3 4 5
S       S   S@   4 3 2 5 6
S       S   -    2 5 3 6 4
 %  x   -   -    2 4 5 6 3
-       S        6 4 3 5 2
S           S@   2 3 4 5 6
--------------------------


[3.02] order 3p [7.40] order 6m (TT) 2 3 4 5 6 7 8 2 4 3 6 5 8 7 * 3 4 2 5 6 7 8 b 3 2 4 6 5 8 7 * 4 2 3 5 6 7 8 b 4 3 2 6 5 8 7 *

A Cyclic-Dihedral palindrome. All the apical transfigures are of kind * making the system tractable. The commonest 3-part system. P/B links are available but not necessary. In the 5,184 Superlative SM below the first apex is at a Bob Before with 3 making the place and 2x4, 5x6, 7x8 swapping; the second is at a Bob at Home, with 2 making the bob and 3x4, 5x6, 7x8 swapping.

  5,184 Superlative S.Major

 B   M   W   H    2 3 4 5 6
 --------------------------
 x           -    2 3 5 6 4
             -    5 2 3 6 4
             -    3 5 2 6 4
     S   S        6 5 2 4 3
@x                5 4 6 3 2
     S   S   -    6 3 4 2 5
             -    4 6 3 2 5
             -    3 4 6 2 5
 x           -@   3 4 2 5 6
 --------------------------
                   3-part


[6.33] order 3p [6.15] order 6m 2 3 4 5 6 7 8 5 6 7 3 4 2 8 3 4 2 6 7 5 8 6 7 5 4 2 3 8 * 4 2 3 7 5 6 8 7 5 6 2 3 4 8

The peal below reproduces exactly for partends the group [6.33] as listed on the left, and the apices are all of the same one available transfigure (hence rounds must issue in one part). No disposable calls are available, so linkage must be asymmetric. It will be found that S for - at one of the positions marked % will shunt to the other position % in another part, hence two P/S Q-sets will link the three parts (not recommended for actual performance!).

                  5,184 Plain Bob Major

  2345678     6325478     8432567     7635284     4387526     6528347
  -------     -------     -------     -------     -------     -------
- 2357486     3567284     4286375     6578342   - 4372865     5864273
  3728564   - 3578642     2647853     5864723     3246758     8457632
-@3786245     5834726     6725438    @8452637     2635487     4783526
  7634852     8452367     7563284     4283576     6528374   - 4732865
- 7645328     4286573   - 7538642     2347865     5867243     7246358
  6572483     2647835     5874326   - 2376458     8754632   S 2765483
  5268734     6723458     8452763     3625784     7483526    %7528634
- 5283647     7365284     4286537     6538247   - 7432865     5873246
  2354876   S 3758642     2643875     5864372     4276358     8354762
S 3247568     7834526     6327458     8457623     2645783   - 8346527
  2736485     8472365   - 6375284     4782536     6528437     3682475
- 2768354     4286753     3568742     7243865     5863274   - 3627854
  7825643     2645837     5834627   - 7236458     8357642     6735248
- 7854236     6523478     8452376     2675384     3784526   S 7654382
  8473562     5367284     4287563     6528743   S 7342865    %6478523
  4386725   - 5378642     2746835     5864237     3276458   - 6482735
  3642857     3854726     7623458     8453672     2635784     4263857
S 6325478     8432567   - 7635284     4387526     6528347     2345678
  -------     -------     -------     -------     -------     -------
        3-part;  Partends 3426758, 4237568, 2345678;  see text


[6.33] order 3p [6.16] order 6m 2 3 4 5 6 7 8 7 6 5 4 3 2 8 * 3 4 2 6 7 5 8 6 5 7 3 2 4 8 * 4 2 3 7 5 6 8 5 7 6 2 4 3 8 *

A Rotating-Sets palindromic system (see the later section on this topic). The tenors for major must be parted, and the group [6.33] is more useful in the field of Stedman and Grandsire but the 9-part palindromic system [6.31]-[6.12], which is an extension, is very useful. In the peal below, the apices are:

  Plain lead at 3526478   (3x5, 2x6, 4x7, 8)   transfigure 6573248
  Mid-lead   at 8365472   (8, 2x7, 3x6, 4x5)   transfigure 7654328

The complex effect of these is to rotate the trios 234, 567.

                     5,088 Bristol Surprise Major

  2345678     4283756     7435862     7456283     2476583     7683245
  -------     -------     -------     -------     -------     -------
  4263857   - 8423756     3784256     5724368   - 7246583   - 8763245
  6482735     2874635     8327645   - 2574368   - 4726583     6827534
  8674523     7268543   - 2837645   - 7254368     2457368     2658473
- 7864523     6752384   - 3287645     5732846   - 5247368   - 5268473
  6758342   - 5672384     8362574     3587624     4532876   - 6528473
  5637284     7536428     6853427    @8365472@    3485627     2645387
 @3526478     3745862     5648732     6843257     8364752     4236758
  2345867   - 4375862     4576283     4628735     6873245   - 3426758
  4283756   - 7435862   - 7456283     2476583   - 7683245     -------
  -------     -------     -------     -------     -------     3-part


[6.33] order 3p [6.32] order 6p 2 3 4 5 6 7 8 2 4 3 5 7 6 8 $ 3 4 2 6 7 5 8 3 2 4 6 5 7 8 $ 4 2 3 7 5 6 8 4 3 2 7 6 5 8 $

The choice of $-apices enables two independent Cyclic-Dihedral systems on the trios 234, 567 to be produced, giving an exact 3-part:

                        5,088 Plain Bob Major

  2345678     8475362     2367485     5876423     2534678     5623847
  -------     -------     -------     -------     -------     -------
- 2357486   - 8456723     3728654     8652734     5427386     6354278
  3728564     4682537     7835246     6283547   - 5478263     3467582
  7836245     6243875     8574362     2364875     4856732   S 4378625
  8674352     2367458   - 8546723     3427658   - 4863527     3842756
  6485723     3725684     5682437     4735286     8342675     8235467
- 6452837     7538246     6253874   - 4758362     3287456   - 8256374
  4263578     5874362     2367548     7846523   S 2375864     2687543
  2347685   - 5846723     3724685     8672435     3526748   - 2674835
  3728456     8652437     7438256     6283754     5634287     6423758
  7835264     6283574     4875362     2365847   - 5648372   S 4635287
  8576342     2367845   - 4856723   - 2354678     6857423     6548372
  5684723     3724658     8642537     3427586     8762534     5867423
- 5642837     7435286     6283475     4738265   S 7823645     8752634
  6253478     4578362     2367854     7846352   S@8734256     7283546
  2367584   S@5486723     3725648     8675423   S 7845362     2374865
  3728645     4652837     7534286     6582734     8576423     3426758
  7834256     6243578   - 7548362     5263847     5682734     -------
  8475362     2367485     5876423   S 2534678   - 5623847     3-part
  -------     -------     -------     -------     -------


Four-Part Palindromic Systems


[4.04] order 4p  [6.21] order 8m    (TT)

 2 3 4 5 6 7 8    2 3 4 5 6 8 7
 3 2 5 4 6 7 8    3 2 5 4 6 8 7 *
 4 5 2 3 6 7 8    4 5 2 3 6 8 7 *
 5 4 3 2 6 7 8    5 4 3 2 6 8 7 *

The choice of apical transfigures make it possible to find two 2-part blocks of Rotating-Sets kind, but group [4.04] has no disposable linkages. A mixed universal set is required in order to find asymmetric link Q-sets. The peal below is given in two 2-part blocks, as only one part, or one block, would not make clear the nature of the partends. 4 is the observation bell and 2356 perform the group [4.04]. The blocks may be linked by S for - at % (the first of the two Bobs Wrong) to shunt to &.

          5,120 Bristol Surprise Major

M   W   H   2 3 4 5 6   M   W   H   3 2 4 6 5
---------------------   ---------------------
S  %2   S   3 4 5 6 2   S   2   S   2 4 6 5 3
-       -@  2 5 4 6 3   -       -@  3 6 4 5 2
    -   S   6 4 2 5 3      &-   S   5 4 3 6 2
2   S   -@  6 5 4 3 2   2   S   -@  5 6 4 2 3
---------------------   ---------------------
S   2   S   5 4 3 2 6   S   2   S   6 4 2 3 5
-       -@  6 3 4 2 5   -       -@  5 2 4 3 6
    -   S   2 4 6 3 5       -   S   3 4 5 2 6
2   S   -@  2 3 4 5 6   2   S   -@  3 2 4 6 5
---------------------   ---------------------


[4.04] order 4p [6.23] order 8p (TT) 2 3 4 5 6 7 8 3 4 5 2 6 8 7 3 2 5 4 6 7 8 5 2 3 4 6 8 7 4 5 2 3 6 7 8 2 5 4 3 6 8 7 $ 5 4 3 2 6 7 8 4 3 2 5 6 8 7 $

The two $-type apices enable two 2-part blocks to be constructed. In the 5,376 below the 2 is fixed bell rather than 6, and 3456 perform group [4.04]. As well as 7x8, 3x4 and 5x6 swap at different apices. To link the blocks, S for - at Home positions % to &.

     5,376 Plain Bob Major

W  M  H    2 3 4 5 6    2 5 6 3 4
--------------------    ---------
      2    3 4 2 5 6    5 6 2 3 4
   2  -%   3 6 4 5 2    5 4 6 3 2
S     -    4 5 6 3 2   &6 3 4 5 2
S  -  S@   6 2 5 4 3    4 2 3 6 5
-  S  -    5 3 6 2 4    3 5 4 2 6
   S  -    6 4 3 2 5    4 6 5 2 3
2     2    2 3 4 6 5    2 5 6 4 3
      S@   2 4 3 6 5    2 6 5 4 3
--------------------    ---------
repeated   2 3 4 5 6    2 5 6 3 4
--------------------    ---------


[4.05] order 4m [4.03] order 8m 2 3 4 5 6 7 8 2 5 4 3 6 7 8 3 4 5 2 6 7 8 3 2 5 4 6 7 8 $ 4 5 2 3 6 7 8 4 3 2 5 6 7 8 5 2 3 4 6 7 8 5 4 3 2 6 7 8 $

This system inevitably parts the tenors, but is very close to the much more musical [4.05]-[6.25]. Use of both apical transfigures will result in a Rotating-Sets 2-part block, with complementary block to be joined asymmetrically. The block below reproduces exactly the figures above. It is a 2-part block 2345 - 4523 - 2345, its complement being 3452 - 5234 - 3452. Linkage may be effected by Single at any position %, shunting to the other position % in the other block, with corresponding return Single after two whole parts. The first apex has 2x5, 3x4 and the second apex 2x3, 4x5 making overall 2x4, 3x5 for the part; 678 make places.

                        5,376 Plain Bob Major

  2345678     6237854     5274863     4583267     4523876     4523867
  -------     -------     -------     -------     -------     -------
  3527486     2765348     2456738     5346872     5347268     5346278
  5738264     7524683   S 4263587     3657428     3756482     3657482
- 5786342    %5478236     2348675     6732584     7638524     6738524
  7654823   S 4583762   S 3287456     7268345   - 7682345   - 6782345
  6472538     5346827     2735864   - 7284653     6274853     7264853
  4263785   S 3562478     7526348     2475836     2465738     2475638
  2348657     5237684     5674283     4523768     4523687     4523786
  3825476    %2758346   - 5648732     5346287     5348276     5348267
  8537264     7824563     6853427     3658472     3857462     3856472
S 5876342   - 7846235   - 6832574     6837524     8736524     8637524
  8654723   S@8763452     8267345   - 6872345   S@7862345   - 8672345
- 8642537   - 8735624   S 2874653     8264753     8274653     6284753
  6283475     7582346     8425736     2485637     2485736     2465837
- 6237854     5274863     4583267     4523876     4523867     4523678
  -------     -------     -------     -------     -------     -------


[4.05] order 4m [6.22] order 8m (TT) 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 4 5 2 6 7 8 3 4 5 2 6 8 7 4 5 2 3 6 7 8 4 5 2 3 6 8 7 * 5 2 3 4 6 7 8 5 2 3 4 6 8 7

This system has only one apical transfigure, hence four one-part blocks are inevitable and asymmetric links are needed.

  5,184 Plain Bob Major

W   B   M   H   2 3 4 5 6      2 4 5 6 3      2 5 6 3 4      2 6 3 4 5
-------------------------      ---------      ---------      ---------
S   1       S   3 5 2 6 4      4 6 2 3 5      5 3 2 4 6      6 4 2 5 3
        -       2 5 4 6 3      2 6 5 3 4      2 3 6 4 5      2 4 3 5 6
        S       3 5 4 6 2      4 6 5 3 2      5 3 6 4 2      6 4 3 5 2
        -       4 5 2 6 3      5 6 2 3 4      6 3 2 4 5      3 4 2 5 6
        -    @  2 5 3 6 4      2 6 4 3 5      2 3 5 4 6      2 4 6 5 3
-               6 2 3 5 4      3 2 4 6 5      4 2 5 3 6      5 2 6 4 3
-               5 6 3 2 4      6 3 4 2 5      3 4 5 2 6      4 5 6 2 3
S           %a  2 6 3 5 4   %b 2 3 4 6 5   %c 2 4 5 3 6   %d 2 5 6 4 3
-           S   5 3 2 6 4      6 4 2 3 5      3 5 2 4 6      4 6 2 5 3
    1   S   %d  2 6 5 4 3   %a 2 3 6 5 4   %b 2 4 3 6 5   %c 2 5 4 3 6
-   2@  -       2 3 4 5 6      2 4 5 6 3      2 5 6 3 4      2 6 3 4 5
-------------------------      ---------      ---------      ---------

In the above blocks, 2 is the fixed bell and 3456 rotate cyclically. The four 1-part blocks may be linked at plain Homes by Singles at pairs of positions %a, %b etc., three pairs being required. The first apex is at plain Home, the second at mid-lead between two Befores. At all apices the pairs crossing are 3x5, 4x6, 7x8, with 2 place-making.


[4.05] order 4m [6.25] order 8m (TT) 2 3 4 5 6 7 8 2 5 4 3 6 8 7 $ 3 4 5 2 6 7 8 5 4 3 2 6 8 7 * 4 5 2 3 6 7 8 4 3 2 5 6 8 7 $ 5 2 3 4 6 7 8 3 2 5 4 6 8 7 *

A Cyclic-Dihedral system. A 4-part block may be produced by using one of each kind of apex, * and $; such as in the 5,376 Superlative given below. A case in which two 2-part blocks exist was noted in Price 1989, in the appendix on palindromes, where a peal of Superlative was quoted using the cyclic group of part-ends [4.05], but there were two blocks each of the Rotating-Sets category [4.07]-[6.27] which could only be joined asymmetrically.

5,376 Superlative S. Major

M    W    H    2 3 4 5 6
------------------------
S         S    6 4 3 5 2
          S    6 3 4 5 2
S    -    -@   4 5 2 3 6
-    S    S    3 6 5 2 4
          S    3 5 6 2 4
     S    S@   2 6 5 3 4
------------------------
         4-part


[4.06] order 4m [4.03] order 8m 2 3 4 5 6 7 8 4 5 3 2 6 7 8 3 2 4 5 6 7 8 5 4 2 3 6 7 8 2 3 5 4 6 7 8 4 5 2 3 6 7 8 $ 3 2 5 4 6 7 8 5 4 3 2 6 7 8 $

This system, with its peal below, is similar to [4.05]-[4.03] above, but the latter's partend group is cyclic, whereas this one is not. The individual 2-part blocks have the same structure.

                        5,376 Plain Bob Major

  2345678     4625738     2453687     2453876     2453678     2487635
  -------     -------     -------     -------     -------     -------
- 2357486   - 4653287     4328576     4327568     4327586     4723856
  3728564     6348572     3847265     3746285     3748265   - 4735268
  7836245   - 6387425     8736452     7638452     7836452     7546382
S@8764352     3762854   - 8765324   S@6785324   - 7865324     5678423
  7485623     7235648     7582643     7562843     8572643   - 5682734
  4572836   - 7254386     5274836     5274638     5284736     6253847
- 4523768     2478563     2453768     2453786     2453867   - 6234578
  5346287     4826735     4326587     4328567     4326578    %2467385
- 5368472     8643257     3648275     3846275    %3647285     4728653
  3857624   - 8635472     6837452     8637452   - 3678452     7845236
  8732546     6587324   - 6875324   - 8675324     6835724     8573462
  7284365   - 6572843     8562743     6582743   - 6852347     5386724
  2476853     5264738     5284637     5264837     8264573   - 5362847
  4625738     2453687     2453876     2453678     2487635     3254678
  -------     -------     -------     -------     -------     -------

This 2-part block has partends 2345-3254-2345 and its complementary block has 2354-3245-2354. One pair of singles, linking the plain lead at any of the four % positions in the first block to one of the other kind of % positions in the complementary block.


[4.06] order 4m [6.24] order 8m (TT) 2 3 4 5 6 7 8 4 5 2 3 6 8 7 * 2 3 5 4 6 7 8 s 4 5 3 2 6 8 7 3 2 4 5 6 7 8 s 5 4 2 3 6 8 7 3 2 5 4 6 7 8 5 4 3 2 6 8 7 *

Using each of the available apical transfigures makes it possible to find two 2-part blocks, each a Rotating-Sets palindrome of system [4.07]-[6.27]. Linking shunts are available. The block given below has apices at bobs Before and at plains Home. It may be doubled to give a peal of 6,912 by singling the Home plain at % in alternate parts, swapping bells 3x6 to produce all four combinations of 3x6 and 4x5 swapping.

3,456 Superlative S. Major

 B   M   W   H   2 3 4 5 6
 -------------------------
 -               3 5 2 6 4
     -   S       6 5 4 2 3
         S       2 5 4 6 3
     S           3 5 4 6 2
@-            %  5 6 3 2 4
         S       2 6 3 5 4
     S           4 6 3 5 2
     S   -       5 2 3 6 4
 -            @  2 6 5 4 3
 -------------------------
         2-part


[4.06] order 4m [6.34] order 8m (TT) 2 3 4 5 6 7 8 2 3 4 5 6 8 7 2 3 5 4 6 7 8 s 2 3 5 4 6 8 7 $ 3 2 4 5 6 7 8 s 3 2 4 5 6 8 7 $ 3 2 5 4 6 7 8 3 2 5 4 6 8 7 *

The block below is a 2-part palindrome, which swaps the pair 5x6. It is doubled by swapping 2x3 at the final call, thus completing the group [4.06].

  5,120 Bristol S. Major

 M    W    H     2 3 4 5 6
--------------------------
      --         3 5 4 2 6
-S-        S@    6 4 5 2 3
     -S-         2 4 5 6 3
 --        -@    2 3 4 6 5
--------------------------
 S for - half-way and end.
4-part             80 CRUs


[6.26] order 4p [6.22] order 8m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 3 4 5 2 7 6 8 3 4 5 2 6 7 8 4 5 2 3 6 7 8 4 5 2 3 7 6 8 * 5 2 3 4 7 6 8 5 2 3 4 6 7 8

As there is only one usable apical transfigure, four 1-part blocks are inevitable. Using a mixed universal set and finding asymmetric links is feasible, but it was found that this system is closely related to the system [4.06] order 4m - [6.24] order 8m, by inverting alternate blocks (thus making 6-7 fixed bells) and composition in that system is much easier. Many unlinkable sets of four blocks were found, and the only ones which were linkable were double palindromes belonging rather to the 8-part system [6.23] - [6.20]. The example given under that system was actually produced by this one (regard only the top four partends).


[6.26] order 4p [6.23] order 8p 2 3 4 5 6 7 8 2 5 4 3 7 6 8 $ 3 4 5 2 7 6 8 3 2 5 4 6 7 8 $ 4 5 2 3 6 7 8 4 3 2 5 7 6 8 $ 5 2 3 4 7 6 8 5 4 3 2 6 7 8 $

The mixed nature of the $-apical transfigures enables 6x7 to swap at one apex and not at the other, giving an exact 4-part; in the peal below 2x4 and 6x7 swap at the first, 2x5 and 3x4 at the second.

                        5,120 Plain Bob Major

  2345678     2345786     2348765     5368274     8567234     8675234
  -------     -------     -------     -------     -------     -------
  3527486     3528467     3826457     3857642   - 8573642     6583742
  5738264     5836274   S 8365274   S 8374526     5384726     5364827
  7856342   - 5867342     3587642     3482765     3452867     3452678
- 7864523     8754623     5734826     4236857     4236578     4237586
  8472635     7482536   S 7542368     2645378     2647385     2748365
  4283756     4273865     5276483     6527483     6728453     7826453
  2345867     2346758   S 2568734     5768234   - 6785234   S@8765234
  3526478     3625487     5823647   - 5783642     7563842     7583642
  5637284     6538274   S@8534276     7354826     5374628     5374826
  6758342   - 6587342     5487362     3472568     3452786     3452768
- 6784523     5764823   S 4576823     4236785     4238567     -------
  7462835     7452638     5642738     2648357     2846375     4-part
  4273658     4273586   S 6523487     6825473     8627453
  2345786     2348765     5368274     8567234   - 8675234
  -------     -------     -------     -------     -------


[6.26] order 4p [6.24] order 8m 2 3 4 5 6 7 8 5 4 3 2 7 6 8 * 3 4 5 2 7 6 8 4 3 2 5 6 7 8 4 5 2 3 6 7 8 3 2 5 4 7 6 8 * 5 2 3 4 7 6 8 2 5 4 3 6 7 8

The two available apices enable a Rotating-Sets 2-part block to be had, requiring asymmetric links. The peal below follows the partheads exactly. One can not get the cyclic partheads except by the use of the links % to & and back, which rotate 2345 and swap 67 in order to join the pair of 2-part blocks. At the first apex (plain Home) the swaps are (2x5 3x4 6x7) and at the second (mid-lead) apex (2x3 4x5 6x7), giving overall the swaps (2x4 3x5) for a Rotating-Sets palindromic block. This system is closely related to the 4-part system [4.06]-[6.24] and any pair of its 2-part blocks are equivalent, by inverting one block, to a pair in the other system, which has disposable links.

                        5,056 Plain Bob Major

  2345678     5234876     5237864     5736824     3765824     6384257
  -------     -------     -------     -------     -------     -------
S 3257486     2457368     2756348   - 5762348     7532648     3465872
  2738564     4726583   - 2764583     7254683     5274386   S 4357628
 %7826345   - 4768235     7428635     2478536     2458763     3742586
  8674253     7843652     4873256     4823765     4826537   - 3728465
  6485732     8375426     8345762     8346257     8643275     7836254
- 6453827     3582764     3586427     3685472    &6387452     8675342
  4362578     5236847     5632874     6537824   - 6375824     6584723
S 3427685     2654378     6257348   - 6572348     3562748     5462837
  4738256     6427583   - 6274583     5264783     5234687   S@4523678@
  7845362   - 6478235     2468735     2458637   - 5248376     -------
  8576423     4863752     4823657     4823576     2857463      4-part
- 8562734     8345627    @8345276     8347265   - 2876534
  5283647     3582476     3587462     3786452     8623745
- 5234876     5237864     5736824   - 3765824     6384257
  -------     -------     -------     -------     -------


[6.27] order 4m [6.21] order 8m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 3 2 5 4 6 7 8 3 2 5 4 7 6 8 * 4 5 2 3 7 6 8 4 5 2 3 6 7 8 $ 5 4 3 2 7 6 8 5 4 3 2 6 7 8 $

Using both $-apices gives a two-part block plus a complementary one, the two to be linked by asymmetric calls. In the peal below, 2 is fixed bell and 3x4 the swapping pair; 5678 permute.

                        5,120 Plain Bob Major

  2345678     4728536     6847235     3624758     8356247     6854732
  -------     -------     -------     -------     -------     -------
  3527486   - 4783265     8763452     6435287     3684572     8463527
- 3578264     7346852     7385624     4568372     6437825     4382675
 %5836742     3675428   - 7352846     5847623     4762358   S@3427856
  8654327     6532784     3274568     8752436   - 4725683     4735268
  6482573   - 6528347   S@2346785   - 8723564     7548236     7546382
- 6427835     5864273     3628457     7386245     5873462     5678423
  4763258    %8457632   - 3685274     3674852     8356724     6852734
  7345682     4783526     6537842     6435728   - 8362547     8263547
  3578426     7342865     5764328   - 6452387     3284675     2384675
- 3582764     3276458   - 5742683     4268573   - 3247856   - 2347856
  5236847     2635784     7258436     2847635     2735468     -------
- 5264378   - 2658347   - 7283564   - 2873456     7526384     4-part
  2457683     6824573     2376845     8325764     5678243     (see
  4728536   - 6847235     3624758   - 8356247     6854732      below)
  -------     -------     -------     -------     -------

The complementary block has partends 2436587 - 2438765 - 2436587, linkage being by Singles between the positions %% in different blocks.


[6.27] order 4m [6.25] order 8m 2 3 4 5 6 7 8 4 5 3 2 6 7 8 3 2 5 4 6 7 8 5 4 2 3 6 7 8 4 5 2 3 7 6 8 2 3 5 4 7 6 8 $ 5 4 3 2 7 6 8 3 2 4 5 7 6 8 $

In the peal below, the pair 23 takes the place of 67. The swaps are: first apex (2x3 6x7), second one (2x3 4x5), giving (4x5 6x7) for the part. S for P at % links to & in the complementary block, swapping three pairs (2x3 4x6 5x7) and generating the curious group [6.27].

                        5,120 Plain Bob Major

  2345678     2437856     2574683     5687243     8675423     7685423
  -------     -------     -------     -------     -------     -------
  3527486     4725368     5428736   - 5674832     6582734     6572834
 %5738264     7546283   S@4583267     6453728     5263847     5263748
  7856342   - 7568432     5346872     4362587     2354678     2354687
- 7864523     5873624   S 3567428   S 3428675     3427586     3428576
  8472635   S 8532746     5732684     4837256     4738265     4837265
  4283756     5284367    &7258346     8745362     7846352     8746352
  2345867     2456873   S 2784563     7586423   - 7865423   - 8765423
  3526478     4627538     7426835   - 7562834     8572634     7582634
  5637284   - 4673285   - 7463258     5273648     5283746     5273846
- 5678342     6348752     4375682     2354786     2354867     2354768
  6854723   S 3685427     3548726     3428567     3426578     -------
  8462537     6532874     5832467     4836275     4637285      4-part
  4283675     5267348   S 8526374     8647352     6748352
S 2437856   S 2574683     5687243   - 8675423   S@7685423
  -------     -------     -------     -------     -------


[6.35] order 4p [6.23] order 8p 2 3 4 5 6 7 8 4 5 2 3 6 7 8 $ 2 3 5 4 7 6 8 4 5 3 2 7 6 8 3 2 4 5 7 6 8 5 4 2 3 7 6 8 3 2 5 4 6 7 8 5 4 3 2 6 7 8 $

This system needs a mixed set of 5,040 leadheads as Singles must be used for the apices. The two available apical transfigures enable a Rotating-Sets block on bells 2345. In the peal below there are two 2-part blocks with 2345 doing this, but the blocks can only be linked asymmetrically. A Single at % will shunt to & in the other block, swapping pairs 6x7 and 3x5, with a corresponding return with a Single, & to %.

                        5,120 Plain Bob Major

  2345678     2345687     2345876     2856734     4673285     7286345
  -------     -------     -------     -------     -------     -------
  3527486     3528476     3527468     8623547   - 4638752   - 7264853
  5738264     5837264     5736284    %6384275     6845327     2475638
  7856342     8756342   - 5768342   - 6347852     8562473     4523786
- 7864523   - 8764523     7854623     3765428   S 5827634     5348267
  8472635     7482635     8472536     7532684     8753246     3856472
  4283756     4273856     4283765   S 5728346   - 8734562     8637524
  2345867     2345768     2346857    &7854263     7486325   - 8672345
  3526478     3526487     3625478   - 7846532     4672853     6284753
  5637284     5638274   S 6357284   S@8763425   S 6425738     2465837
  6758342     6857342     3768542   - 8732654     4563287     4523678
S@7684523   - 6874523     7834625     7285346     5348672     -------
  6472835     8462735   - 7842356   S 2754863     3857426     4-part
  4263758     4283657     8275463     7426538     8732564     (see
  2345687     2345876   S 2856734     4673285     7286345       text)
  -------     -------     -------     -------     -------


[6.35] order 4p [6.25] order 8m 2 3 4 5 6 7 8 4 5 2 3 7 6 8 * 2 3 5 4 7 6 8 4 5 3 2 6 7 8 3 2 4 5 7 6 8 5 4 2 3 6 7 8 3 2 5 4 6 7 8 5 4 3 2 7 6 8 *

The two available apical transfigures enable two 2-part blocks of Rotating-Sets kind on 2345 to be made, but as there are no available disposable calls, asymmetric links must be found. In the peal below, S for plain at % will shunt to & in the other block, swapping 2x3, 6x7, and the corresponding return Single is needed, & to %. This system is related to the system [4.05]-[6.25] by block inversion.

                        5,184 Plain Bob Major

  2345678     6328457     4673285     7846532     8746532     7648532
  -------     -------     -------     -------     -------     -------
  3527486   S 3685274     6348752   - 7863425   - 8763425   S 6783425
  5738264    &6537842   - 6385427     8372654     7382654     7362854
  7856342   - 6574328     3562874     3285746     3275846     3275648
  8674523     5462783     5237648     2534867    @2534768@    2534786
  6482735     4258637     2754386     5426378     5426387     5428367
- 6423857   -@4283576   - 2748563     4657283     4658273     4856273
  4365278     2347865     7826435     6748532     6847532     8647532
  3547682     3726458     8673254   S 7683425   - 6873425   - 8673425
  5738426   - 3765284     6385742     6372854     8362754     6382754
- 5782364     7538642     3564827     3265748     3285647     3265847
 %7256843   S 5784326     5432678     2534687     2534876   - 3254678
  2674538     7452863   - 5427386     5428376     5427368     -------
- 2643785   - 7426538     4758263     4857263     4756283     4-part
  6328457     4673285     7846532     8746532     7648532     (see
  -------     -------     -------     -------     -------       text)


[6.35] order 4p [6.34] order 8m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 2 3 5 4 7 6 8 2 3 5 4 6 7 8 3 2 4 5 7 6 8 3 2 4 5 6 7 8 3 2 5 4 6 7 8 3 2 5 4 7 6 8 *

This system, having only one apical transfigure available, would give 1-part blocks, and with no direct links. It is equivalent to system [4.06]-[6.34] by block inversion.


[6.36] order 4m [6.21] order 8m 2 3 4 5 6 7 8 4 5 2 3 6 7 8 $ 3 2 5 4 6 7 8 5 4 3 2 6 7 8 $ 2 3 4 5 7 6 8 s 4 5 2 3 7 6 8 * 3 2 5 4 7 6 8 5 4 3 2 7 6 8 *

In the peal below, the first apex at a plain lead swaps 2x4, 3x5, 6x7; and the second at mid-lead swaps 2x3, 4x5, 6x7 thus making a Rotating-Sets palindrome on 2345 in two 2-part blocks. A Single at apex %@ prevents 6x7 swapping and links the blocks. The pairs swapping from one partend to the next are 2x5, 3x4 as "nicer" partends are prevented by falsity. This system is equivalent to system [4.04]-[6.21] by block inversion.

                        5,184 Plain Bob Major

 @2345678@    6732854     6285473     6285734     8256734     8253746
  -------     -------     -------     -------     -------     -------
  3527486     7265348     2567834   - 6253847     2683547   S 2834567
  5738264   - 7254683     5723648     2364578     6324875     8426375
S 7586342     2478536     7354286     3427685   - 6347258   - 8467253
  5674823     4823765   - 7348562     4738256     3765482     4785632
- 5642738     8346257     3876425     7845362     7538624   - 4753826
  6253487     3685472     8632754     8576423     5872346     7342568
- 6238574     6537824     6285347     5682734     8254763   - 7326485
  2867345     5762348     2564873   - 5623847     2486537     3678254
- 2874653   - 5724683     5427638     6354278     4623875   S 6385742
  8425736     7458236     4753286     3467582   - 4637258     3564827
S 4853267     4873562   - 4738562     4738625     6745382    @5432678@
  8346572   - 4836725     7846325     7842356     7568423     -------
  3687425     8642357   %@8672453     8275463     5872634     4-part*
  6732854     6285473     6285734   - 8256734     8253746   S at % in
  -------     -------     -------     -------     -------   pts 2 & 4


[6.36] order 4m [6.34] order 8m 2 3 4 5 6 7 8 2 3 5 4 6 7 8 3 2 5 4 6 7 8 3 2 4 5 6 7 8 2 3 4 5 7 6 8 s 2 3 5 4 7 6 8 $ 3 2 5 4 7 6 8 3 2 4 5 7 6 8 $

The two apical transfigures available enable one to swap 2x3 at one apex and 3x4 at the next, creating a 2-part block (which is not Rotating-Sets). 6x7 may then be swapped at a disposable call. In the peal below, the swapping pairs are 2x4, 3x5 and 6x7; falseness prevents a "nice" start. This system is equivalent to the system [4.06]-[6.21] by block inversion.

                        5,120 Plain Bob Major

  2345678     7425386     2874536     4258637     4852763     3486725
  -------     -------     -------     -------     -------     -------
  3527486     4578263     8423765   S@2483576     8246537     4632857
- 3578264   - 4586732     4386257     4327865     2683475   - 4625378
S@5386742     5643827     3645872     3746258     6327854     6547283
- 5364827   - 5632478     6537428     7635482   - 6375248    $5768432
  3452678     6257384   - 6572384     6578324     3564782     7853624
  4237586   - 6278543     5268743     5862743     5438627   - 7832546
- 4278365     2864735     2854637     8254637     4852376     8274365
  2846753     8423657   - 2843576   - 8243576     8247563     2486753
  8625437     4385276     8327465     2387465     2786435   - 2465837
- 8653274     3547862     3786254     3726854   - 2763854     4523678
  6387542     5736428     7635842   - 3765248     7325648     -------
 $3764825   - 5762384     6574328     7534682   - 7354286     4-part
  7432658     7258643     5462783     5478326     3478562 
  -------     -------     -------     -------     ------- 
  S for P at one position $ in parts 2, 4.


Five-Part Palindromic Systems



[5.05] order 5p  [5.04] order 10p

 2 3 4 5 6 7 8    2 6 5 4 3 7 8 $
 3 4 5 6 2 7 8    4 3 2 6 5 7 8 $
 4 5 6 2 3 7 8    6 5 4 3 2 7 8 $
 5 6 2 3 4 7 8    3 2 6 5 4 7 8 $
 6 2 3 4 5 7 8    5 4 3 2 6 7 8 $

All apices must be at Singles, 7 and 8 making a place as well as one of the working bells. The latter perform the Cyclic-Dihedral Group. The tenors cannot be kept together, and the next system is much to be preferred.

                        5,120 Plain Bob Major

  2345678     8634725     2785643     8457362     2345687     7654823
  -------     -------     -------     -------     -------     -------
  3527486     6482357     7524836     4786523     3528476     6472538
- 3578264     4265873     5473268   - 4762835     5837264     4263785
  5836742     2547638   - 5436782     7243658   - 5876342     2348657
  8654327   - 2573486     4658327     2375486     8654723     3825476
  6482573     5328764   - 4682573     3528764     6482537     8537264
  4267835   - 5386247     6247835     5836247     4263875   - 8576342
  2743658     3654872   - 6273458     8654372     2347658     5684723
  7325486   - 3647528     2365784     6487523     3725486     6452837
- 7358264     6732485     3528647   - 6472835     7538264     4263578
  3876542     7268354     5834276     4263758   S@5786342     -------
  8634725   S@2785643     8457362     2345687     7654823      5-part
  -------     -------     -------     -------     -------


[5.05] order 5p [7.15] order 10m (TT) 2 3 4 5 6 7 8 2 6 5 4 3 8 7 * 3 4 5 6 2 7 8 4 3 2 6 5 8 7 * 4 5 6 2 3 7 8 6 5 4 3 2 8 7 * 5 6 2 3 4 7 8 3 2 6 5 4 8 7 * 6 2 3 4 5 7 8 5 4 3 2 6 8 7 *

A Cyclic-Dihedral palindrome. As 5 is prime, searches will produce one 5-part block, or five 1-part blocks, the latter requiring asymmetric links which might not be available. Refer to the section below on Cyclic-Dihedral palindromes: At the mid-lead apex in Middleton's peal below, pairs 2x3 4x5 (and 7x8) swap and 6 makes the place, while at the Home apex 2x5 and 4x6 swap and 3 makes the place. These each correspond to a turning over of the regular pentagon in 3-dimensions about a 2-fold axis of symmetry, the overall result being a rotation of the pentagon in its own plane.

 5,600 Cambridge S. Major

M     W     H     2 3 4 5 6
---------------------------
-                 4 3 6 5 2
-  @  -           5 6 2 3 4
      -     -     2 3 5 6 4
            -@    5 2 3 6 4
            -     3 5 2 6 4
---------------------------
   5-part by Middleton


Six-Part Palindromes


[3.01] order 6m   [5.06] order 12m    (TT)

 2 3 4 5 6 7 8     2 3 4 5 6 8 7
 3 2 4 5 6 7 8 s   3 2 4 5 6 8 7 $
 3 4 2 5 6 7 8 b   3 4 2 5 6 8 7
 4 3 2 5 6 7 8 s   4 3 2 5 6 8 7 $
 4 2 3 5 6 7 8 b   4 2 3 5 6 8 7
 2 4 3 5 6 7 8 s   2 4 3 5 6 8 7 $

The apices must both be of kind $ but this is able to rotate the trio of working bells, which in the example given below are bells 456, while 2 and 3 are fixed bells, both making places at the apices. To extend to 6 parts, two of 456 must be swapped half-way and end, for instance S for - at %. This happens to be quite a reasonable peal, as the 2nd never gets into 5ths or 6ths. Two courses may be cut out (making a 5,152) by calling S for - at &, preferably in the first part when the 6th would otherwise be away from the tenors.

5,376 Plain Bob Major

W   M   H   2 3 4 5 6
---------------------
-      %-   4 5 2 3 6
    -       2 5 6 3 4
-      &-   6 3 2 5 4
        S@  6 2 3 5 4
        -   3 6 2 5 4
    -       2 6 4 5 3
-       -   4 5 2 6 3
    -   S@  2 3 5 6 4
---------------------
       6-part


[3.01] order 6m [7.36] order 12m (TT) 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ 3 2 4 5 6 7 8 s 3 2 4 6 5 8 7 * 3 4 2 5 6 7 8 b 3 4 2 6 5 8 7 4 3 2 5 6 7 8 s 4 3 2 6 5 8 7 * 4 2 3 5 6 7 8 b 4 2 3 6 5 8 7 2 4 3 5 6 7 8 s 2 4 3 6 5 8 7 *

This is the more usual kind of 6-part, with two * apices rotating the three working bells. Unlike the previous system, 5-6 swap at both apices. To extend the given 3-part block, S at the Home apex halfway and end.

  5,088 Plain Bob Major

W   B   M   H   2 3 4 5 6
-------------------------
-               5 2 4 3 6
-   -       -   3 5 2 6 4
    -       -   3 5 6 4 2
      @     -   6 3 5 4 2
    -       -   6 3 4 2 5
    -   -       6 2 4 5 3
        -    @  4 2 3 5 6
-------------------------
          6-part


[5.07] order 6p [7.35] order 12p (TT) 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ 3 2 4 6 5 7 8 3 2 4 5 6 8 7 $ 3 4 2 5 6 7 8 b 3 4 2 6 5 8 7 4 3 2 6 5 7 8 4 3 2 5 6 8 7 $ 4 2 3 5 6 7 8 b 4 2 3 6 5 8 7 2 4 3 6 5 7 8 2 4 3 5 6 8 7 $

The group is +ve but the only apices available must have singles, hence the universal set must be mixed. The available links are P/B Q-sets, hence one cannot link two 3-part blocks, one has to find three 2-part blocks, which can be done if the pair of bells 5-6 cross at one apex and not at the other. With these restrictions as well as falsity it was found very difficult to compose treble-bob. In the peal below, 3x4 and 7x8 cross at the first apex, 5x6 and 7x8 at the second one, giving a 2-part block which may be extended by rotating bells 234, a bob in alternate parts at either of the two positions %.

5,376 Plain Bob Major

W   M   H   2 3 4 5 6
---------------------
    -       4 3 6 5 2
    -       6 3 2 5 4
-   -   -   4 2 6 3 5
 %  -   S@  6 5 2 3 4
-    %  -   2 3 6 5 4
-   -       6 2 4 3 5
-           3 6 4 2 5
-       S@  2 4 3 6 5
---------------------
        6-part


[5.07] order 6p [7.38] order 12m (TT) 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 2 4 6 5 7 8 3 2 4 6 5 8 7 * 3 4 2 5 6 7 8 b 3 4 2 5 6 8 7 4 3 2 6 5 7 8 4 3 2 6 5 8 7 * 4 2 3 5 6 7 8 b 4 2 3 5 6 8 7 2 4 3 6 5 7 8 2 4 3 6 5 8 7 *

The *-type apices will give a 3-part Cyclic-Dihedral block with 234 rotating. Asymmetric links will be needed to join the two 3-part blocks. In the peal below 23 swap and 456 rotate. A single at % (Middle) will shunt to & in other block, with a swap of 23 and a pair of the trio 456.

                   5,184 Plain Bob Major in 6 parts

  2345678     2647583     7824536     8372456     2456837     2385764
  -------     -------     -------     -------     -------     -------
S 3257486     6728435     8473265     3285764   S 4263578     3526847
  2738564     7863254     4386752   S 2356847     2347685   S 5364278
  7826345     8375642    %3645827     3624578     3728456     3457682
- 7864253     3584726     6532478   S 6347285     7835264     4738526
  8475632   - 3542867   S 5627384     3768452   - 7856342     7842365
  4583726    @5236478     6758243     7835624   -@7864523     8276453
S 5432867   - 5267384     7864532     8572346   - 7842635     2685734
  4256378     2758643   - 7843625     5284763     8273456   S 6253847
  2647583     7824536     8372456    &2456837     2385764     2364578
  -------     -------     -------     -------     -------     -------


[5.08] order 6m [5.06] order 12m 2 3 4 5 6 7 8 2 4 3 5 6 7 8 3 4 2 5 6 7 8 b 3 2 4 5 6 7 8 4 2 3 5 6 7 8 b 4 3 2 5 6 7 8 2 3 4 6 5 7 8 s 2 4 3 6 5 7 8 $ 3 4 2 6 5 7 8 3 2 4 6 5 7 8 $ 4 2 3 6 5 7 8 4 3 2 6 5 7 8 $

The apices must be at Singles and 78 cannot be kept together. In the peal below, 456 are the rotating trio, and 23 must be swapped by disposable calls, which may be done in every part.

                        5,184 Plain Bob Major

  2345678     5367284     5386742     3628574     4867352     7542836
  -------     -------     -------     -------     -------     -------
  3527486     3758642     3654827     6837245   - 4875623     5273468
  5738264   S 7384526   - 3642578     8764352     8542736     2356784
  7856342     3472865    %6237485   - 8745623     5283467     3628547
- 7864523    %4236758     2768354     7582436     2356874     6834275
  8472635   - 4265387   S 7285643     5273864     3627548     8467352
  4283756     2548673     2574836     2356748     6734285   - 8475623
  2345867     5827436   S 5243768     3624587     7468352     4582736
  3526478   S@8573264     2356487     6438275   S@4785623     5243867
S 5367284     5386742     3628574     4867352     7542836     2356478
  -------     -------     -------     -------     -------     -------
  6-part.  S for P at one of % to swap 23, every part or 3rd & 6th.


[5.08] order 6m [7.37] order 12m (TT) 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 4 2 5 6 7 8 b 3 4 2 5 6 8 7 4 2 3 5 6 7 8 b 4 2 3 5 6 8 7 2 3 4 6 5 7 8 s 2 3 4 6 5 8 7 $ 3 4 2 6 5 7 8 3 4 2 6 5 8 7 4 2 3 6 5 7 8 4 2 3 6 5 8 7

As there is only one $ apex available, rounds after one part is inevitable. In the peal below, note the swaps 5x6 and 7x8 at both apices. Two kinds of extra calls are required: At one place only of %, a Bob will rotate the trio 234; and at one place only of & a Single will swap 56. Each of these could be inserted in every part to give an exact 6-part, which however would not be an exact palindrome.

5,376 Plain Bob Major

W   M   H   2 3 4 5 6
---------------------
    -   S   4 6 3 5 2
     %  -   3 4 6 5 2
    -       6 4 2 5 3
 &  S   S@  3 2 4 5 6
S    &      5 2 4 3 6
-       -   4 3 5 2 6
 %      S   4 5 3 2 6
-       S@  2 3 4 5 6
---------------------


[5.08] order 6m [7.38] order 12m (TT) 2 3 4 5 6 7 8 2 4 3 5 6 8 7 $ 3 4 2 5 6 7 8 b 3 2 4 5 6 8 7 $ 4 2 3 5 6 7 8 b 4 3 2 5 6 8 7 $ 2 3 4 6 5 7 8 s 2 4 3 6 5 8 7 * 3 4 2 6 5 7 8 3 2 4 6 5 8 7 * 4 2 3 6 5 7 8 4 3 2 6 5 8 7 *

Having both apices of kind * will effect rotation of the trio of working bells 234 (as well as swapping 5x6, 7x8) while the available s-type disposable call will swap 5x6 half-way and end, here S for Plain at one of %. The first apex is at mid-lead between the two Before calls.

 5,088 Plain Bob Major

W  B  M  H   2 3 4 5 6
----------------------
-      %     5 2 4 3 6
-  x         5 2 3 6 4
-     -  -   4 3 5 2 6
   2@    -   3 2 6 5 4
-     -      6 3 4 2 5
   x  -      6 2 4 5 3
 %    -   @  4 2 3 5 6
----------------------
     6-part


[6.15] order 6m [6.13] order 12m 2 3 4 5 6 7 8 2 7 6 5 4 3 8 $ 3 4 5 6 7 2 8 7 6 5 4 3 2 8 * 4 5 6 7 2 3 8 6 5 4 3 2 7 8 $ 5 6 7 2 3 4 8 5 4 3 2 7 6 8 * 6 7 2 3 4 5 8 4 3 2 7 6 5 8 $ 7 2 3 4 5 6 8 3 2 7 6 5 4 8 *

A Cyclic-Dihedral Palindrome. 6 is not prime, but good luck might produce a 6-part block. For a 6-part block there must be one each of the apical transfigures * and $, from considerations of parity. An example is the peal of 5,184 Bristol given below. Attempts to obtain the obvious cyclic partends 3456728, 4567238 etc. failed, for some hidden reason probably concerned with falsity.

                     5,184 Bristol Surprise Major

  2 3 4 5 6 7 8     3 5 7 4 2 6 8     4 8 2 6 5 7 3     7 5 8 3 6 2 4
  -------------     -------------     -------------     -------------
  4 2 6 3 8 5 7     7 3 2 5 8 4 6   - 2 4 8 6 5 7 3   - 8 7 5 3 6 2 4
  6 4 8 2 7 3 5     2 7 8 3 6 5 4     8 2 5@4 3 6 7     5 8 6 7 4 3 2
S@6 8 4 2 7 3 5     8 2 6 7 4 3 5     5 8 3 2 7 4 6     6 5 4 8 2 7 3
  4 6 7 8 5 2 3   - 6 8 2 7 4 3 5   - 3 5 8 2 7 4 6     4 6 2 5 3 8 7
  7 4 5 6 3 8 2   - 2 6 8 7 4 3 5   - 8 3 5 2 7 4 6   - 2 4 6 5 3 8 7
  5 7 3 4 2 6 8     8 2 4 6 5 7 3     5 8 7 3 6 2 4     6 2 3 4 7 5 8
- 3 5 7 4 2 6 8   - 4 8 2 6 5 7 3   - 7 5 8 3 6 2 4     -------------
  -------------     -------------     -------------        6-part


[6.16] order 6m [6.13] order 12m 2 3 4 5 6 7 8 2 4 3 7 6 5 8 $ 3 4 2 6 7 5 8 3 2 4 5 7 6 8 $ 4 2 3 7 5 6 8 4 3 2 6 5 7 8 $ 7 6 5 4 3 2 8 5 6 7 3 4 2 8 6 5 7 3 2 4 8 6 7 5 4 2 3 8 * 5 7 6 2 4 3 8 7 5 6 2 3 4 8

This curious system has two trios of working bells rotating in tandem, then swapping places. If all $-type apices are used, two 3-part blocks can result, to be joined by asymmetric links, as there are no disposable calls. The peal below has 2 as fixed bell, and trios 345, 678. S for - at the positions %% will be found to effect the necessary linkage, S in one position linking to the other position in the other block. The block has two parallel 3-part cyclic-dihedral palindromes.

                        5,184 Plain Bob Major

  2345678     6834572     7563842     4275386     5876432     6387524
  -------     -------     -------     -------     -------     -------
- 2357486     8467325   - 7534628     2548763     8653724     3762845
  3728564   - 8472653   - 7542386     5826437     6382547   - 3724658
-%3786245     4285736   - 7528463     8653274     3264875   -%3745286
  7634852   - 4253867     5876234     6387542     2437658   - 3758462
- 7645328   - 4236578     8653742     3764825   - 2475386     7836524
  6572483     2647385     6384527     7432658     4528763   - 7862345
S@5628734     6728453     3462875   S@4725386     5846237     8274653
  6853247   - 6785234     4237658     7548263     8653472     2485736
- 6834572     7563842   - 4275386     5876432     6387524   - 2453867
  -------     -------     -------     -------     -------     -------


[6.32] order 6p [6.13] order 12m 2 3 4 5 6 7 8 5 6 7 2 3 4 8 * 3 2 4 6 5 7 8 6 5 7 3 2 4 8 * 3 4 2 6 7 5 8 6 7 5 3 4 2 8 4 3 2 7 6 5 8 7 6 5 4 3 2 8 * 4 2 3 7 5 6 8 7 5 6 4 2 3 8 2 4 3 5 7 6 8 5 7 6 2 4 3 8 *

By careful choice of apices (the topmost one differs essentially from the others and must not be used) one can achieve a 3-part Rotating-Sets palindrome, which must be doubled by asymmetric links. In the peal below, the 2 is fixed bell. Trios 345 and 678 rotate; they are made to permute by a pair of S for P at %&, S at % shunting to & in the other block.

                        5,088 Plain Bob Major

  2345678     8635274     5674283     3874526     3586427     6287435
  -------     -------     -------     -------     -------     -------
  3527486     6587342     6458732     8432765     5632874     2763854
  5738264   S 5674823     4863527     4286357     6257348     7325648
- 5786342     6452738   S 8432675     2645873     2764583   - 7354286
- 5764823     4263587     4287356   S 6257438     7428635   -@7348562@
  7452638     2348675    @2745863     2763584    &4873256   - 7386425
  4273586   S 3287456     7526438     7328645   - 4835762   - 7362854
  2348765     2735864   S 5763284    %3874256   - 4856327     3275648
  3826457     7526348     7358642   S 8345762     8642573     2534786
  8635274     5674283     3874526     3586427     6287435     -------
  -------     -------     -------     -------     -------     6-part


[7.39] order 6p [7.31] order 12p 2 3 4 5 6 7 8 2 3 4 7 8 5 6 $ 3 4 2 5 6 7 8 b 3 4 2 7 8 5 6 4 2 3 5 6 7 8 b 4 2 3 7 8 5 6 2 3 4 6 5 8 7 2 3 4 8 7 6 5 $ 3 4 2 6 5 8 7 3 4 2 8 7 6 5 4 2 3 6 5 8 7 4 2 3 8 7 6 5

There are two different $-kind apices which operate on 5678, so a Rotating-Sets 2-part block may be found, with 234 fixed. b-type disposable calls may be available for rotating 234. In the peal below, at the first apex (5x8 6x7) swap and at the second (5x6 7x8) giving a 2-part Rotating-Sets palindrome with (5x7 6x8). The rotation of 234 is achieved by Bob for Plain at one position %, in alternate parts.

                        5,184 Plain Bob Major

  2345678     5738426     3472856   S 3427658     3645278     8364572
  -------     -------     -------     -------     -------     -------
  3527486     7852364    %4235768   S 4375286   S 6357482     3487625
  5738264     8276543     2546387     3548762     3768524     4732856
  7856342     2684735   - 2568473     5836427   - 3782645   - 4725368
  8674523   - 2643857     5827634   - 5862374     7234856     7546283
  6482735     6325478   - 5873246     8257643   - 7245368     5678432
- 6423857   - 6357284     8354762   - 8274536     2576483     6853724
  4365278     3768542     3486527     2483765     5628734     8362547
  3547682   S 7384625   S 4362875    %4326857     6853247     3284675
  5738426     3472856   S@3427658     3645278     8364572   S@2347856
  -------     -------     -------     -------     -------     -------
  - for P in one of the positions %, in alternate parts.      6-part


[7.39] order 6p [7.32] order 12m 2 3 4 5 6 7 8 2 4 3 7 8 5 6 * 3 4 2 5 6 7 8 b 3 2 4 7 8 5 6 * 4 2 3 5 6 7 8 b 4 3 2 7 8 5 6 * 2 3 4 6 5 8 7 2 4 3 8 7 6 5 * 3 4 2 6 5 8 7 3 2 4 8 7 6 5 * 4 2 3 6 5 8 7 4 3 2 8 7 6 5 *

This system parts the tenors, but otherwise works very well. All apices swap three pairs, and in the peal below the apical transfigures do two different things: they swap (5x7 6x8), (5x8 6x7) thus producing a Rotating-Sets 2-part palindrome on bells (5x6 7x8), and also swaps for a Cyclic-Dihedral rotation of 234. The combined result is an exact 6-part. Disposable calls are not required.

                      5,088 Plain Bob Major

  2345678     3625874     5862437     3827546     7486235   - 4623758
  -------     -------     -------     -------     -------     -------
  3527486     6537248     8253674     8734265     4673852   - 4635287
  5738264     5764382   - 8237546     7486352     6345728   - 4658372
-@5786342@    7458623     2784365    @4675823     3562487     6847523
- 5764823   - 7482536     7426853     6542738   - 3528674     8762435
  7452638   - 7423865     4675238     5263487     5837246     7283654
  4273586   - 7436258     6543782   - 5238674     8754362     2375846
  2348765     4675382     5368427     2857346     7486523   - 2354768
- 2386457     6548723     3852674     8724563     4672835     3426587
  3625874     5862437   - 3827546     7486235   - 4623758     -------
  -------     -------     -------     -------     -------     6-part


[7.39] order 6p [7.35] order 12p 2 3 4 5 6 7 8 2 4 3 5 6 8 7 $ 3 4 2 5 6 7 8 b 3 2 4 5 6 8 7 $ 4 2 3 5 6 7 8 b 4 3 2 5 6 8 7 $ 2 3 4 6 5 8 7 2 4 3 6 5 7 8 $ 3 4 2 6 5 8 7 3 2 4 6 5 7 8 $ 4 2 3 6 5 8 7 4 3 2 6 5 7 8 $

The peal below is apparently on the same plan as that for system [7.39]-[7.32], but bells 2345 do not perform Rotating-Sets; the pairs 2x3, 4x5 swap at separate apices (678 in the peal doing the rotating). As in the other system, disposable calls are not needed. The partend group is positive, but as the apices require Singles, the universal set of leadheads used must be mixed.

                        5,184 Plain Bob Major

  2345678     2537864     7286345     5624738     4358672     8732546
  -------     -------     -------     -------     -------     -------
  3527486     5726348     2674853   - 5643287     3847526     7284365
  5738264     7654283     6425738     6358472     8732465     2476853
- 5786342     6478532   - 6453287     3867524     7286354     4625738
  7654823   - 6483725     4368572     8732645     2675843   - 4653287
  6472538     4362857     3847625     7284356     6524738     6348572
  4263785     3245678     8732456     2475863   - 6543287     3867425
  2348657   - 3257486     7285364     4526738     5368472   S 8372654
S@3285476     2738564     2576843   S@5463287     3857624     3285746
  2537864   S 7286345     5624738     4358672     8732546   - 3254867
  -------     -------     -------     -------     -------     -------
                                                              6-part

[7.39] order 6p [7.36] order 12m 2 3 4 5 6 7 8 2 4 3 5 6 7 8 3 4 2 5 6 7 8 b 3 2 4 5 6 7 8 4 2 3 5 6 7 8 b 4 3 2 5 6 7 8 2 3 4 6 5 8 7 2 4 3 6 5 8 7 * 3 4 2 6 5 8 7 3 2 4 6 5 8 7 * 4 2 3 6 5 8 7 4 3 2 6 5 8 7 *

As every apical transfigure contains 5x6, 7x8 these four bells must necessarily be fixed in every part, though 234 may rotate. Hence asymmetric links will be needed. P/B Q-sets preserve parity and will not unite two blocks, hence singles will be needed, and a mixed universal set of lead-heads. In the peal below both apices are at a mid-lead. To join the 3-part block with its fellow, a pair of Singles are called, at % in one part shunting to & in the other block

                        5,184 Plain Bob Major

  2345678     8537264     5364728     5432876     2456387     4782635
  -------     -------     -------     -------     -------     -------
  3527486   - 8576342     3452687     4257368   S 4268573     7243856
  5738264     5684723     4238576   S 2476583     2847635     2375468
- 5786342     6452837   - 4287365     4628735   S 8273456     3526784
  7654823     4263578    &2746853     6843257     2385764     5638247
  6472538     2347685     7625438     8365472     3526847     6854372
  4263785     3728456     6573284     3587624     5634278     8467523
  2348657   S 7385264     5368742   - 3572846     6457382   - 8472635
  3825476    %3576842   S 3584627     5234768     4768523     4283756
  8537264   S 5364728    @5432876@    2456387   - 4782635    @2345867@
  -------     -------     -------     -------     -------     -------
                                                              6-part

[7.40] order 6m [7.32] order 12m 2 3 4 5 6 7 8 2 3 4 7 8 5 6 $ 3 2 4 6 5 8 7 3 2 4 8 7 6 5 * 3 4 2 5 6 7 8 b 3 4 2 7 8 5 6 4 3 2 6 5 8 7 4 3 2 8 7 6 5 * 4 2 3 5 6 7 8 b 4 2 3 7 8 5 6 2 4 3 6 5 8 7 2 4 3 8 7 6 5 *

By using one $-type apex and one *-type, 5678 may perform a 2-part Rotating-Sets palindrome, while a pair of 234 swap. In the peal below, 2x4 swap at the mid-lead apex. This gives three 2-part blocks which may be united by a P/B disposable Q-set at one of the two positions marked %, the bob being inserted in alternate parts.

                        5,088 Plain Bob Major

  2345678     5738426     7456823     6725438     6853247     3287645
  -------     -------     -------     -------     -------     -------
  3527486     7852364   - 7462538     7563284     8364572     2734856
  5738264     8276543     4273685     5378642     3487625   - 2745368
  7856342     2684735    %2348756   - 5384726     4732856     7526483
  8674523   S 6243857     3825467     3452867   S 7425368     5678234
  6482735     2365478   - 3856274    %4236578     4576283     6853742
- 6423857     3527684     8637542     2647385     5648732     8364527
  4365278     5738246     6784325   - 2678453     6853427     3482675
  3547682   - 5784362   -@6742853@    6825734     8362574   S@4327856
  5738426     7456823   - 6725438   - 6853247     3287645     -------
  -------     -------     -------     -------     -------     6-part


[7.40] order 6m [7.36] order 12m 2 3 4 5 6 7 8 2 3 4 6 5 8 7 $ 3 2 4 6 5 8 7 3 2 4 5 6 7 8 3 4 2 5 6 7 8 b 3 4 2 6 5 8 7 4 3 2 6 5 8 7 4 3 2 5 6 7 8 4 2 3 5 6 7 8 b 4 2 3 6 5 8 7 2 4 3 6 5 8 7 2 4 3 5 6 7 8

As there is only one apical transfigure available, kind $, rounds in one part is inevitable. A b-type P/B Q-set might rotate the trio 234, giving two 3-part blocks, but otherwise asymmetric calls will have to be invoked. In the peal below, Bob for plain at one of the positions $ in every part will rotate 234 to give two 3-part blocks; and Singles at % and & in the separate blocks will link them together.

                        5,184 Plain Bob Major

  2345678     8537264     7524368     4238756     6823475     7362845
  -------     -------     -------     -------     -------     -------
  3527486   - 8576342     5476283     2845367     8367254     3274658
  5738264     5684723   S 4568732   S 8256473   - 8375642    $2435786
- 5786342     6452837    &5843627     2687534     3584726     4528367
  7654823   S 4623578   S 8532476     6723845     5432867     5846273
 %6472538     6347285     5287364   S 7634258   S 4526378     8657432
  4263785     3768452     2756843     6475382     5647283   - 8673524
 $2348657   - 3785624   S 7264538   S 4658723     6758432     6382745
  3825476     7532846     2473685     6842537   - 6783524     3264857
  8537264   - 7524368   S@4238756   - 6823475     7362845   S@2345678
  -------     -------     -------     -------     -------     -------


[7.40] order 6m [7.38] order 12m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 2 4 6 5 8 7 3 2 4 6 5 7 8 $ 3 4 2 5 6 7 8 b 3 4 2 5 6 8 7 4 3 2 6 5 8 7 4 3 2 6 5 7 8 $ 4 2 3 5 6 7 8 b 4 2 3 5 6 8 7 2 4 3 6 5 8 7 2 4 3 6 5 7 8 $

The three available $-type apical transfigures swap 5x6 but not 7x8 (in the peal below these roles are reversed) and a pair of 234, so that a 3-part rotation of 234 may be achieved, but 7x8 may only be swapped by asymmetric calls. The peal is given as two 3-part blocks, linked by a pair of singles at % and & in different blocks.

                        5,184 Plain Bob Major

  2345678   S 3754286     8435726     4253867     7435682     2853764
  -------     -------     -------     -------     -------     -------
  3527486     7438562   S 4852367   S 2436578     4578326     8326547
- 3578264   - 7486325    %8246573     4627385     5842763     3684275
  5836742     4672853     2687435   - 4678253   S 8526437   - 3647852
  8654327     6245738     6723854     6845732     5683274     6735428
- 8642573   S@2653487   - 6735248   S 8653427   S@6537842   S 7652384
  6287435     6328574     7564382     6382574     5764328     6278543
  2763854     3867245     5478623   - 6327845   S 7542683   - 6284735
  7325648   - 3874652   - 5482736     3764258     5278436     2463857
S 3754286     8435726     4253867    &7435682     2853764   S 4235678
  -------     -------     -------     -------     -------     -------


Seven-Part Palindromes


[7.07] order 7p  [7.06] order 14m

 2 3 4 5 6 7 8    2 8 7 6 5 4 3 *
 3 4 5 6 7 8 2    4 3 2 8 7 6 5 *
 4 5 6 7 8 2 3    6 5 4 3 2 8 7 *
 5 6 7 8 2 3 4    8 7 6 5 4 3 2 *
 6 7 8 2 3 4 5    3 2 8 7 6 5 4 *
 7 8 2 3 4 5 6    5 4 3 2 8 7 6 *
 8 2 3 4 5 6 7    7 6 5 4 3 2 8 *

A Cyclic-Dihedral Palindrome. The system is appropriate to in-course peals with bobs only. A tree search will have a probability of 6/7ths of producing one 7-part block. The peal given below has cyclical partends.

                     5,152 Bristol Surprise Major

  2 3 4 5 6 7 8    5 7 2 8 4 6 3    6 4 3 5 8 2 7    4 5 6 2 3 7 8
  -------------    -------------    -------------    -------------
- 4 2 3 5 6 7 8    2 5 4 7 3 8 6    3 6 8 4 7 5 2  - 6 4 5 2 3 7 8
  3 4 6 2 8 5 7  - 4 2 5 7 3 8 6    8 3 7 6 2 4 5    5 6 3 4 8 2 7
  6 3 8 4 7 2 5    5 4 3 2 6 7 8    7 8 2 3 5 6 4    3 5 8@6 7 4 2
  8 6 7 3 5 4 2  -@3 5 4 2 6 7 8    2 7 5 8 4 3 6    8 3 7 5 2 6 4
  7 8 5 6 2 3 4    4 3 6 5 8 2 7    5 2 4 7 6 8 3    7 8 2 3 4 5 6
  5 7 2 8 4 6 3  - 6 4 3 5 8 2 7    4 5 6 2 3 7 8    -------------
  -------------    -------------    -------------        7-part


Eight-Part Palindromes


[4.03] order 8m  [6.20] order 16m

 2 3 4 5 6 7 8     2 3 4 5 7 6 8
 3 4 5 2 6 7 8     3 4 5 2 7 6 8
 4 5 2 3 6 7 8     4 5 2 3 7 6 8 *
 5 2 3 4 6 7 8     5 2 3 4 7 6 8
 5 4 3 2 6 7 8     5 4 3 2 7 6 8 *
 4 3 2 5 6 7 8 s   4 3 2 5 7 6 8 $
 3 2 5 4 6 7 8     3 2 5 4 7 6 8 *
 2 5 4 3 6 7 8 s   2 5 4 3 7 6 8 $

It seems possible to find two 4-part blocks of Cyclic-Dihedral kind, or four 2-part blocks of Rotating-Sets kind, but the dihedral group is a tricky one to handle as a part-plan. The block below is a 2-part Rotating-Sets palindrome, with 5x6 and 7x8 swapping at the first mid-lead apex (also 3x4) in the first lead, 5x7 and 6x8 at the second mid-lead apex, overall effect 5x8 and 6x7. 5678 will describe the dihedral group [4.03]. A Single instead of a Bob at % will swap 6x8 which is within the part plan, and calling S for B in parts 1, 2, 3, 5, 6, 7 produces a peal. The other bob in the part may be singled instead, consistently. This peal has an all-time low of 12 CRUs! The system is related to system [6.23]-[6.20].

                    5,120 Bristol S. Major

   2 3 4@5 6 7 8         5 3 8 4 2 7 6       3 5 6 7 8 4 2
   -------------         -------------       -------------
   4 2 6 3 8 5 7         8 5 2 3 6 4 7       6 3 8 5 2 7 4
   6 4 8 2 7 3 5         2 8 6@5 7 3 4     - 8 6 3 5 2 7 4
%- 8 6 4 2 7 3 5         6 2 7 8 4 5 3       3 8 2 6 4 5 7
   4 8 7 6 5 2 3         7 6 4 2 3 8 5       2 3 4@8 7 6 5
 S 4 7 8 6 5 2 3       S 7 4 6 2 3 8 5       -------------
   8 4 5 7 3 6 2         6 7 3 4 5 2 8          8-part
   5 8 3 4 2 7 6         3 6 5 7 8 4 2
 S 5 3 8 4 2 7 6       S 3 5 6 7 8 4 2
   -------------         -------------


[6.21] order 8m [6.20] order 16m 2 3 4 5 6 7 8 2 5 4 3 6 7 8 3 2 5 4 6 7 8 3 4 5 2 6 7 8 4 5 2 3 6 7 8 4 3 2 5 6 7 8 5 4 3 2 6 7 8 5 2 3 4 6 7 8 2 3 4 5 7 6 8 s 2 5 4 3 7 6 8 $ 3 2 5 4 7 6 8 3 4 5 2 7 6 8 4 5 2 3 7 6 8 4 3 2 5 7 6 8 $ 5 4 3 2 7 6 8 5 2 3 4 7 6 8

The two available apices are (3x5 6x7), (2x4 6x7) which can give a 2-part palindrome of (2x4 3x5) (not Rotating-Sets) in 4 blocks. The disposable Single can link pairs by 6x7, but asymmetric calls will be needed to complete the linkage. In the peal below, 2 is the fixed bell and 78 swap, while 3456 perform the pairs-of-pairs group [4.04]. Swaps are Apex 1: (3x4 7x8), Apex 2: (5x6 7x8) giving at the partend (3x4 5x6). S for P at % will shunt to &, effecting (4x6 3x5). The resulting two blocks may be joined by S for P at one of the positions +, swapping 7x8.

                        5,120 Plain Bob Major

  2345678     6785234     2754638     6754238     2574638     6734258
  -------     -------     -------     -------     -------     -------
  3527486     7563842     7423586    %7463582     5423786    &7465382
S 5378264     5374628     4378265     4378625   S 4538267     4578623
  3856742   S 3542786     3846752     3842756     5846372   S 5482736
  8634527     5238467     8635427     8235467     8657423     4253867
- 8642375   - 5286374     6582374     2586374    +6782534   S@2436578
- 8627453   - 5267843   S@5627843   - 2567843   - 6723845     -------
 +6785234     2754638     6754238   - 2574638   - 6734258     8-part
  -------     -------     -------     -------     -------


[6.22] order 8m [6.20] order 16m 2 3 4 5 6 7 8 5 4 3 2 6 7 8 $ 3 4 5 2 6 7 8 4 3 2 5 6 7 8 4 5 2 3 6 7 8 3 2 5 4 6 7 8 $ 5 2 3 4 6 7 8 2 5 4 3 6 7 8 2 3 4 5 7 6 8 s 5 4 3 2 7 6 8 * 3 4 5 2 7 6 8 4 3 2 5 7 6 8 $ 4 5 2 3 7 6 8 3 2 5 4 7 6 8 * 5 2 3 4 7 6 8 2 5 4 3 7 6 8 $

This system has a good selection of apical transfigures, and it happens that some */$ apices are at disposable calls. By a judicious juggling the cyclic bells 2345 may give a 4-part Cyclic-Dihedral palindrome, while 6x7 may be swapped by a disposable call at an apex. In the peal below, 5678 are the cyclic set, and 3x4 swap with a single at the end of the 4th and 8th parts.

                        5,120 Plain Bob Major

  2345678     2743658     6547238     2835674     6834257     6837245
  -------     -------     -------     -------     -------     -------
  3527486     7325486     5763482     8527346     8465372     8764352
- 3578264   - 7358264   - 5738624     5784263     4587623     7485623
  5836742     3876542     7852346   - 5746832   - 4572836   - 7452836
  8654327     8634725     8274563     7653428     5243768     4273568
  6482573     6482357   - 8246735     6372584     2356487    @2346785
  4267835   S 4625873   S@2863457   S 3628745     3628574     -------
  2743658     6547238   - 2835674     6834257     6837245     8-part
  -------     -------     -------     -------     -------


[6.23] order 8p [6.20] order 16m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 3 4 5 2 7 6 8 3 4 5 2 6 7 8 4 5 2 3 6 7 8 4 5 2 3 7 6 8 * 5 2 3 4 7 6 8 5 2 3 4 6 7 8 5 4 3 2 6 7 8 5 4 3 2 7 6 8 * 4 3 2 5 7 6 8 4 3 2 5 6 7 8 3 2 5 4 6 7 8 3 2 5 4 7 6 8 * 2 5 4 3 7 6 8 2 5 4 3 6 7 8

6x7 swap at every available apex hence must be fixed in any round block, but the other four working bells may perform a Rotating-Sets palindrome and give four 2-part blocks. As there are no disposable calls, asymmetric links will be needed. The four blocks of 2-part palindromes below belong to this system. Linking singles are possible at aa, bb etc. but this requires "block inversion", the second and fourth blocks being rung backwards to put them out-of-course (i.e. with -ve leadheads). Note that inverting a palindrome does not alter the calling sequence. However, doing this puts the assembly in another 8-part system, [4.03]-[6.20], of which a much better example is given. Note that the mid-lead apices are not at the calls to be altered, so that the resulting peal is irregular, with isolated linking singles and also -S- substituted for a pair of bobs.

       5,120 Bristol Surprise Major

  2345678    2436785    2347856    2438567
  -------    -------    -------    -------
 -4235678   -3246785   -4237856   -3248567
  3462857    4372568    3482675    4352786
 -6342857   -7432568   -8342675   -5432786
  4683725    3754826    4863527    3574628
  8476532    5387642    6458732    7365842
 -7846532   -8537642   -5648732   -6735842
  4758263    3865274    4576283    3687254
 -5478263   -6385274   -7456283   -8367254
  7524386    8623457    5724368    6823475
a-2754386@ b-2863457@ c-2574368@ d-2683475@
p-5274386   -6283457   -7254368   -8263475
  7532648    8642735    5732846    6842537
 -3752648   -4862735   -3572846   -4682537
  5367824    6478523    7385624    8456723
 -6537824   -7648523   -8735624   -5846723
  3685472    4756382    3867452    4578362
  8346257    5437268    6348275    7435286
 -4836257   -3547268   -4638275   -3745286
  3428765    4325876    3426587    4327658
 -2348765   -2435876   -2346587   -2437658
  -------    -------    -------    -------
 -4238765   -3245876   -4236587   -3247658
  3472586    4382657    3452768    4362875
 -7342586   -8432657   -5342768   -6432875
  4753628    3864725    4573826    3684527
  5467832    6378542    7485632    8356742
 -6547832   -7638542   -8745632   -5836742
  4685273    3756284    4867253    3578264
 -8465273   -5376284   -6487253   -7358264
  6824357    7523468    8624375    5723486
d-2684357@ a-2753468@ b-2864375@ c-2573486@
 -8264357  p-5273468   -6284375   -7253486
  6832745    7542836    8632547    5742638
 -3682745   -4752836   -3862547   -4572638
  8376524    5487623    6358724    7465823
 -7836524   -8547623   -5638724   -6745823
  3758462    4865372    3576482    4687352
  5347286    6438257    7345268    8436275
 -4537286   -3648257   -4735268   -3846275
  3425678    4326785    3427856    4328567
 -2345678   -2436785   -2347856   -2438567
  -------    -------    -------    -------


[6.24] order 8m [6.20] order 16m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 3 4 5 2 7 6 8 3 4 5 2 6 7 8 4 5 2 3 6 7 8 4 5 2 3 7 6 8 * 5 2 3 4 7 6 8 5 2 3 4 6 7 8 2 5 4 3 6 7 8 s 2 5 4 3 7 6 8 $ 5 4 3 2 7 6 8 5 4 3 2 6 7 8 $ 4 3 2 5 6 7 8 s 4 3 2 5 7 6 8 $ 3 2 5 4 7 6 8 3 2 5 4 6 7 8 $

As with system [6.22]-[6.20] it is possible to manoeuvre the apices to obtain two 4-part Cyclic-Dihedral blocks with bells 2345 rotating, but this time the rotations are in opposite senses. In the peal below 5678 are the set of four and the block is doubled by swapping 5x7 in parts 1 and 5 with, for instance, a Single at % (it would be 6x8 in the next part) which has the effect of reversing the rotation of 5678. 3x4 look after themselves.

                        5,120 Plain Bob Major

  2345678     5368274     6523874     8524637     2365847     6235784
  -------     -------     -------     -------     -------     -------
S 3257486   - 5387642     5367248     5483276     3524678     2568347
  2738564    %3754826     3754682   - 5437862     5437286   S 5284673
S 7286345     7432568   - 3748526     4756328     4758362     2457836
  2674853     4276385     7832465     7642583   - 4786523   S 4273568
- 2645738     2648753     8276354     6278435     7642835   S@2436785
  6523487   - 2685437   S@2865743   - 6283754     6273458     -------
  5368274     6523874     8524637     2365847   - 6235784     8-part
  -------     -------     -------     -------     -------


[6.25] order 8m [6.20] order 16m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 3 4 5 2 6 7 8 3 4 5 2 7 6 8 4 5 2 3 6 7 8 4 5 2 3 7 6 8 * 5 2 3 4 6 7 8 5 2 3 4 7 6 8 2 5 4 3 7 6 8 2 5 4 3 6 7 8 3 2 5 4 7 6 8 3 2 5 4 6 7 8 $ 4 3 2 5 7 6 8 4 3 2 5 6 7 8 5 4 3 2 7 6 8 5 4 3 2 6 7 8 $

By choice of apices * and $ one can achieve a Rotating-Sets two-part block on 2345 and also the swapping of 67 in the block. Then asymmetric links must be found to give a cyclic 4-part shunt. In the peal below, the fixed bell is 2 and the swapping pair is 34, with 5678 generating the dihedral group. The 2-part block contains one 34 and one 43. S for P at % will shunt to position & in another block with a 4-part shift of bells 5678, 3 pairs of Singles being needed to link 4 blocks.

                        5,120 Plain Bob Major

  2345678     2743658     2547863     3645287     5347268     7548236
  -------     -------     -------     -------     -------     -------
  3527486     7325486     5726438   - 3658472   - 5376482     5873462
- 3578264   - 7358264   - 5763284     6837524     3658724     8356724
 %5836742     3876542    &7358642   - 6872345   - 3682547   - 8362547
  8654327   - 3864725   - 7384526     8264753     6234875     3284675
  6482573     8432657     3472865     2485637     2467358     2437856
  4267835     4285376   S 4326758     4523876     4725683     -------
  2743658     2547863     3645287     5347268     7548236     8-part
  -------     -------     -------     -------     -------


[6.34] order 8m [6.20] order 16m 2 3 4 5 6 7 8 4 5 2 3 6 7 8 $ 2 3 5 4 6 7 8 s 4 5 3 2 6 7 8 3 2 4 5 6 7 8 s 5 4 2 3 6 7 8 3 2 5 4 6 7 8 5 4 3 2 6 7 8 $ 2 3 4 5 7 6 8 s 4 5 2 3 7 6 8 * 2 3 5 4 7 6 8 4 5 3 2 7 6 8 3 2 4 5 7 6 8 5 4 2 3 7 6 8 3 2 5 4 7 6 8 5 4 3 2 7 6 8 *

The available apices are limited to (2x4 5x3) and (2x5 3x4) but this will give four 2-part blocks of Rotating-Sets kind, 2x3, 4x5. 6x7 work independently. In the peal below, 5678 perform Rotating Sets, 3x4 swap every part. To join the blocks, S at % will swap 5x7 in alternate parts, and S at & will swap 6x8 every 4 parts.

                        5,120 Plain Bob Major

  2345678     2743658     4768325     3275486     7842356     7548236
  -------     -------     -------     -------     -------     -------
  3527486   S 7235486     7842653     2538764     8275463     5873462
- 3578264    %2578364     8275436   - 2586347   - 8256734     8356724
  5836742   - 2586743   - 8253764     5624873    &2683547   - 8362547
  8654327     5624837     2386547     6457238   S 6234875     3284675
  6482573     6453278   - 2364875     4763582     2467358    @2437856
  4267835   - 6437582   S@3247658   - 4738625     4725683     -------
  2743658     4768325   - 3275486     7842356     7548236     8-part
  -------     -------     -------     -------     -------


Nine-Part Palindromes


[6.31] order 9p  [6.12] order 18m

 2 3 4 5 6 7 8     5 6 7 2 3 4 8 *
 3 4 2 6 7 5 8     6 7 5 4 2 3 8 *
 4 2 3 7 5 6 8     7 5 6 3 4 2 8 *
 2 3 4 6 7 5 8 b   5 6 7 4 2 3 8
 3 4 2 7 5 6 8     6 7 5 3 4 2 8
 4 2 3 5 6 7 8 b   7 5 6 2 3 4 8
 2 3 4 7 5 6 8 b   5 6 7 3 4 2 8
 3 4 2 5 6 7 8 b   6 7 5 2 3 4 8
 4 2 3 6 7 5 8     7 5 6 4 2 3 8

An extension of the Rotating-Sets Palindrome [6.33]-[6.16]. Results of a tree search may produce nine 1-part blocks or three 3-part blocks; disposable calls are available as linkages. The two trios of bells 234, 567 rotate independently. This system is particularly suited to composing long lengths of methods (and particularly mx methods) because:

  1. The group [6.31] is positive, needing only the 2,520 +ve leadheads, hence the false array is less extensive which economises in computer storage required by the arrays.
  2. The number of parts is high, cutting down tree search time.
  3. A mixture of 4ths and 6th place Bobs is possible in the positive group, and moreover yields disposable calls because of the 3-part transpositions.
  4. The group [6.31] does not fall foul of the falsity of methods to the extent that many even-parted groups do.
  5. The parity of the number of blocks is odd, making linkage easier.

This is illustrated by the following long peals, which exceed the records in the methods. The first, of Kent Treble Bob Major, is longer than the record length (17,824 by Thomas Worsley, rung at Heptonstall on Easter Monday 1927) by well over 3,000:

                21,600 Kent or Oxford Treble Bob Major

   2345678     4756382     8267453     5873264  6- 5782364  6- 2783564
   -------     -------     -------     -------     -------     -------
4- 4235678  4- 5476382  4- 6827453  4- 7583264  4- 8572364  4- 8273564
4- 3425678  6- 7534682  6- 2648753  6- 8725364  6- 7835264  6- 7852364
6- 2364578  6- 3765482  4- 4268753  6- 2837564  4- 3785264     5738426
   6253847  6- 6347582  6- 6472853  6- 3258764  4- 8375264     3547682
   5682734  4- 4637582     7684325  6- 5372864  6- 7823564     4365278
   8576423  4- 3467582     8736542  4- 7532864  6- 2758364  6- 6423578
4- 7856423     6354278     3857264  4- 3752864  6- 5237864  4- 2643578
4- 5786423     5623847  6- 5328764  6- 5387264  6- 3582764  6- 4256378
6- 8547623     2586734  4- 2538764  4- 8537264  4- 8352764  4- 5426378
6- 4865723  6- 8275634  6- 3275864  6- 3825764  4- 5832764  *  2534867
6- 6478523  4- 7825634  6- 7382564  6-@2378564@ 6- 3578264     -------
4- 7648523  4- 2785634  6- 8753264  6- 7253864  6- 7325864      9-part
@  4756382     8267453  4- 5873264  6- 5782364  6- 2783564   * 6- in
   -------     -------     -------     -------     -------   pts 3,6,9

Both the above, and the long peal of Bristol Surprise Major below, are rather too saturated with calls, but that's the way the cookie crumbles!

                    28,512 Bristol Surprise Major

   2345678     4762385     7234865     6837254     3865427     8245736
   -------     -------     -------     -------     -------     -------
6- 4263578  4- 6472385  4- 3724865  4- 3687254  4- 6385427  4- 4825736
   6452837     7634528     2387546    @8326475     8643752     2478653
4-@5642837@ 4- 3764528  4- 8237546  4- 2836475  4- 4863752  4- 7248653
4- 4562837  6- 6357428  4- 3827546     3248567  4- 6483752  4- 4728653
   6485723  4- 5637428  6- 2358746     4352786  6- 8674352     2467385
6- 8674523  4- 3567428  4- 5238746  6- 5473286  6- 7836452     6234578
   7856342  6- 6345728  4- 3528746  6- 7524386  4- 3786452  4- 3624578
6- 5738642  4- 4635728  6- 2375846  4- 2754386  4- 8376452  6- 2356478
4- 3578642     3476852  6- 7283546  6- 5237486     7843265     5243867
4- 7358642  4- 7346852     8752634     3542678  4- 4783265  6- 4582367
6- 5763842  4- 4736852     5867423     4365827  4- 8473265  4- 8452367
   6587234     3487265  6- 6548723  6- 6483527     7824536  4- 5842367
6- 8625734  4- 8347265  4- 4658723  6- 8654327  4- 2784536  6- 4538267
4- 2865734  4- 4837265  6- 5476823  4- 5864327  6- 8257436  %% 3425786
   6278453  6- 3428765  6- 7584623  4- 6584327  4- 5827436     -------
   7642385  6- 2374865     8765342  6- 8635427  4- 2587436     9-part
4- 4762385  4- 7234865     6837254  4- 3865427  6- 8245736
   -------     -------     -------     -------     -------
         6th place Bob for Plain lead at %% in parts 3, 6, 9.

The peal of Real Superlative below is over a thousand changes longer than the longest length of Superlative so far rung (which was not recognised as a record peal):

                18,432 Real Superlative Surprise Major

H F 2345678   H F 5843627   H F 5623874   H F 7568324   H F 4857236
-----------   -----------   -----------   -----------   -----------
    5738264   S   5287364       3764582     - 5824736     S 7836425
S  *5674823   S S 5264738       4872356   -  @8356472       6285743
  S 4623587       4328576     S 2856437   - - 8372645       5423678
S   4867352   S   4736852     S 6837245       2435867     - 4378562
S   4582736     S 6752483       7485623   S   2647583     S 8362457
S S 4536278     - 7283645       5243768   S   2863754   -   3587246
  S 6578423     S 3245768     S 3268574     S 3854276   S   3456728
  - 5823647       5628374     S 8274356   -   8736425   S S 3428675
-  @8457362   S S 5674832   S   8526437   S S 8725643   S   3745862
- - 8462735   S   5362487   - S 5837642     S 5743862     S 5762384
  S 2435876   S   5837246       7482563       3672584     - 7284536
  S 5476283   - S 8546723   S   7643258       2864357     S 4236758
S   5843627     - 5623874   S   7568324     S 4857236   -----------
-----------   -----------   -----------   -----------      9-part

Full-Lead Bob at * in 1st, 4th, 7th parts to rotate 567.
Half-Lead: Bob 58, Single 5678;     Full-Lead: Bob 14, Single 1234.
CRUs 64;  78s at backstroke 243;  87s at backstroke 99.   14/8/1997


[6.31] order 9p [6.30] order 18p 2 3 4 5 6 7 8 2 4 3 5 7 6 8 $ 3 4 2 6 7 5 8 3 2 4 6 5 7 8 $ 4 2 3 7 5 6 8 4 3 2 7 6 5 8 $ 2 3 4 6 7 5 8 b 2 4 3 6 5 7 8 $ 3 4 2 7 5 6 8 3 2 4 7 6 5 8 $ 4 2 3 5 6 7 8 b 4 3 2 5 7 6 8 $ 2 3 4 7 5 6 8 b 2 4 3 7 6 5 8 $ 3 4 2 5 6 7 8 b 3 2 4 5 7 6 8 $ 4 2 3 6 7 5 8 4 3 2 6 5 7 8 $

This system has only $-type apices, so that singles are essential. The peal below has a basic 3-part block of two separate Cyclic-Dihedral rotations on 234, 567 and is extended to a 9-part by bobs at one of the locations % every three parts.

                        5,184 Plain Bob Major

  2345678     4237586     3458672     5367248     6523784     2645837
  -------     -------     -------     -------     -------     -------
  3527486     2748365     4837526     3754682     5368247     6523478
- 3578264   S 7286453   - 4872365     7438526     3854672     5367284
S@5386742     2675834     8246753   S@4782365     8437526     3758642
- 5364827     6523748     2685437     7246853   - 8472365   S 7384526
  3452678     5364287     6523874     2675438     4286753     3472865
 %4237586     3458672     5367248     6523784     2645837    %4236758
  -------     -------     -------     -------     -------     -------

Ten-part Palindromes


[7.14] order 10m   [7.13] order 20m

 2 3 4 5 6 7 8      6 5 4 3 2 7 8 $
 3 4 5 6 2 7 8      5 4 3 2 6 7 8 $
 4 5 6 2 3 7 8      4 3 2 6 5 7 8 $
 5 6 2 3 4 7 8      3 2 6 5 4 7 8 $
 6 2 3 4 5 7 8      2 6 5 4 3 7 8 $
 2 3 4 5 6 8 7 s    6 5 4 3 2 8 7 *
 3 4 5 6 2 8 7      5 4 3 2 6 8 7 *
 4 5 6 2 3 8 7      4 3 2 6 5 8 7 *
 5 6 2 3 4 8 7      3 2 6 5 4 8 7 *
 6 2 3 4 5 8 7      2 6 5 4 3 8 7 *

There is hope of a 5-part with the extra pair swapping also, giving a 10-part without any disposable calls or links. There is a good selection of apical transfigures. The peal below realised the hope! 23456 perform a Cyclic-Dihedral palindrome, doubled by 7x8 swapping in every part.

                        5,120 Plain Bob Major

  2345678     4263857     6382574     5246837     4682753     3475862
  -------     -------     -------     -------     -------     -------
  3527486   S 2435678   -@6327845   S 2563478     6245837     4536728
  5738264     4527386     3764258     5327684   - 6253478   - 4562387
  7856342   - 4578263     7435682     3758246     2367584     -------
  8674523     5846732     4578326     7834562     3728645     10-part
  6482735     8653427   - 4582763     8476325   S@7384256   
  4263857     6382574     5246837     4682753     3475862   
  -------     -------     -------     -------     ------- 


[7.15] order 10m [7.13] order 20m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 4 5 6 2 7 8 3 4 5 6 2 8 7 4 5 6 2 3 7 8 4 5 6 2 3 8 7 5 6 2 3 4 7 8 5 6 2 3 4 8 7 6 2 3 4 5 7 8 6 2 3 4 5 8 7 2 6 5 4 3 8 7 2 6 5 4 3 7 8 $ 3 2 6 5 4 8 7 3 2 6 5 4 7 8 $ 4 3 2 6 5 8 7 4 3 2 6 5 7 8 $ 5 4 3 2 6 8 7 5 4 3 2 6 7 8 $ 6 5 4 3 2 8 7 6 5 4 3 2 7 8 $

The $-type apices available keep bells 78 and one of 23456 making places, enabling a 5-part Cyclic-Dihedral palindrome on 23456. In the peal below, the given block is such a palindrome. S for - at % will shunt to & in the complementary 5-part block, effecting swaps of 78 and two pairs of the rotating bells, having the effect of reversing the cycle 23456 and completing their dihedral group.

                        5,120 Plain Bob Major

  2345678     4572836     6724358     5467283     2358674     2583746
  -------     -------     -------     -------     -------     -------
-%2357486   - 4523768   S 7645283     4758632   S 3287546     5324867
  3728564     5346287     6578432   S@7483526     2734865     3456278
  7836245     3658472   S 5683724     4372865   S 7246358     -------
S@8764352   - 3687524   - 5632847     3246758   - 7265483     10-part
  7485623     6732845   S 6524378   S 2365487     2578634   
  4572836   - 6724358     5467283   -&2358674   - 2583746   
  -------     -------     -------     -------     -------   


Twelve-Part Palindromes


[4.02] order 12p  [6.18] order 24m    (TT)

 2 3 4 5 6 7 8     2 3 4 5 6 8 7
 2 4 5 3 6 7 8 b   2 4 5 3 6 8 7
 2 5 3 4 6 7 8 b   2 5 3 4 6 8 7
 3 2 5 4 6 7 8     3 2 5 4 6 8 7 *
 3 4 2 5 6 7 8 b   3 4 2 5 6 8 7
 3 5 4 2 6 7 8 b   3 5 4 2 6 8 7
 4 2 3 5 6 7 8 b   4 2 3 5 6 8 7
 4 3 5 2 6 7 8 b   4 3 5 2 6 8 7
 4 5 2 3 6 7 8     4 5 2 3 6 8 7 *
 5 2 4 3 6 7 8 b   5 2 4 3 6 8 7
 5 3 2 4 6 7 8 b   5 3 2 4 6 8 7
 5 4 3 2 6 7 8     5 4 3 2 6 8 7 *

In the peal below, 2 is the fixed bell and 3456 permute to all +ve perms. The block given is 2-part so that the peal has six such two-part blocks to be joined. There are 4 plain positions %%&& at which a Bobbed Q-set will link three of these blocks, so two such Q-sets will give two blocks overall. The Q-set parity law makes further linkage by P/B Q-set impossible; an asymmetric pair of singles will link, which can be % to %, or & to &, between the blocks in different positions. One good feature of this peal is that the 2nd is never in 5ths or 6ths.

5,376 Plain Bob Major

W   M   H   2 3 4 5 6
---------------------
-    &  -@  4 5 2 3 6
 %  -   -   6 2 5 3 4
S    &  -@  5 3 2 6 4
 %  S   -   2 4 3 6 5
---------------------
  12-part (see text)


[5.06] order 12m [7.34] order 24m (TT) 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 2 4 5 6 7 8 s 3 2 4 5 6 8 7 $ 3 4 2 5 6 7 8 b 3 4 2 5 6 8 7 4 3 2 5 6 7 8 s 4 3 2 5 6 8 7 $ 4 2 3 5 6 7 8 b 4 2 3 5 6 8 7 2 4 3 5 6 7 8 s 2 4 3 5 6 8 7 $ 2 3 4 6 5 7 8 2 3 4 6 5 8 7 $ 3 2 4 6 5 7 8 3 2 4 6 5 8 7 * 3 4 2 6 5 7 8 3 4 2 6 5 8 7 4 3 2 6 5 7 8 4 3 2 6 5 8 7 * 4 2 3 6 5 7 8 4 2 3 6 5 8 7 2 4 3 6 5 7 8 2 4 3 6 5 8 7 *

In the simple peal below 456 are the bells permuting and 23 swap. At the first Home apex 2x3, 4x5 and 7x8 swap, while at the Bob Home apex 2x3, 4x6 and 7x8. Thus 456 perform a Cyclic-Dihedral palindrome. By altering - to S in one of the two positions % every 3rd part, a pair of 456 are swapped, achieving the extent on 456; and S for - at halfway and end of the peal reverses 2x3.

 5,376 Plain Bob Major

 W   M   H   2 3 4 5 6
 ---------------------
    %-       4 3 6 5 2
     -    @  6 3 2 5 4
 -           5 6 2 3 4
%-       -@  2 3 5 6 4
 ---------------------
   12-part (see text)


The curious group [6.14], with its three gyrating pairs, occurs as partend group in the three different palindromic systems below. They illustrate how arcane the application of group structure can be.


[6.14] order 12p  [6.09] order 24p

 2 3 4 5 6 7 8     2 3 6 7 4 5 8 $
 2 3 5 4 7 6 8     2 3 7 6 5 4 8 $
 3 2 4 5 7 6 8     3 2 6 7 5 4 8
 3 2 5 4 6 7 8     3 2 7 6 4 5 8
 4 5 6 7 2 3 8     4 5 2 3 6 7 8 $
 4 5 7 6 3 2 8     4 5 3 2 7 6 8
 5 4 6 7 3 2 8     5 4 2 3 7 6 8
 5 4 7 6 2 3 8     5 4 3 2 6 7 8 $
 6 7 2 3 4 5 8     6 7 4 5 2 3 8 $
 6 7 3 2 5 4 8     6 7 5 4 3 2 8
 7 6 2 3 5 4 8     7 6 4 5 3 2 8 $
 7 6 3 2 4 5 8     7 6 5 4 2 3 8

The transpositions of the group [6.14] are at most of order 3, so four 3-part blocks are the best aim. All apices are $-type. In the block below, trios 247 and 356 each perform Cyclic-Dihedral systems (one of each making the place at an apex) giving a 3-part block. The three other 3-part blocks are reached by asymmetric calls, S for B at % shunting to & in another block. All three positions % in one block will link to the three other blocks.

                        5,376 Plain Bob Major

  2345678     8475623     6725843     4825763     3824765     8724635
  -------     -------     -------     -------     -------     -------
- 2357486   S@4852736   - 6754238     8546237   - 3846257     7483256
  3728564     8243567     7463582   S 5863472     8635472   - 7435862
  7836245   - 8236475   - 7438625     8357624   S 6857324     4576328
- 7864352     2687354     4872356   -&8372546     8762543     -------
  8475623     6725843   -%4825763   S@3824765   - 8724635     12-part
  -------     -------     -------     -------     -------


[6.14] order 12p [6.10] order 24m 2 3 4 5 6 7 8 2 3 6 7 5 4 8 2 3 5 4 7 6 8 2 3 7 6 4 5 8 3 2 4 5 7 6 8 3 2 6 7 4 5 8 * 3 2 5 4 6 7 8 3 2 7 6 5 4 8 * 4 5 6 7 2 3 8 4 5 2 3 7 6 8 * 4 5 7 6 3 2 8 4 5 3 2 6 7 8 5 4 6 7 3 2 8 5 4 2 3 6 7 8 5 4 7 6 2 3 8 5 4 3 2 7 6 8 * 6 7 2 3 4 5 8 6 7 4 5 3 2 8 6 7 3 2 5 4 8 6 7 5 4 2 3 8 * 7 6 2 3 5 4 8 7 6 4 5 2 3 8 7 6 3 2 4 5 8 7 6 5 4 3 2 8 *

The peal below is very similar to the one above in execution, but the apical transfigures are essentially different. Only the fixed bell 8 makes a place. The partends are identical - [6.14] - but the trios 247, 356 perform a Rotating-Sets palindrome. As a mid-lead apex is now possible, the peal is shorter. Three pairs of S for P at % will join on the other three 3-part blocks at & in the same way as in the above peal.

                        5,184 Plain Bob Major

  2345678     6485723     7386524     4587623     7684523     5287463
  -------     -------     -------     -------     -------     -------
- 2357486    %4562837     3672845    &5742836     6472835     2756834
  3728564   - 4523678   - 3624758   -@5723468@    4263758     7623548
  7836245     5347286     6435287   - 5736284   - 4235687     -------
 @8674352     3758462     4568372     7658342     2548376     12-part
  6485723   S 7386524   - 4587623   - 7684523   S 5287463   
  -------     -------     -------     -------     -------


[6.14] order 12p [6.11] order 24m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 2 3 5 4 7 6 8 2 3 5 4 6 7 8 3 2 4 5 7 6 8 3 2 4 5 6 7 8 3 2 5 4 6 7 8 3 2 5 4 7 6 8 * 4 5 6 7 2 3 8 4 5 6 7 3 2 8 4 5 7 6 3 2 8 4 5 7 6 2 3 8 5 4 6 7 3 2 8 5 4 6 7 2 3 8 5 4 7 6 2 3 8 5 4 7 6 3 2 8 6 7 2 3 4 5 8 6 7 2 3 5 4 8 6 7 3 2 5 4 8 6 7 3 2 4 5 8 7 6 2 3 5 4 8 7 6 2 3 4 5 8 7 6 3 2 4 5 8 7 6 3 2 5 4 8

This system has only one apical transfigure available, hence twelve 1-part blocks are inevitable, which makes it unsuitable for major methods. But it lends itself to the composition of Stedman and Erin Triples. The number of apical six-types of [6.14] is just that required for a 12-part palindrome (in triples all the apical sixes must be used in a peal). Also, group [6.11] has transfigures of kind (2,1,1,1,1,1) so that asymmetric P/S and B/S linkages are easy to find. Linking blocks is tricky, but in the Stedman peal below the inversion of a whole section of one side of the palindrome for singling-in produces an approximation to a 24-part peal, and that with a fixed observation bell (the treble). The author produced the following peal in 1992:

                  5,040 Stedman Triples in 12 parts

   2314567                     7613452                     4516327
   -------     4725361         -------     3472651         -------
   3426175     -------         6375124     -------         5642173
 - 3461275     7546213       - 6351724     4235716       - 5621473
   -------   - 7562413         -------   - 4257316         -------
 - 4132675     -------      $S 3167542     -------       - 6154273
   4127356   S 5274631         3174625   S 2743561         6147532
   -------     5243716         -------     2736415         -------
 - 1743256     -------       - 1436725     -------       - 1765432 (s)
   1735462   - 2357416         1462357   - 7624315      *- 1754632
   -------     2371564         -------     7641253         -------
 - 7514362     -------       - 4213657     -------         7413526
   7546123   S 3125746         4235176   S 6172435      +S 7435162
   -------     3154267         -------     6123754         -------
 S 5671432     -------       S 2541367     -------       - 4571362
   5613724     1436572         2516473     1365247         4516723 (q)
   -------     1467325         -------     1354672         -------
 S 6357142     -------       S 5624137     -------         12-part
   6374521     4712653         5643271     3417526
   -------   S 4726135         -------   S 3475162   Call S for - at *
 - 3465721     -------       - 6352471     -------   in parts 1,4,7,10
 - 3457621   - 7641235       - 6324571   - 4531762   to turn 2-3, and
   -------     7613452         -------     4516327   in parts 5,11 to
   4732516     -------         3467215     -------   turn 4-5.
 S 4725361                   S 3472651
   -------                     -------

Never more than two consecutive calls. The single at $ effects a repetition of sections on one side of the basic palindrome (fortunately very long sections, making the peal an approximate 24-part) and the single at + cuts out the corresponding sections which would be reverses of the introduced sections. The apices are: slow six (s) and quick six (q). The treble is observation bell throughout.


[7.29] order 12m [7.22] order 24m 2 3 4 5 6 7 8 2 5 4 3 6 7 8 3 4 5 2 6 7 8 3 2 5 4 6 7 8 $ 4 5 2 3 6 7 8 4 3 2 5 6 7 8 5 2 3 4 6 7 8 5 4 3 2 6 7 8 $ 2 3 4 5 7 8 6 b 2 5 4 3 7 8 6 3 4 5 2 7 8 6 3 2 5 4 7 8 6 4 5 2 3 7 8 6 4 3 2 5 7 8 6 5 2 3 4 7 8 6 5 4 3 2 7 8 6 2 3 4 5 8 6 7 b 2 5 4 3 8 6 7 3 4 5 2 8 6 7 3 2 5 4 8 6 7 4 5 2 3 8 6 7 4 3 2 5 8 6 7 5 2 3 4 8 6 7 5 4 3 2 8 6 7

The two available $-type apices can together give a Rotating-Sets 2-part palindrome (2x4 3x5) and the peal is thus in six 2-part blocks. Since 678 have to make places at both apices, it is difficult to get them at the front for a P/B disposable call. The asymmetric link %S-&S below has the effect of rotating both 2345 (backwards) and 678, and the blocks may be chained together by 5 such pairs of P/S linkages.

                        5,376 Plain Bob Major

  2345678     6482573     7836542     5376482     2475836   S 8672345
  -------     -------     -------     -------     -------     -------
  3527486    &4267835   S@8764325     3658724     4523768     6284753
- 3578264     2743658     7482653     6832547     5346287     2465837
 %5836742     7325486   S 4725836     8264375     3658472     4523678
  8654327   S 3758264     7543268     2487653     6837524     -------
  6482573     7836542     5376482   - 2475836   S@8672345     12-part
  -------     -------     -------     -------     -------


[7.29] order 12m [7.24] order 24m 2 3 4 5 6 7 8 2 5 4 3 6 8 7 $ 3 4 5 2 6 7 8 3 2 5 4 6 8 7 * 4 5 2 3 6 7 8 4 3 2 5 6 8 7 $ 5 2 3 4 6 7 8 5 4 3 2 6 8 7 * 2 3 4 5 7 8 6 b 2 5 4 3 7 6 8 $ 3 4 5 2 7 8 6 3 2 5 4 7 6 8 * 4 5 2 3 7 8 6 4 3 2 5 7 6 8 $ 5 2 3 4 7 8 6 5 4 3 2 7 6 8 * 2 3 4 5 8 6 7 b 2 5 4 3 8 7 6 $ 3 4 5 2 8 6 7 3 2 5 4 8 7 6 * 4 5 2 3 8 6 7 4 3 2 5 8 7 6 $ 5 2 3 4 8 6 7 5 4 3 2 8 7 6 *

This system makes it possible for an exact 12-part, by giving separate Cyclic-Dihedral palindromes on the four bells 2345 and on the three bells 678. The apices in the first part of the peal below are:

    First:  [2] 3x5 [4]    [6] 8x7
    Second:  2x5  3x4      6x8 [7]

thus creating two independent systems, 4-part and 3-part (the second apex is at mid-lead).

                        5,184 Plain Bob Major

  2345678     7482635     5682437     2685437     4835627     8375624
  -------     -------     -------     -------     -------     -------
  3527486   - 7423856     6253874   - 2653874   - 4852376     3582746
  5738264   - 7435268   - 6237548     6327548   - 4827563    @5234867@
  7856342     4576382   S 2674385   - 6374285     8746235     -------
S 8764523     5648723   S@6248753     3468752   S 7863452     12-part
  7482635   - 5682437   S 2685437     4835627     8375624   
  -------     -------     -------     -------     -------   


[7.29] order 12m [7.26] order 24m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 4 5 2 6 7 8 3 4 5 2 6 8 7 * 4 5 2 3 6 7 8 4 5 2 3 6 8 7 5 2 3 4 6 7 8 5 2 3 4 6 8 7 2 3 4 5 7 8 6 b 2 3 4 5 7 6 8 3 4 5 2 7 8 6 3 4 5 2 7 6 8 * 4 5 2 3 7 8 6 4 5 2 3 7 6 8 5 2 3 4 7 8 6 5 2 3 4 7 6 8 2 3 4 5 8 6 7 b 2 3 4 5 8 7 6 3 4 5 2 8 6 7 3 4 5 2 8 7 6 * 4 5 2 3 8 6 7 4 5 2 3 8 7 6 5 2 3 4 8 6 7 5 2 3 4 8 7 6

All three available apices have 2x4, 3x5 hence 2345 must remain fixed in any block; but 678 may perform a 3-part Cyclic-Dihedral palindrome. Hence there might be four 3-part blocks, to be linked by asymmetric calls. In the peal below, 234 are the rotating trio, while 5678 rely on asymmetric P/S links. A single at % will shunt to & in another block, effecting rotation of both 5678 and 234 (the latter not being necessary). But the rotation of 5678 in this peal is not as simple as appears; 2345678 is an apex, and the pairs 5x6, 7x8 must be opposite, not adjacent, in the rotation order, hence the partends commencing 234 are 2345678, 2347865, 2346587, 2348756. Three pairs of S/P links will chain the four blocks together.

                        5,184 Plain Bob Major

  2345678     6472538     3876542     5723486     4235786     6384725
  -------     -------     -------     -------     -------     -------
  3527486     4263785    &8634725     7358264     2548367     3462857
  5738264     2348657     6482357   - 7386542     5826473    @4235678
- 5786342     3825476    @4265873@  - 7364825     8657234     -------
  7654823   - 3857264     2547638     3472658   - 8673542     12-part
 %6472538   - 3876542     5723486     4235786     6384725   
  -------     -------     -------     -------     -------


[7.30] order 12p [7.23] order 24p 2 3 4 5 6 7 8 2 5 4 3 6 8 7 $ 3 4 5 2 6 8 7 3 2 5 4 6 7 8 $ 4 5 2 3 6 7 8 4 3 2 5 6 8 7 $ 5 2 3 4 6 8 7 5 4 3 2 6 7 8 $ 2 3 4 5 7 8 6 b 2 5 4 3 7 6 8 $ 3 4 5 2 7 6 8 3 2 5 4 7 8 6 4 5 2 3 7 8 6 4 3 2 5 7 6 8 $ 5 2 3 4 7 6 8 5 4 3 2 7 8 6 2 3 4 5 8 6 7 b 2 5 4 3 8 7 6 $ 3 4 5 2 8 7 6 3 2 5 4 8 6 7 4 5 2 3 8 6 7 4 3 2 5 8 7 6 $ 5 2 3 4 8 7 6 5 4 3 2 8 6 7

By careful choice of apices, it is possible to obtain a 4-part block in which 2345 perform a Cyclic-Dihedral palindrome while a pair of 678 swap, and three 4-part blocks result; the disposable P/B calls on 678 provide a Q-set to link them. In the peal below, 234 are the permuting trio and 5678 rotate cyclically. The blocks are linked by bobs at the ends of parts 4, 8, 12, which rotate bells 234.

                        5,760 Plain Bob Major

  2345678     3642578     5482736     7865432     3657482     4527386
  -------     -------     -------     -------     -------     -------
  3527486   - 3627485     4253867     8573624   - 3678524   - 4578263
S 5378264     6738254   S@2436578     5382746   S 6382745   S@5486732
- 5386742     7865342     4627385   - 5324867     3264857   - 5463827
  3654827     8574623     6748253   - 5346278     2435678     4352678
- 3642578     5482736     7865432     3657482     4527386     3247586
  -------     -------     -------     -------     -------     -------


[7.30] order 12p [7.25] order 24m 2 3 4 5 6 7 8 2 5 4 3 6 7 8 3 4 5 2 6 8 7 3 2 5 4 6 8 7 * 4 5 2 3 6 7 8 4 3 2 5 6 7 8 5 2 3 4 6 8 7 5 4 3 2 6 8 7 * 2 3 4 5 7 8 6 b 2 5 4 3 7 8 6 3 4 5 2 7 6 8 3 2 5 4 7 6 8 * 4 5 2 3 7 8 6 4 3 2 5 7 8 6 5 2 3 4 7 6 8 5 4 3 2 7 6 8 * 2 3 4 5 8 6 7 b 2 5 4 3 8 6 7 3 4 5 2 8 7 6 3 2 5 4 8 7 6 * 4 5 2 3 8 6 7 4 3 2 5 8 6 7 5 2 3 4 8 7 6 5 4 3 2 8 7 6 *

At all available apices, two pairs of 2345 swap, making it impossible to obtain a Cyclic-Dihedral 4-part, but possible to get a Rotating-Pairs 2-part. 678 may be persuaded to rotate, thus giving two 6-part blocks needing an asymmetrical link. In the peal below 5678 are the rotating bells and 234 the permuting ones. S for P at % will provide the necessary shunt to position & between the two 6-part blocks.

                        5,184 Plain Bob Major

  2345678     6472538     4672538     3724568     4723568     8263547
  -------     -------     -------     -------     -------     -------
 %3527486   - 6423785     6243785     7436285   - 4736285     2384675
  5738264   - 6438257    &2368457   S 4768352     7648352    @3427856
- 5786342   - 6485372   -@2385674@  - 4785623     6875423     -------
  7654823   - 6457823   - 2357846   - 4752836   - 6852734     12-part
  6472538   S 4672538     3724568   - 4723568     8263547
  -------     -------     -------     -------     -------


[7.30] order 12p [7.26] order 24m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 4 5 2 6 8 7 3 4 5 2 6 7 8 4 5 2 3 6 7 8 4 5 2 3 6 8 7 * 5 2 3 4 6 8 7 5 2 3 4 6 7 8 2 3 4 5 7 8 6 b 2 3 4 5 7 6 8 3 4 5 2 7 6 8 3 4 5 2 7 8 6 4 5 2 3 7 8 6 4 5 2 3 7 6 8 * 5 2 3 4 7 6 8 5 2 3 4 7 8 6 2 3 4 5 8 6 7 b 2 3 4 5 8 7 6 3 4 5 2 8 7 6 3 4 5 2 8 6 7 4 5 2 3 8 6 7 4 5 2 3 8 7 6 * 5 2 3 4 8 7 6 5 2 3 4 8 6 7

The three available *-apices all have pairs 2x4, 3x5 swapping so that any block produced must have these four bells fixed. But permuting bells 678 have the freedom of a Cyclic-Dihedral 3-part block. Thus there will be four 3-part blocks to link, and the P/B disposable calls are useless as they rotate 678. The peal below has 234 permuting and 5678 rotating, the latter being done indirectly by asymmetric pairs of singles % to & and back. The chaining requires three such pairs. These links effect a transfigure of kind (4,2,1).

                        5,376 Plain Bob Major

  2345678     2645738     7482365     3482675     5872643     8672543
  -------     -------     -------     -------     -------     -------
S 3257486     6523487   - 7426853   S 4327856   - 5824736    &6284735
  2738564     5368274     4675238     3745268   -@5843267     2463857
S 7286345    @3857642   S 6453782     7536482   - 5836472   S 4235678
  2674853    %8734526     4368527   - 7568324     8657324     -------
- 2645738     7482365   S 3482675     5872643   - 8672543     12-part
  -------     -------     -------     -------     -------


[7.31] order 12p [7.23] order 24p 2 3 4 5 6 7 8 2 3 5 4 6 8 7 $ 3 2 5 4 6 7 8 3 2 4 5 6 8 7 $ 4 5 2 3 6 7 8 4 5 3 2 6 8 7 5 4 3 2 6 7 8 5 4 2 3 6 8 7 2 3 4 5 7 8 6 b 2 3 5 4 7 6 8 $ 3 2 5 4 7 8 6 3 2 4 5 7 6 8 $ 4 5 2 3 7 8 6 4 5 3 2 7 6 8 5 4 3 2 7 8 6 5 4 2 3 7 6 8 2 3 4 5 8 6 7 b 2 3 5 4 8 7 6 $ 3 2 5 4 8 6 7 3 2 4 5 8 7 6 $ 4 5 2 3 8 6 7 4 5 3 2 8 7 6 5 4 3 2 8 6 7 5 4 2 3 8 7 6

All available apices swap one of the pairs 23 or 45, and one pair of 678. Hence it is possible to make 2345 2-part (3254, 2345) and 678 a 3-part Cyclic-Dihedral palindrome, giving a 6-part block such as the example below, where 234 rotate and 5x7, 6x8. But all attempts to find a pair of linking singles have failed! Such a shunt would swap 5x6 7x8 or 5x8 6x7, perhaps also inconsequentially rotating 234.

                6,144 Plain Bob Major (in two blocks)

  2345678     8563427     6485372     7624853     7486235     8263547
  -------     -------     -------     -------     -------     -------
S 3257486     5382674     4567823   S 6745238   S 4763852     2384675
  2738564     3257846     5742638     7563482   - 4735628     3427856
  7826345   S 2374568     7253486   S@5738624     7542386     -------
  8674253     3426785   - 7238564     7852346     5278463
S@6845732     4638257   S 2786345   S 8724563     2856734   
  8563427   S 6485372     7624853     7486235   S 8263547   
  -------     -------     -------     -------     -------


[7.31] order 12p [7.27] order 24m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 2 5 4 6 7 8 3 2 5 4 6 8 7 * 4 5 2 3 6 7 8 4 5 2 3 6 8 7 * 5 4 3 2 6 7 8 5 4 3 2 6 8 7 * 2 3 4 5 7 8 6 b 2 3 4 5 7 6 8 3 2 5 4 7 8 6 3 2 5 4 7 6 8 * 4 5 2 3 7 8 6 4 5 2 3 7 6 8 * 5 4 3 2 7 8 6 5 4 3 2 7 6 8 * 2 3 4 5 8 6 7 b 2 3 4 5 8 7 6 3 2 5 4 8 6 7 3 2 5 4 8 7 6 * 4 5 2 3 8 6 7 4 5 2 3 8 7 6 * 5 4 3 2 8 6 7 5 4 3 2 8 7 6 *

It is possible to get bells 2345 perform a 2-part Rotating-Sets palindrome, while 678 do Cyclic-Dihedral, giving a 6-part block. The complementary block must be joined by an asymmetric pair of singles. The peal below reproduces the partends on the left exactly. At the mid-lead apex (the first lead) 2x3,4x5 swap and at the plain lead apex 3x4, 2x5 giving overall 2x4, 3x5; while 678 rotate. S for P at % will link to & in the other block, swapping (in the first part) 2x3 and 4x5 and inconsequentially rotating 678 also.

                        5,568 Plain Bob Major

 @2345678@    5642738     7342658     6432578     8436572     6238574
  -------     -------     -------     -------     -------     -------
  3527486   - 5623487   - 7325486    &4267385     4687325     2867345
 %5738264     6358274     3578264   S 2478653   - 4672853   S 8274653
S 7586342   - 6387542   S 5386742     4825736     6245738     2485736
  5674823     3764825     3654827   - 4853267   - 6253487    @4523867@
- 5642738   S 7342658    @6432578   S 8436572   - 6238574     -------
  -------     -------     -------     -------     -------     12-part


[7.32] order 12m [7.24] order 24m 2 3 4 5 6 7 8 2 3 5 4 6 8 7 $ 3 2 5 4 6 7 8 3 2 4 5 6 8 7 $ 4 5 2 3 6 8 7 4 5 3 2 6 7 8 5 4 3 2 6 8 7 5 4 2 3 6 7 8 2 3 4 5 7 8 6 b 2 3 5 4 7 6 8 $ 3 2 5 4 7 8 6 3 2 4 5 7 6 8 $ 4 5 2 3 7 6 8 4 5 3 2 7 8 6 5 4 3 2 7 6 8 5 4 2 3 7 8 6 2 3 4 5 8 6 7 b 2 3 5 4 8 7 6 $ 3 2 5 4 8 6 7 3 2 4 5 8 7 6 $ 4 5 2 3 8 7 6 4 5 3 2 8 6 7 5 4 3 2 8 7 6 5 4 2 3 8 6 7

The conditions are similar to system [7.31]-[7.23] in regard to apices, but [7.32] is mixed and 678 permute in a linked fashion, so that the asymmetric links must swap a pair of 678. In the peal below, 234 take the place of 678. The first apex swaps 6x7 and the second 5x8, giving partends 8765, 5678 while 234 rotate; this gives a 6-part block. S for B at % shunts to & in the complementary 6-part, swapping 5x6, 7x8 to complete the group [4.04] on 5678, while 3x4 swap to throw the trio 234 into -ve permutations.

                        5,760 Plain Bob Major

  2345678     3842675     5473682     5673482     8275463     2738465
  -------     -------     -------     -------     -------     -------
  3527486     8237456   S 4538726     6358724     2586734   -&2786354
S 5378264     2785364   S@5482367     3862547   S 5263847   S@7265843
  3856742     7526843   S 4526873     8234675     2354678   - 7254638
- 3864527   - 7564238     5647238   - 8247356   S 3247586     2473586
- 3842675     5473682   - 5673482   -%8275463     2738465   S 4238765
  -------     -------     -------     -------     -------     -------
                                                              12-part

[7.32] order 12m [7.27] order 24m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 2 5 4 6 7 8 3 2 5 4 6 8 7 * 4 5 2 3 6 8 7 4 5 2 3 6 7 8 $ 5 4 3 2 6 8 7 5 4 3 2 6 7 8 $ 2 3 4 5 7 8 6 b 2 3 4 5 7 6 8 3 2 5 4 7 8 6 3 2 5 4 7 6 8 * 4 5 2 3 7 6 8 4 5 2 3 7 8 6 5 4 3 2 7 6 8 5 4 3 2 7 8 6 2 3 4 5 8 6 7 b 2 3 4 5 8 7 6 3 2 5 4 8 6 7 3 2 5 4 8 7 6 * 4 5 2 3 8 7 6 4 5 2 3 8 6 7 5 4 3 2 8 7 6 5 4 3 2 8 6 7

It was discovered that all the disposable-call types which are not at $-apices cannot be accessed because of self-false types, hence disposable calls cannot be used. The best block is a 3-part one with bells 678 rotating and 2345 fixed; then there are difficulties with the asymmetric links; one type of link alone refuses to join up the four 3-part blocks. This is not a practicable system, one is forced to conclude!


[7.33] order 12p [7.28] order 24p 2 3 4 5 6 7 8 2 3 5 4 7 6 8 $ 2 4 5 3 7 8 6 2 4 3 5 8 7 6 $ 2 5 3 4 8 6 7 2 5 4 3 6 8 7 $ 3 2 5 4 6 7 8 3 2 4 5 7 6 8 $ 3 4 2 5 8 6 7 3 4 5 2 6 8 7 3 5 4 2 7 8 6 3 5 2 4 8 7 6 4 2 3 5 7 8 6 4 2 5 3 8 7 6 4 3 5 2 8 6 7 4 3 2 5 6 8 7 $ 4 5 2 3 6 7 8 4 5 3 2 7 6 8 5 2 4 3 8 6 7 5 2 3 4 6 8 7 5 3 2 4 7 8 6 5 3 4 2 8 7 6 $ 5 4 3 2 6 7 8 5 4 2 3 7 6 8

The $-type apices available all have one pair of 2345 swapping and one pair of 678. One of the two possible strategies, in the peal below, is to have three of 2345, and also 678, rotating, giving a 3-part block. Swaps:

        First  S@    2 4 3x5    7 6x8
        Second S@    2 3x4 5    6 7x8

Hence 2 is fixed in this block, 345 and 678 rotate. S for P at % shunts to &; in this first part the shunt is to the block with 3 fixed. Shunts in all three parts of the block will link up the other three 3-part blocks.

                        5,184 Plain Bob Major

  2345678     6452378     3586472     2856473     6572483     6782453
  -------     -------     -------     -------     -------     -------
  3527486     4267583   - 3567824     8627534   S@5628734     7265834
- 3578264   S@2478635   - 3572648     6783245     6853247    &2573648
- 3586742    %4823756     5234786   - 6734852   S 8634572   - 2534786
  5634827     8345267   - 5248367   S 7645328   - 8647325     -------
  6452378     3586472     2856473     6572483     6782453     12-part
  -------     -------     -------     -------     -------


[7.35] order 12p [7.23] order 24p 2 3 4 5 6 7 8 2 3 4 7 8 5 6 $ 2 3 4 6 5 8 7 2 3 4 8 7 6 5 $ 3 2 4 5 6 8 7 3 2 4 8 7 5 6 3 2 4 6 5 7 8 3 2 4 7 8 6 5 3 4 2 5 6 7 8 b 3 4 2 7 8 5 6 3 4 2 6 5 8 7 3 4 2 8 7 6 5 4 3 2 5 6 8 7 4 3 2 8 7 5 6 4 3 2 6 5 7 8 4 3 2 7 8 6 5 4 2 3 5 6 7 8 b 4 2 3 7 8 5 6 4 2 3 6 5 8 7 4 2 3 8 7 6 5 2 4 3 5 6 8 7 2 4 3 8 7 5 6 2 4 3 6 5 7 8 2 4 3 7 8 6 5

The two available apices enable a 2-part Rotating-Sets palindrome on 5678. This leaves 234 to permute. Disposable calls P/B will link 6 blocks into 2, leaving an asymmetric pair of singles to join up. The peal below follows the figures on the left exactly; B for P at one of the positions + in alternate parts will give a 6-part block, while S for P at % will shunt to & in the complementary block, swapping a pair of 234 and also 5x6 or 7x8.

                        6,528 Plain Bob Major

  2345678     8567423     6825374     3268547     7853624     5768342
  -------     -------     -------     -------     -------     -------
- 2357486     5782634    &8567243   - 3284675     8372546   S 7584623
- 2378564     7253846     5784632    +2437856     3284765     5472836
  3826745     2374568   S 7543826     4725368   S@2346857     4253768
  8634257   S@3246785     5372468     7546283     3625478    +2346587
S 6845372     2638457     3256784   S 5768432     6537284     -------
 %8567423     6825374   - 3268547     7853624     5768342     12-part
  -------     -------     -------     -------     -------

[7.35] order 12p [7.24] order 24m 2 3 4 5 6 7 8 3 2 4 7 8 5 6 * 2 3 4 6 5 8 7 3 2 4 8 7 6 5 * 3 2 4 5 6 8 7 2 3 4 8 7 5 6 3 2 4 6 5 7 8 2 3 4 7 8 6 5 3 4 2 5 6 7 8 b 4 3 2 7 8 5 6 * 3 4 2 6 5 8 7 4 3 2 8 7 6 5 * 4 3 2 5 6 8 7 3 4 2 8 7 5 6 4 3 2 6 5 7 8 3 4 2 7 8 6 5 4 2 3 5 6 7 8 b 2 4 3 7 8 5 6 * 4 2 3 6 5 8 7 2 4 3 8 7 6 5 * 2 4 3 5 6 8 7 4 2 3 8 7 5 6 2 4 3 6 5 7 8 4 2 3 7 8 6 5

Apices as in the previous system but with an extra pair of swaps on a pair of 234, making it not necessary to use disposables. In the peal below, 5x7 and 6x8 are the swapping pairs. The swaps in the apices are:

   Mid-lead     3 2x4    5x8 6x7
   Partend      2x3 4    5x6 7x8

giving a 6-part block with Cyclic-Dihedral on 234 and Rotating-Sets on 5678. S for P at % shunts to & in the complementary 6-part block, swapping (in the first part) bells 5x7 and 2x3 and permuting 234.

                        5,568 Plain Bob Major

  2345678     6452738     4852736     7564832     5267834     8675234
  -------     -------     -------     -------     -------     -------
  3527486   S 4623587     8243567     5473628     2753648   S 6853742
  5738264   S 6438275   S 2836475   S 4532786   S 7234586     8364527
S 7586342     4867352     8627354     5248367   S 2748365     3482675
 %5674823   - 4875623    @6785243@  - 5286473     7826453    @4237856
  6452738   - 4852736    &7564832   - 5267834     8675234     -------
  -------     -------     -------     -------     -------     12-part


[7.35] order 12p [7.34] order 24m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 2 3 4 6 5 8 7 2 3 4 6 5 7 8 3 2 4 5 6 8 7 3 2 4 5 6 7 8 3 2 4 6 5 7 8 3 2 4 6 5 8 7 * 3 4 2 5 6 7 8 b 3 4 2 5 6 8 7 3 4 2 6 5 8 7 3 4 2 6 5 7 8 4 3 2 5 6 8 7 4 3 2 5 6 7 8 4 3 2 6 5 7 8 4 3 2 6 5 8 7 * 4 2 3 5 6 7 8 b 4 2 3 5 6 8 7 4 2 3 6 5 8 7 4 2 3 6 5 7 8 2 4 3 5 6 8 7 2 4 3 5 6 7 8 2 4 3 6 5 7 8 2 4 3 6 5 8 7 *

As all three available apices swap 5x6 and 7x8, these are fixed in any block and only 234 may rotate, leaving four 3-part blocks for linking asymmetrically. Two different such links are needed. In the peal below, the link S for P at % will shunt to a position & swapping 2x3 and 7x8; while S for B at * will shunt to a position $ elsewhere exchanging 5x6 and 7x8 (incidentally rotating 234). One of one kind of link and two of the other are needed.

                        6,336 Plain Bob Major

  2345678     5287436     6475823     3624758     2356487     6278534
  -------     -------     -------     -------     -------     -------
 %3527486   - 5273864     4562738     6435287     3628574   - 6283745
-*3578264     2356748     5243687     4568372   - 3687245     2364857
  5836742     3624587    &2358476   - 4587623     6734852    @3425678
- 5864327     6438275   -@2387564@  - 4572836   S 7645328     -------
S 8542673   -$6487352   - 2376845     5243768   - 7652483     12-part
  5287436   - 6475823     3624758     2356487     6278534   
  -------     -------     -------     -------     -------


[7.36] order 12m [7.27] order 24m 2 3 4 5 6 7 8 2 3 4 7 8 5 6 $ 2 3 4 6 5 8 7 2 3 4 8 7 6 5 $ 3 2 4 5 6 7 8 s 3 2 4 7 8 5 6 * 3 2 4 6 5 8 7 3 2 4 8 7 6 5 * 3 4 2 5 6 7 8 b 3 4 2 7 8 5 6 3 4 2 6 5 8 7 3 4 2 8 7 6 5 4 3 2 5 6 7 8 s 4 3 2 7 8 5 6 * 4 3 2 6 5 8 7 4 3 2 8 7 6 5 * 4 2 3 5 6 7 8 b 4 2 3 7 8 5 6 4 2 3 6 5 8 7 4 2 3 8 7 6 5 2 4 3 5 6 7 8 s 2 4 3 7 8 5 6 * 2 4 3 6 5 8 7 2 4 3 8 7 6 5 *

Regarding bells 5678, the apical swaps enable a Rotating-Sets 2-part block. The *-type apices also swap a pair of 234, making possible a Cyclic-Dihedral 3-part rotation. Hence a 6-part block is possible, to be doubled by a disposable P/S call. In the peal below the apical swaps are:

   Bells:         234       5678
   Mid-lead:     2x4 3    5x7 6x8
   Final Plain:  2x3 4    5x6 7x8

Thus 5x8, 6x7 swap overall. S/P turning calls at partends 6, 12.

                        5,184 Plain Bob Major

  2345678     5482673     3685742     7823465     4257368     7354628
  -------     -------     -------     -------     -------     -------
  3527486     4257836     6534827     8376254     2746583     3472586
- 3578264     2743568     5462378   - 8365742     7628435    @4238765
  5836742     7326485    @4257683@    3584627   S 6783254     -------
S 8564327     3678254     2748536     5432876     7365842     12-part
  5482673   - 3685742     7823465     4257368   - 7354628   
  -------     -------     -------     -------     -------


[7.36] order 12m [7.34] order 24m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 2 3 4 6 5 8 7 2 3 4 6 5 7 8 3 2 4 5 6 7 8 s 3 2 4 5 6 8 7 $ 3 2 4 6 5 8 7 3 2 4 6 5 7 8 $ 3 4 2 5 6 7 8 b 3 4 2 5 6 8 7 3 4 2 6 5 8 7 3 4 2 6 5 7 8 4 3 2 5 6 7 8 s 4 3 2 5 6 8 7 $ 4 3 2 6 5 8 7 4 3 2 6 5 7 8 $ 4 2 3 5 6 7 8 b 4 2 3 5 6 8 7 4 2 3 6 5 8 7 4 2 3 6 5 7 8 2 4 3 5 6 7 8 s 2 4 3 5 6 8 7 $ 2 4 3 6 5 8 7 2 4 3 6 5 7 8 $

The $-type apices swap a pair of the trio 234, and either 5x6 or 7x8. If the two apices swap different pairs of 234, and also one swaps 5x6 and the other 7x8, a 6-part block results, which can be doubled by a disposable S/P (or S/B) pair of calls. In the peal below, the swapping pairs are 5x8 and 6x7, thus giving either 5678 or 8765 at the partends. It remains to call S for P at one of the positions *, halfway and end.

                        5,376 Plain Bob Major

  2345678     6482573     5862473     4635278     8235746     7365842
  -------     -------     -------     -------     -------     -------
  3527486   - 6427835     8257634   - 4657382   S@2854367   - 7354628
- 3578264   S@4673258   - 8273546     6748523   - 2846573     3472586
  5836742   - 4635782     2384765     7862435     8627435    *4238765
  8654327     6548327    *3426857     8273654     6783254     -------
  6482573     5862473     4635278   - 8235746     7365842     12-part
  -------     -------     -------     -------     -------


[7.37] order 12m [7.22] order 24m 2 3 4 5 6 7 8 2 3 4 7 8 5 6 $ 2 3 4 5 6 8 7 s 2 3 4 7 8 6 5 2 3 4 6 5 7 8 s 2 3 4 8 7 5 6 2 3 4 6 5 8 7 2 3 4 8 7 6 5 $ 3 4 2 5 6 7 8 b 3 4 2 7 8 5 6 3 4 2 5 6 8 7 3 4 2 7 8 6 5 3 4 2 6 5 7 8 3 4 2 8 7 5 6 3 4 2 6 5 8 7 3 4 2 8 7 6 5 4 2 3 5 6 7 8 b 4 2 3 7 8 5 6 4 2 3 5 6 8 7 4 2 3 7 8 6 5 4 2 3 6 5 7 8 4 2 3 8 7 5 6 4 2 3 6 5 8 7 4 2 3 8 7 6 5

The two $-type apices enable a Rotating-Sets block 6587, 5678. A disposable pair P/S on either 5x6 or 7x8 will give 4 parts with 234 fixed, and a P/B Q-set on 234 will give 12 parts. In the peal below, 5x8 and 6x7 take the place of 5x6, 7x8 quoted on the left. S for P at * in alternate parts will turn 6x7 (there are other places where 6x7 or 5x8 may be turned) and B for P at one of the positions % in every fourth part will rotate 234.

                        5,184 Plain Bob Major

  2345678     8472635     6758342     4528367     3274865     5763824
  -------     -------     -------     -------     -------     -------
  3527486     4283756   - 6784523     5846273    %2436758     7352648
  5738264    %2345867     7462835   - 5867432     4625387     3274586
  7856342     3526478     4273658     8753624     6548273   S@2348765
- 7864523     5637284   S@2435786     7382546   - 6587432     -------
  8472635     6758342     4528367     3274865    *5763824     12-part
  -------     -------     -------     -------     -------


[7.37] order 12m [7.25] order 24m 2 3 4 5 6 7 8 2 4 3 7 8 5 6 * 2 3 4 5 6 8 7 s 2 4 3 7 8 6 5 2 3 4 6 5 7 8 s 2 4 3 8 7 5 6 2 3 4 6 5 8 7 2 4 3 8 7 6 5 * 3 4 2 5 6 7 8 b 3 2 4 7 8 5 6 * 3 4 2 5 6 8 7 3 2 4 7 8 6 5 3 4 2 6 5 7 8 3 2 4 8 7 5 6 3 4 2 6 5 8 7 3 2 4 8 7 6 5 * 4 2 3 5 6 7 8 b 4 3 2 7 8 5 6 * 4 2 3 5 6 8 7 4 3 2 7 8 6 5 4 2 3 6 5 7 8 4 3 2 8 7 5 6 4 2 3 6 5 8 7 4 3 2 8 7 6 5 *

This system is similar to the previous one [7.37]-[7.22], but a pair of 234 also swaps at each apex. This will give a 6-part block without the need for using disposable calls to rotate 234. The turning calls for either 5x6 or 7x8 are required. In the peal below, as before, 5x8 and 6x7 are the swapping pairs. % is a position for turning calls S/P to swap 67 every 6 parts. As mid-lead apices are now possible the peal is shorter.

                        5,184 Plain Bob Major

  2345678     8475623     3728645     6523748     7246358     3457628
  -------     -------     -------     -------     -------     -------
- 2357486     4582736     7834256     5364287    %2675483   - 3472586
  3728564     5243867   S 8745362     3458672     6528734    @4238765
  7836245     2356478    @7586423@  - 3487526   - 6583247     -------
- 7864352   - 2367584     5672834     4732865     5364872     12-part
  8475623     3728645   S 6523748     7246358     3457628   
  -------     -------     -------     -------     -------


[7.37] order 12m [7.34] order 24m 2 3 4 5 6 7 8 2 4 3 5 6 7 8 2 3 4 5 6 8 7 s 2 4 3 5 6 8 7 $ 2 3 4 6 5 7 8 s 2 4 3 6 5 7 8 $ 2 3 4 6 5 8 7 2 4 3 6 5 8 7 * 3 4 2 5 6 7 8 b 3 2 4 5 6 7 8 3 4 2 5 6 8 7 3 2 4 5 6 8 7 $ 3 4 2 6 5 7 8 3 2 4 6 5 7 8 $ 3 4 2 6 5 8 7 3 2 4 6 5 8 7 * 4 2 3 5 6 7 8 b 4 3 2 5 6 7 8 4 2 3 5 6 8 7 4 3 2 5 6 8 7 $ 4 2 3 6 5 7 8 4 3 2 6 5 7 8 $ 4 2 3 6 5 8 7 4 3 2 6 5 8 7 *

Compared with the two systems on [7.37] above, 5678 cannot now give Rotating-Sets but can give a 6-part block by independent swaps, and with 234 rotating. In the peal below, 5x8 and 6x7 swap:

    Bells:       234      5678
    Apex S@:    2x4 3    5 8 6x7 
    Mid-lead:   2 3x4    5x8 6x7
    Overall:    rotate   5x8 6 7

This gives a 6-part block, to be doubled by swapping 6x7, S for B in one position % every 6 parts (or in alternate parts instead).

                        5,184 Plain Bob Major

  2345678     7642538     3487562     6583742     4826753     3256748
  -------     -------     -------     -------     -------     -------
  3527486     6273485     4736825     5364827   - 4865237   S 2364587
  5738264     2368754     7642358     3452678    @8543672@    3428675
- 5786342     3825647   -%7625483     4237586     5387426     -------
  7654823   S@8354276     6578234   S 2478365   - 5372864     12-part
-%7642538     3487562   - 6583742     4826753     3256748   
  -------     -------     -------     -------     -------


[7.38] order 12m [7.34] order 24m 2 3 4 5 6 7 8 2 3 4 6 5 7 8 2 3 4 5 6 8 7 s 2 3 4 6 5 8 7 $ 3 2 4 6 5 7 8 3 2 4 5 6 7 8 3 2 4 6 5 8 7 3 2 4 5 6 8 7 $ 3 4 2 5 6 7 8 b 3 4 2 6 5 7 8 3 4 2 5 6 8 7 3 4 2 6 5 8 7 4 3 2 6 5 7 8 4 3 2 5 6 7 8 4 3 2 6 5 8 7 4 3 2 5 6 8 7 $ 4 2 3 5 6 7 8 b 4 2 3 6 5 7 8 4 2 3 5 6 8 7 4 2 3 6 5 8 7 2 4 3 6 5 7 8 2 4 3 5 6 7 8 2 4 3 6 5 8 7 2 4 3 5 6 8 7 $

The first $-apex swaps 5x6 7x8 if one of each kind is chosen and is different from the rest; the result is a 2-part block which can have disposable calls for direct linkage. In the peal below, the apices swap (2x4 7X8) and (5x6 7x8). B for P at + in alternate parts will rotate 234 thus permuting 234 over 6 parts (56 will adjust themselves) while S for B at * twice, at half-peal intervals, will swap 7x8 and generate group [7.38].

                        6,144 Plain Bob Major

  2345678     4263785     7482653     8254376     3472568     5384726
  -------     -------     -------     -------     -------     -------
  3527486    +2348657   S 4725836   S 2847563     4236785     3452867
  5738264     3825476     7543268     8726435     2648357   S@4326578
- 5786342   - 3857264     5376482   - 8763254     6825473     -------
  7654823     8736542   S@3568724     7385642     8567234     12-part
  6472538   -*8764325     5832647   - 7354826   - 8573642     (see
  4263785     7482653     8254376     3472568     5384726       text)
  -------     -------     -------     -------     -------   


Eighteen-part Palindromes


[6.12] order 18m   [6.08] order 36m

 2 3 4 5 6 7 8     7 6 5 4 3 2 8 *
 3 4 5 6 7 2 8     6 5 4 3 2 7 8 $
 4 5 6 7 2 3 8     5 4 3 2 7 6 8 *
 5 6 7 2 3 4 8     4 3 2 7 6 5 8 $
 6 7 2 3 4 5 8     3 2 7 6 5 4 8 *
 7 2 3 4 5 6 8     2 7 6 5 4 3 8 $
 2 7 4 3 6 5 8 b   5 6 3 4 7 2 8
 7 4 3 6 5 2 8     6 3 4 7 2 5 8 $
 4 3 6 5 2 7 8 b   3 4 7 2 5 6 8
 3 6 5 2 7 4 8     4 7 2 5 6 3 8 $
 6 5 2 7 4 3 8     7 2 5 6 3 4 8
 5 2 7 4 3 6 8     2 5 6 3 4 7 8 $
 2 5 4 7 6 3 8 b   3 6 7 4 5 2 8
 5 4 7 6 3 2 8     6 7 4 5 2 3 8 $
 4 7 6 3 2 5 8     7 4 5 2 3 6 8
 7 6 3 2 5 4 8     4 5 2 3 6 7 8 $
 6 3 2 5 4 7 8 b   5 2 3 6 7 4 8
 3 2 5 4 7 6 8     2 3 6 7 4 5 8 $

The system incorporates as a subgroup 6-part [6.15]-[6.13] hence a 6-part Cyclic-Dihedral palindrome is possible, using the first three $-type apices only. The linking of the three resulting 6-part blocks is effected by the P/B disposable links. Cyclic partends are very difficult to achieve and the peal below is longer, which helps the task of composition. The rotation order of the part-ends is 2-5-3-7-4-6-2 which puts trios 234, 567 alternate, and these trios come together at the partends, 234/567 or 567/234 making it simple to turn one trio relative to the other by a disposable bob.

                        5,760 Plain Bob Major

  2345678     7654823     3685742     5768234     4236785   18-part
  -------     -------     -------     -------     -------
  3527486   - 7642538     6534827   - 5783642     2648357  Bob at the
  5738264   - 7623485   - 6542378     7354826   -@2685473  ends of the
- 5786342     6378254   - 6527483     3472568     6527834  6th, 12th,
  7654823   S@3685742     5768234     4236785     5763248  18th parts.
  -------     -------     -------     -------     -------


[6.29] order 18m [6.28] order 36m 2 3 4 5 6 7 8 2 3 4 5 7 6 8 2 3 4 6 7 5 8 b 2 3 4 6 5 7 8 2 3 4 7 5 6 8 b 2 3 4 7 6 5 8 3 2 4 6 7 5 8 3 2 4 6 5 7 8 $ 3 2 4 5 6 7 8 s 3 2 4 5 7 6 8 $ 3 2 4 7 5 6 8 3 2 4 7 6 5 8 $ 3 4 2 5 6 7 8 b 3 4 2 5 7 6 8 3 4 2 6 7 5 8 3 4 2 6 5 7 8 3 4 2 7 5 6 8 3 4 2 7 6 5 8 4 3 2 5 6 7 8 s 4 3 2 5 7 6 8 $ 4 3 2 6 7 5 8 4 3 2 6 5 7 8 $ 4 3 2 7 5 6 8 4 3 2 7 6 5 8 $ 4 2 3 5 6 7 8 b 4 2 3 5 7 6 8 4 2 3 6 7 5 8 4 2 3 6 5 7 8 4 2 3 7 5 6 8 4 2 3 7 6 5 8 2 4 3 5 6 7 8 s 2 4 3 5 7 6 8 $ 2 4 3 6 7 5 8 2 4 3 6 5 7 8 $ 2 4 3 7 5 6 8 2 4 3 7 6 5 8 $

The $-type apices all swap one pair each of the trios 234, 567 hence it is possible to produce a 3-part block with simultaneous Cyclic-Dihedral palindromes on the two trios. Moreover, the trio 234 which permutes has P/B and P/S disposable calls which enable this trio to be permuted relative to the rotating trio 567. In the peal below, 5 is the fixed bell (instead of 8) and 678 is the rotating trio. 234 come to the front at the partends, where the usual --S--S at the ends of the blocks permute them.

                        5,376 Plain Bob Major

  2345678     5364827     2846753     5784362     6358247    18-part
  -------     -------     -------     -------     -------
  3527486     3452678   - 2865437   S@7546823   - 6384572   --S--S at
S 5378264     4237586     8523674     5672438     3467825   ends of
S@3586742   - 4278365   - 8537246   - 5623784   - 3472658   pts 3,6,9,
S 5364827     2846753     5784362     6358247     4235786   12,15,18.
  -------     -------     -------     -------     -------


[6.30] order 18p [6.08] order 36m 2 3 4 5 6 7 8 5 6 7 2 3 4 8 ** 2 3 4 6 7 5 8 b 5 6 7 3 4 2 8 2 3 4 7 5 6 8 b 5 6 7 4 2 3 8 3 2 4 5 7 6 8 6 5 7 2 4 3 8 3 2 4 6 5 7 8 6 5 7 3 2 4 8 * 3 2 4 7 6 5 8 6 5 7 4 3 2 8 3 4 2 5 6 7 8 b 6 7 5 2 3 4 8 3 4 2 6 7 5 8 6 7 5 3 4 2 8 3 4 2 7 5 6 8 6 7 5 4 2 3 8 ** 4 3 2 5 7 6 8 7 6 5 2 4 3 8 4 3 2 6 5 7 8 7 6 5 3 2 4 8 4 3 2 7 6 5 8 7 6 5 4 3 2 8 * 4 2 3 5 6 7 8 b 7 5 6 2 3 4 8 4 2 3 6 7 5 8 7 5 6 3 4 2 8 ** 4 2 3 7 5 6 8 7 5 6 4 2 3 8 2 4 3 5 7 6 8 5 7 6 2 4 3 8 * 2 4 3 6 5 7 8 5 7 6 3 2 4 8 2 4 3 7 6 5 8 5 7 6 4 3 2 8

The 9-part system [6.31]-[6.12] is a subset of this one, and by use of the *-type apices marked ** a 9-part palindrome may be had (using P/B disposable calls); the complementary 9-part block requires one link of kind (2,2,1,1,1) which can only be achieved by asymmetric singles. Thus although [6.30] is a +ve group, and the link is also of +ve parity, singles are needed. In the peal below (shorter ones are more difficult to find) a bob is called at the partend * at the ends of parts 3, 6, 9 and one pair of linking singles is required, % to &, which swap two pairs of working bells and shunt to the complementary 9-pt. Block

                        9,504 Plain Bob Major

  2345678     4573826     7465238     6734852     6487325     8362754
  -------     -------     -------     -------     -------     -------
S 3257486     5342768     4573682   S 7645328     4762853     3285647
  2738564     3256487     5348726     6572483   -@4725638@  S 2354876
  7826345     2638574     3852467   S 5628734   - 4753286    *3427568
S 8764253   -@2687345   S 8326574     6853247     7348562     -------
  7485632     6724853    %3687245   S 8634572    &3876425     18-part
  4573826     7465238     6734852     6487325   S 8362754     (see
  -------     -------     -------     -------     -------       text)


Twenty-part Palindromes


[5.03] order 20m   [7.11] order 40m   (TT)

 2 3 4 5 6 7 8     2 3 4 5 6 8 7
 3 4 5 6 2 7 8     3 4 5 6 2 8 7
 4 5 6 2 3 7 8     4 5 6 2 3 8 7
 5 6 2 3 4 7 8     5 6 2 3 4 8 7
 6 2 3 4 5 7 8     6 2 3 4 5 8 7
 2 6 5 4 3 7 8     2 6 5 4 3 8 7 *
 3 2 6 5 4 7 8     3 2 6 5 4 8 7 *
 4 3 2 6 5 7 8     4 3 2 6 5 8 7 *
 5 4 3 2 6 7 8     5 4 3 2 6 8 7 *
 6 5 4 3 2 7 8     6 5 4 3 2 8 7 *
 2 4 6 3 5 7 8     2 4 6 3 5 8 7
 3 5 2 4 6 7 8     3 5 2 4 6 8 7
 4 6 3 5 2 7 8     4 6 3 5 2 8 7
 5 2 4 6 3 7 8     5 2 4 6 3 8 7
 6 3 5 2 4 7 8     6 3 5 2 4 8 7
 2 5 3 6 4 7 8     2 5 3 6 4 8 7
 3 6 4 2 5 7 8     3 6 4 2 5 8 7
 4 2 5 3 6 7 8     4 2 5 3 6 8 7
 5 3 6 4 2 7 8     5 3 6 4 2 8 7
 6 4 2 5 3 7 8     6 4 2 5 3 8 7

The four 5-part blocks in the peal below may be linked asymmetrically by S for P, from % to & and back.

 6,400 Plain Bob Major

  2345678      3652478
  -------      -------
 %3527486     &6237584
  5738264    - 6278345
- 5786342      2864753
- 5764823    - 2845637
  7452638      8523476
- 7423586    - 8537264
  4378265    - 8576342
- 4386752      5684723
  3645827      6452837
-@3652478     @4263578
  -------      -------
       20-part


[7.12] order 20p [7.11] order 40m 2 3 4 5 6 7 8 2 3 4 5 6 8 7 3 4 5 6 2 7 8 3 4 5 6 2 8 7 4 5 6 2 3 7 8 4 5 6 2 3 8 7 5 6 2 3 4 7 8 5 6 2 3 4 8 7 6 2 3 4 5 7 8 6 2 3 4 5 8 7 2 6 5 4 3 7 8 2 6 5 4 3 8 7 * 3 2 6 5 4 7 8 3 2 6 5 4 8 7 * 4 3 2 6 5 7 8 4 3 2 6 5 8 7 * 5 4 3 2 6 7 8 5 4 3 2 6 8 7 * 6 5 4 3 2 7 8 6 5 4 3 2 8 7 * 2 4 6 3 5 8 7 2 4 6 3 5 7 8 3 5 2 4 6 8 7 3 5 2 4 6 7 8 4 6 3 5 2 8 7 4 6 3 5 2 7 8 5 2 4 6 3 8 7 5 2 4 6 3 7 8 6 3 5 2 4 8 7 6 3 5 2 4 7 8 2 5 3 6 4 8 7 2 5 3 6 4 7 8 3 6 4 2 5 8 7 3 6 4 2 5 7 8 4 2 5 3 6 8 7 4 2 5 3 6 7 8 5 3 6 4 2 8 7 5 3 6 4 2 7 8 6 4 2 5 3 8 7 6 4 2 5 3 7 8

This peal is similar in structure to the above, as the two systems are very close. They only differ in the nature of the link calls. Here, the links effect a transfigure of type (4,2,1) which swaps 7x8 and is a +ve one, the group [7.12] being +ve; whereas the links above are -ve of type (4,1,1,1) as [5.03] is mixed. The first apex (a mid-lead one) has swaps (2x6 3x5 7x8 4) and the second one (2x3 4x5 7x8 6) giving (just as in the above peal) a 5-part Cyclic-Dihedral palindromic block. As usual, S for P at a position % will shunt to & in another of the four blocks. The tenors are reversed at each linkage.

                   8,000 Plain Bob Major

  2345678   - 7642538   S 3842567   S 5743682   - 2843675
  -------     -------     -------     -------     -------
  3527486   S 6723485     8236475   S 7538426     8327456
 %5738264     7368254   &@2687354@    5872364   - 8375264
- 5786342     3875642     6725843     8256743     3586742
  7654823   S 8354726     7564238   S 2864537     5634827
- 7642538   S 3842567   S 5743682   - 2843675    @6452378
  -------     -------     -------     -------     -------


Twenty-one part Palindromes

[7.05] order 21p   [7.04] order 42m

 2 3 4 5 6 7 8     8 7 6 5 4 3 2 *
 3 4 5 6 7 8 2     7 6 5 4 3 2 8 *
 4 5 6 7 8 2 3     6 5 4 3 2 8 7 *
 5 6 7 8 2 3 4     5 4 3 2 8 7 6 *
 6 7 8 2 3 4 5     4 3 2 8 7 6 5 *
 7 8 2 3 4 5 6     3 2 8 7 6 5 4 *
 8 2 3 4 5 6 7     2 8 7 6 5 4 3 *
 2 4 6 8 3 5 7     7 5 3 8 6 4 2
 4 6 8 3 5 7 2     5 3 8 6 4 2 7
 6 8 3 5 7 2 4     3 8 6 4 2 7 5
 8 3 5 7 2 4 6     8 6 4 2 7 5 3
 3 5 7 2 4 6 8     6 4 2 7 5 3 8
 5 7 2 4 6 8 3     4 2 7 5 3 8 6
 7 2 4 6 8 3 5     2 7 5 3 8 6 4
 2 6 3 7 4 8 5     5 8 4 7 3 6 2
 6 3 7 4 8 5 2     8 4 7 3 6 2 5
 3 7 4 8 5 2 6     4 7 3 6 2 5 8
 7 4 8 5 2 6 3     7 3 6 2 5 8 4
 4 8 5 2 6 3 7     3 6 2 5 8 4 7
 8 5 2 6 3 7 4     6 2 5 8 4 7 3
 5 2 6 3 7 4 8     2 5 8 4 7 3 6

To obtain the desired cyclic partends and links, the above peal was long. The first apex is at mid-lead, (2x3 5x7 4x8 6) and the second (6x7 2x4 5x8 3) at a bob, giving a Cyclic-Dihedral 7-part block which is one of three. Asymmetric block links are S for P at % shunting to & in another block, giving a transfigure of kind (3,3,1) within the partend group [7.05].

      7,728 Plain Bob Major

  2345678    4236785    7452836
  -------    -------    -------
-@2357486@   2648357   &4273568
- 2378564  - 2685473    2346785
 %3826745  - 2657834  S 3268457
S 8364257    6723548  S 2385674
  3485672  -@6734285    3527846
S 4357826    7468352  S 5374268
S 3472568  - 7485623    3456782
  4236785  - 7452836    -------
  -------    -------    21-part


Twenty-four part Palindrome: just one system of many.

[4.01] order 24m   [6.17] order 48m

 x x x x 6 7 8      x x x x 6 8 7

In each column, xxxx denotes all the 24 permutations of bells 2345. This system is given as the most tractable of a considerable number of 24-part systems. The peal below is a good example of this system; with less changes it is difficult to obtain a "nice start" with 2345 as working bells and 6 fixed, 7x8 swapping at the apices. The tenors remain together, but for "5ths and 4ths" calls. The pairs of 2345 which swap at the apices are (2x3 4x5) and (3x5) giving a 4-part Cyclic-Dihedral palindromic block. At partends 4, 8, 12, 16, 20, 24 the usual --S--S will permute 234, giving all the perms of 2345.

                     6,192 Plain Bob Major

     2 3 4 5 6 7 8       2 3 4 5 8 6 7       6 3 4 7 2 8 5     
     -------------       -------------       -------------
     3 5 2 7 4 8 6       3 5 2 6 4 7 8     - 6 3 7 8 4 5 2     
     5 7 3 8 2 6 4       5 6 3 7 2 8 4     - 6 3 8 5 7 2 4
     7 8 5 6 3 4 2     - 5 6 7 8 3 4 2       3 5 6 2 8 4 7     
   -@7 8 6 4 5 2 3     - 5 6 8 4 7 2 3       5 2 3 4 6 7 8     
     8 4 7 2 6 3 5       6 4 5 2 8 3 7       -------------
     4 2 8 3 7 5 6     S@4 6 2 3 5 7 8          24-part
     2 3 4 5 8 6 7       6 3 4 7 2 8 5     
     -------------       -------------

Next Section Categories of Palindromic Systems which Extend
Previous Section One-Part Palindromes
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