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.
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:
*-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.
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 ----------------------
[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.
[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 ------- ------- ------- ------- -------
[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.
[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
[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 ------- ------- ------- ------- ------- -------
[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
[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 ------- ------- ------- ------- -------
[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:
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 ------- ------- ------- ------- ------- -------
[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 ------- ------- ------- ------- -------
[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) ------- ------- ------- ------- -------
[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)
[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-partTwenty-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 ------------- -------------