(Mathematics of Campanology)
Brian D. Price
January 2006
Brian D. Price
If the calling of a particular Qset be altered, all the other Qsets remaining fixed in state, and the numbers of round blocks before and after the alteration be compared, thenIt is conditional that there is oneway traffic only along routes.
 if the Qset is of odd order, the parity of the number of blocks remains unaltered;
or
 if the Qset is of even order, the parity of the number of blocks is changed.
The Qset Parity Law, and not the phrase The Qset Law which I used over fifty years ago, is the title of this paper because some have thought the latter one to be concerned merely with the existence of the Qset structure. The aim of this paper is to review, hopefully to explain and occasionally to criticise, published material on the proof of it.
If a bob be called at the end of a plain course of Plain Bob Minor, the course 42356 is produced. If this is to be part of a touch, and no call be made beforehand, then a bob must be called at the end of the course 42356, producing the course 34256. A bob at the end of that course will bring the touch round. The effect of the three bobs is to link together three full courses. In composers’ jargon, three Pblocks have been linked by a P/B (PlainBob) Qset. This is the simplest example of a Qset, one of order 3. Each of the Pblocks has five different leadends where a call may be made, the option Plain
(i.e. no call by the conductor) at every leadend signified by the P
. Singles also form Qsets, generally of order 2 because they swap two bells. If a single be called at the end of a plain course, another single after the second course will bring rounds, and two Pblocks have been linked by a P/S Qset. If the 120 different Pblocks are written down and every leadend labelled with its Qset serial number (there will be 120 of them) the result is difficult to comprehend, and various graphical ways of simplifying the situation have been used. The use of a computer for composition since the 1960s copes with the complexity admirably, but requires mathematical knowledge for the programming of it. A ringer once said to me Why do you use a computer to compose peals, why don’t you do it yourself?
which showed a basic misunderstanding of the situation. The computer is merely a tool for handling complexities very rapidly, to the composer’s own instructions.
The Qset parity law is concerned with a special situation, in which a number of Pblocks have to be linked by just one kind of call, all the leadheads in the extent being true with one another. It does not apply to any minor method (plain or surprise &c) because owing to the nature of the rows (incourse or outofcourse), a lead such as 23456 in Plain Bob is false with the lead 32546, so only half the possible leadheads may be used.
This problem is discussed later on page 9. However a triples method using bobs only does fit in with the conditions, as the false leadheads are outofcourse and not reachable by bobs. It is no accident that W.H.Thompson first coined the term Qset
in his work on Grandsire Triples, the most popular of the triples methods. The 360 possible incourse leadheads of Grandsire form 72 Pblocks. They also form 120 Bblocks (3lead touches with a bob at every leadend) and the whole assembly is linked by 72 Qsets, each with five leadends giving, by plain or bob, the same five leadheads. Separate from his proof of the Qset Parity Law, Thompson exploited a graphical method of composing peals of Grandsire Triples in parts. [1]
With Grandsire Triples, as in the great majority of other methods, Plaining
every Qset produces an even number of Pblocks, likewise with Bobbing
, so that in either way one may start with an even number of round blocks.
The Grandsire Qsets are of order 5, an odd number, hence the Qset Parity Law establishes the impossibility of just one round block being produced. One cannot get a peal of Grandsire Triples with bobs alone.
On a much higher plane of difficulty there are a few cases, using a 21part or 7part Group for composition, in which one may start with an odd number of blocks and achieve a peal with bobs. For example, it is not difficult to produce a 21part peal of spliced Stedman and Erin Triples with bobs only (though the change of method may be thought of as a second type of call). The Manchester prototype computer attempted to solve a problem in changeringing in the early 1950s trying to produce a peal of Stedman Triples with bobs only, which was found to be impossible in 21 parts [2]. That problem proved much more difficult to solve.
I pointed out the duality of transpositions and transfigures in a 1969 article [3], which no doubt was triggered off by my starting programming in Fortran in early September 1968 with a course at Imperial College for their Schools Computing Project. Immediately I was writing changeringing programs, with their necessity for a clear distinction between the two concepts. This Schools Project started in 1964. Pupils submitted Fortran programs to IC Computer Centre by post on portapunch Hollerith cards, and received chainprinted output on continuous stationery by post. They were processed on a vastsized IBM7904 in the basement of Electrical Engineering (under the outside of the concourse). Before 1964 there was I believe no nationwide oppportunity for computer programming.
But mathematical literature is at variance with itself over the distinction. Burnside [4] starts off his treatise with a notation for permutations, which is of transfigures (although his use of letters makes this term rather nonsensical). Thus the permutation
a b c d e to c b d a e is given as (acd)(b)(e)meaning
a is replaced by c, c is replaced by d …etc., whereas Fletcher [5] on p.620 states
2 1 3 6 5 4 to 5 1 4 2 6 3 is given as (145)(36)
denoting a transposition, the bell in the 1^{st} position goes to the 4^{th} position, the 4^{th} to the 5^{th} …
etc.
In either case, constituent cycles are bracketed. Using Burnside’s notation this second example would be (1)(256)(34), or just (256)(34), rendered more awkward by the fact that rounds is not the first of the two given rows – a situation more likely to be met by the change ringer. Fletcher’s use of the word transition
begs the question of the distinction.
In any case, these bracketed cycles do not make for easy computer programming.
Using Fletcher’s two stated rows:
either
Transfigure: 1 5 4 3 6 2 (figure 1 remains 1, figure 2 is replaced by 5, figure 3 by 4, … )
or
Transposition: 4 2 6 5 1 3 (1^{st} figure becomes the 4^{th}, 2^{nd} figure remains 2^{nd}, 3^{rd} figure becomes 6^{th} …)
Using Pascal notation, if a row (a onedimensional array) X is said to be altered to row Y, using the transfigure/transposition Z, then
{Z a transfigure} for i:= 1 to n do Y[n]:=Z[X[n]];
{Z a transposition} for i:= 1 to n do Y[n]:=X[Z[n]];
illustrating clearly the dual relationship [6] between the two different treatments.
In the remaining part of this section, in order to avoid the ambiguity of row, I use perm for permutation. The above dual relationship surfaces in the distinction between the leftcosets
and rightcosets
of mathematical literature [7]. For instance White [8] p.727 gives Bob Minimus:
1 2 3 4 2 1 4 3 2 4 1 3 4 2 3 1 4 3 2 1 3 4 1 2 3 1 4 2 1 3 2 4 
1 3 4 2 3 1 2 4 3 2 1 4 2 3 4 1 2 4 3 1 4 2 1 3 4 1 2 3 1 4 3 2 
1 4 2 3 4 1 3 2 4 3 1 2 3 4 2 1 3 2 4 1 2 3 1 4 2 1 3 4 1 2 4 3 
which is a microcosm of the principle of part composition. The three columns are leftcosets of the Dihedral Group D4 (including D4 itself in column 1) whereas the eight rows are the rightcosets of the first row, which is the cyclic Alternating Group A3. In simpler parlance, the perms in the top row form a group (though this has been shown to have rare exceptions) which defines transfigures, and the remaining rows of perms exhibit the same transfigures with each other (in this case the cyclic rotation of bells 2–3–4); whereas the first column defines a series of transpositions (dictated by the method, principle or calls) which same transpositions are found correspondingly in the other columns. The whole structure is an extension of the duality of transfigures and transpositions.
In the above example, both the first row (Group A3) and the first column (Group D4) are Groups, and this is true in cases where the plain hunt (a Dihedral Group of perms) is involved, but generally in composition it is the top row of perms (the partheads) which (pragmatically always) forms a Group, whereas the first column is selected by some method so its transpositions contribute to a true peal. There are more difficult cases, including Stedman, in which the first column is a Group (e.g. the Stedman six) which is capable of various forms with the same content of perms.
But as to which are left and which rightcosets depends on (a) how a permutation is defined and (b) the convention of multiplication of operations, whereas the distinction between the more pedestrian terms transfigure and transposition is more easily understood.
The idea of a Group in change ringing is not all that simple, as it is the relations between the perms (either as transpositions or as transfigures) which form the Group, and not the perms themselves. Rounds
itself is an arbitrary startandfinish for ringers, but 123456 on six bells, regarded either as a transposition or as a transfigure, is technically the identity opertion
, i.e. that which leaves a perm unaltered, and a Group necessarily contains an identity operation. Technically the first column above regarded as a set of transpositions forms the Group D4, and the other two columns are cosets of D4 in the extent, but in a looser sense the other columns are forms of the same Group, in that their own perms are interrelated by operations forming the same Group, the identity operation being the relation of any perm with itself. As Fletcher says in [5] p.620, The [perms] … by numbering the bells … which result could be regarded as the elements of a … group, but it is more logical to regard the transitions between [perms] as constituting the group …
and he proceeds to explain a transition
as a transposition. Here I edit Fletcher’s wording to perms
in order to avoid the confusion of the usuallyaccepted meaning of change
as the restricted transposition of one perm to the next – Fletcher is implying the transitions between any two of the perms.
I state in the preceding paragraph (either as transpositions or as transfigures)
because the fundamental relation between the two, together with the closure property of a Group, ensure that either implies the validity of the other.
By a process of exhaustion, Thompson considered all the topological ways in which the five exits of a particular Qset of order 5 may be connected (by means of a circuit through other Qsets) to its five entrances. In each case, the nochange in parity of the number of circuits involved in that Qset was checked when the calling of the particular Qset was altered. There are indeed 28 essentially different situations of a Qset of order 5, of which 5 are rotationally symmetrical. It is not particularly easy to enumerate them. This enumeration would be increaingly complex for higherorder Qsets, and on Thompson’s proof method each order would have to be checked separately. Having constructed the 28 logically, I predict that the number of topologically different Qset situations of order seven is 7 + (6!  1) = 726.
It is worthwhile examining the 28 cases of the disposition of a single Qset, which Thompson enumerated as part of his proof [9]. They were later enumerated, in an explanation of Thompson’s proof, by Rev. C.D.P.Davies in [11]. Refer to the page 13 of diagrams. Thompson was concerned with the Qsets of Grandsire Triples of order 5. In the top lefthand corner is a conventional diagram of such a Qset, which has five entrances (distinguished by heavy dots on the circle) and five exits. Logically, the 360 incourse leadheads identify with the routes connecting the Qsets, the heavy dots at the Qset entries focusing attention on them. Any Qset has either all its entrances Plained, or all Bobbed. Oneway traffic is assumed. Next to that is a diagram of the kinds of outside circuits
from the exits to the entrances, to be catalogued by the Pascal program below. These outside circuits will be involved in other Qsets also, but we are concerned here only with the dispositions of the round blocks through the sample Qset under consideration (and finally, how many there are for parity checking). There are likely to be other round blocks which do not traverse the sample Qset at all.
To interpret the Pascal output, in all the other 28 diagrams on page 13 a start is made at the dot of the uppermost entrance. Proceeding clockwise, the near exit is considered first, and has five different ways of linking externally to an entrance. These five are labelled 1 to 5 in clockwise order of entrances. To avoid confusion in the diagrams, No.3 (which links two diametricallyopposed points) is given as two short straight lines, to be notionally linked. Nos.4 and 5 appear to be anticlockwise, as again if they were drawn clockwise the lines would be longer and the diagrams more confusing. Similarly for clarity, the five internal Bob links of each Qset are omitted, so that the number of round blocks may more easily be counted when the Qsets are Plained
.
Program Qset (input,output);
Type row = array[1..5] of integer;
Var x,y: row; i,j: integer; xfile: text;
Procedure IndArr (c: integer; var b: row);
var d: row; e: array[1..6] of integer; m,n: integer;
begin
d[1]:= (c + 23) div 24; c:= c  24*d[1] + 24;
d[2]:= (c + 5) div 6; c:= c  6*d[2] + 6;
d[3]:= (c + 1) div 2; d[4]:= c  2*d[3] + 2;
for m:= 1 to 6 do e[m]:= m;
for m:= 1 to 4 do
begin
b[m]:= e[d[m]]; for n:= d[m] to 5 do e[n]:=e[n+1]
end;
b[5]:= 15  b[1]  b[2]  b[3]  b[4]
end;
{procedure converts indices, in quasinumerical order, to arrays}
Begin
assign(xfile,'a:out.dat'); rewrite(xfile);
for i:= 1 to 120 do {i to be index of all 120 permutations}
begin
IndArr(i,x); {array x runs through all 120 permutations}
{x will denote the five fixed reentry points}
for j:= 1 to 5 do y[j]:= (x[j] + 5  j) mod 5 + 1; {CALCULATION}
{y becomes the routespans, from five exits in cyclic order}
for j:= 1 to 5 do write(xfile,y[j]:2); writeln(xfile)
end;
close(xfile)
End.
The output from this program was a series of 120 sets of external links:
1 1 1 1 1, 1 1 1 2 5, 1 1 2 5 1, 1 1 2 2 4, 1 1 3 5 5 …
In further explanation, consider the fifth solution, 11355. This catalogues the external linkage of the Qset diagram in the top righthand cell of
page 13. Starting at the top dot and moving clockwise, the first two exits are 1, 1 that is, they are linked to the next clockwise entrance; but the third exit (at the lowest point) must have an external link of type 3, diametrically across to the starting dot. The last two exits are 5, 5 both of which are external links to the fifth dot clockwise, i.e. the preceding dot. The net result is three round blocks passing through the Qset when it is in state Plain
, and this number is recorded by the large 3
in the centre of the Qset. The 28 external dispositions are all of a Qset in state Plain
.
Five of these 120 solutions had rotational symmetry. The remaing 115 duplicated cyclically (for instance, 11224 is identical with 12241, 22411 &c) and I reduced these 115 to 23 different kinds, choosing the lowest of each set of five (regarding them as 5digit quasiintegers), and sorting them by hand. The resulting 28 were:
11111, 22222, 33333, 44444, 55555. 11125, 11224, 11314, 11355, 12223, 12313, 12345, 12525, 13344, 13353, 13524, 14145, 14244, 14253, 14555, 22335, 22425, 23334, 23424, 24455, 25355, 34445, 34535.
These 28 occupy the twentyeight cells of page 13 of diagrams. A further relation was found between them, on considering the change occuring when the Qset is Bobbed. When the internal linkage of each Qset assemby is altered from Plain to Bob, the resulting diagram is either the same assembly, or topologically another of the 28. To check this, one has to resort to pencilandpaper. The eight diagrams along the top and righthand side of page 13 result in themselves, and the other twenty pair off (as on the page across fainter lines) in that Bobbing
one gives the other.
A curious fact is that nearly all of the 28 diagram have an axis of bilateral symmetry. The exceptions are the two labelled asymmetric, and they are mirror images of one another, each qualifying as giving itself topologically (by reflection) when bobbed, but the two forming a pair, because altering the calling of either gives the other directly.
The topological comparisons usually reverse the direction of tracing round blocks, and of course swap Plain
for Bob
, but the important parity of the number of round blocks is unaffected. Five of the diagrams have also 5part rotational symmetry, as one might expect.
Under each of the 28 diagrams is given also the Davies labelling of the case. For instance,
11314 AC/BOOO
signifies, to the right of the Pascal output 11314, that in the Plained state the two resulting round blocks are A and C, of which A in the keydiagram of Rev. C.D.P.Davies’s paper [11] — reproduced on
page 15 — denotes using an adjacent pair of Qset entrances, and C a triangle of three consecutive entrances. In the Bobbed state the round blocks are B, a nonadjacent pair of entrances, plus OOO, three loops each containing but one Qset entrance, making four round blocks in all (and conserving parity). This tallies with the case’s opposite number
of 25355 (C and D differing only in the rotational sense of visiting the triangle corners). G and H signify trapezia, KLMN crossed trapezia, etc. The notation is involed with the order of occurence in the round block, linked to directional arrows on the geometrical figures of the keydiagram. All related pairs preserve the parity of the number of round blocks through a Qset.
Davies’ paper gives geometrical diagrams (reproduced on page 15) to help explain the different 28 cases, whereas Thompson’s paper does not. The orders in which the cases are tabulated differ a little. Thompson’s paper is more austere and illustrates his legal mind, whereas Davies’s style make one wonder about the length of his sermons!
Rankin proves a more general theorem, of which a special case is the Qset parity law of interest to change ringers.
The article is wellclothed in the jargon of advanced mathematics, in fact it is doublewrapped
in that the indexes of letters, representing members of a more generalized Qset under consideration, themselves have suffixes. Dickinson [14] has extracted for us the crux of Rankin’s argument (he says so in his article) as it affects ringers, in a more understandable form less formidably wrapped. His article was a reply to Fletcher’s [5] in the same Journal, as apparently Fletcher was unaware of any proof of the Law other than Thompson’s one of 1886. Dickinson stated the proof as for Grandsire, but Rankin’s proof is generalized, for all Qset orders and with further deductions not of ringing interest. The editor of the Proceedings [12] received Rankin’s paper for publication on 20 November 1946.
Rev. C.F.D. Moule was Dean of Clare College, Cambridge 19441951 and a Fellow from 1944, whereas Robert Alexander Rankin was a Research Fellow of Clare College 19391947. The two must have known one another. Rev. Moule was a member of the Cambridge University Guild of Change Ringers (we met regularly in his rooms for handbell ringing, though he was so busy he hardly ever rang with us) and possibly Moule had interested Rankin in the changeringing parity problem.
I proved the law by mathematical induction. Assuming the law is true for Qsets of order n, a Qset of order n+1 is then considered as a combination of a Qset of order n, together with one of order 2. The behaviour of a Qset of order 2 has to be found by inspection, there being only one possible configuration of circuits (refer to Figure 1 on page 14). Parity relations are deduced, and the law proved by induction. Though given only in print, the proof is a diagrammatic one well with the capability of an Alevel student to understand:
The Ringing World leader (half of the front page) of 10 June 1949, No.1992.
The term Qset
was given to us by W.H. Thompson in his pamphlet (1886) on Grandsire Triples. In this pamphlet
(reproduced in Snowdon’s Grandsire
, 1905, p.197) he proves the Law for Qsets of order 5. Dr. Rankin, in his paper on Group Theory (Proc. Camb. Phil. Soc., Vol.44, Pt.1, p.17), proves the Law, but the method of his proof is beyond the reach of most of us.
In order to clarify ideas, we may define a Qset as follows. A, A′, B, B′, C, C′ etc. as far as N, N′ are points in order on a circle. The number of points may be considered as 2n, there being n points A, B, C, etc., separated by the n points A′, B′, C′, etc. Any letters may be supplied between C′ and N, and there are not necessarily 28 points in all.
Routes lead to points A, B, C, etc., and away from points A′, B′, C′, etc. The problem of composing a peal is reduced to that of finding one continuous circuit around a number of interlaced Qsets, where routes lead from dashed letters A′, B′, etc., on each Qset to plain letters A, B, etc., on the same or on different Qsets.
When a Qset is in the state plain
each letter is linked to its partner — A to A′, B to B′, etc. When in state bob
each is linked to its other partner — N′ to A, A′ to B, etc. The order
of a Qset is the number of such links (equal to n).
Each Qset in the extent may be plain
or bob
at will. If the state of each is fixed, then the aggregate of routes will form a number of circuits
(round blocks). The Qset Law concerns the oddness or evenness (parity) of this number of circuits, as follows:—
If a single Qset of odd (even) order is altered in state, the others being fixed in state, the parity of the total number of circuits is not (is) changed.
There are two complementary parts to the Law — either miss out, or substitute, the words in parentheses. Two important conditions are, firstly, every point must be used (this cuts out arguments about Stedman Triples where all sixends are not used), and secondly, every route must have oneway traffic (this disqualifies Treble Bob Minor, where a lead may be used backwards).
The Law may be proved by induction. We assume the Law to be true for all Qsets up to order n, and then show it to be true for the next order (n+1). Since the Law is obviously true for n=1 or n=2, it must therefore be true for all values of n.
Suppose, therefore, that the Law is true for the Qset order n:—
A, A′, B, B′, C, C′, …… N, N′.
Consider the Qset of order n+1:—
A–A′, B–B′, C–C′, …… N–N′, X–X′.
It is in the state plain
. We proceed to bob
it by several steps.
First, connect up the letters as follows:—
A′–B, B′–C, …… M′–N, N′–A, X–X′.
The pair X–X′ has been left alone. The change otherwise is the same as bobbing a Qset or order n. Now consider the four letters in order: A, N′, X, X′. For the purpose of proof they may be considered as forming a Qset of order 2. At present they are in the state N′–A, X′–X. Alter the state of this Qset to X–N′, A–X′. The whole situation is now:—
A′–B, B′–C, …… M′–N, N′–X, X′–A,
which is the Qset of order n+1 in state bob
. The bobbing of this Qset has been carried out in two steps, equivalent to bobbing Qsets of order n and 2. Since bobbing a Qset of order 2 changes the parity of the total number of circuits, it will be seen that the Law then follows for Qsets of order n+1. For example, if n is even, then bobbing the Qset of order n changes the parity of the number of circuits, but so does altering the Qset of order 2. Therefore the effect of bobbing the Qset of order n+1 is not to change the parity. A similar argument holds if n is odd. By application of this argument, starting from n=2 (a case which must be proved by inspection), the Law is proved for any value of n.
Roger Bailey has pointed out to me that the term partner
is confusing, and I agree that it would have been better expressed as neighbour
, each letter in the circle having two neighbours. That was the only time since my proof was published over fifty years ago in which anyone has commented on it, verbally or in writing.
In a previous letter to the Ringing World editor [15] I pointed out that oneway traffic along routes was essential to the proof of the Qset parity law, as I had been aware of the cases of twoway traffic through Qsets in which the law did not apply. Such a practical changeringing matter would not have suggested itself to Rankin, who was not a changeringer.
Dickinson offers the proof as a special case of Rankin’s more general conclusions, using his logic. He considers a particular Qset for altering, the others remaining undisturbed. The extent will be in a number of circuits, some of which involve the particular Qset, which has its entries labelled 1 to 5 in some cyclic order. (So far the proof is identical to Thompson’s.) The involved circuits through a Qset create a permutation by the way in which they link the exits to the entries, each circuit creating a constituent cycle in the transfigure. The Qset is then altered in calling, the result being expressed as the product of that permutation with the cyclic permutation (12345), which in turn is broken down into four single
swaps. Hence the nochange in parity is deduced. As expressed, it is for (Grandsire) Qsets of order 5 only; but unlike Thompson’s proof, the technique as given applies for any Qset order.
As an illustration of Dickinson’s (simplifying Rankin’s) logic, consider the pair of Qsets given in Figure 3 on page 14. They represent a Qset of order 5 at random from page 13, in the states Plained
and Bobbed
.
The five routes external to the Qset remain unchanged – Rankin’s segments of chains
. The internal change from Plained
to Bobbed
produces a relinking of the segments, and we are interested in any parity change of the number of routes.
Here, each has 2 routes.
For each of the two states of the Qset, a permutation is constructed which gives the succession of the five segments among the several routes [round blocks] with which it is involved. (These routes do not necessarily comprise all the routes covering the extent of Triples.) The upper diagram, the Qset in Plained state, of kind 14244 [notation of Pascal output] gives the substitution
as from [ 1 2 3 4 5 ] to [ 2 1 5 3 4 ]
or to put it in Burnside’s notation (12)(354), in terms of the two cycles of digits which it contains. Clearly this is the transfigure form of a permutation – figure 1 proceeds to 2, 2 to 1, 3 to 5 …
.
The digits 1 to 5 refer to Rankin’s five segments
(which form two chains
) outside the Qset.
In a similar way, the Qset with its external segments unaltered, but now Bobbed
, gives kind 22425 (that is, reckoned topologically, with a reflection of route direction, and a rotation) with substitution
as from [ 1 2 3 4 5 ] to [ 4 2 1 5 3 ]
The crux of the proof is in the nature of the substitution which
transforms [ 2 1 5 3 4 ] into [ 4 2 1 5 3 ]
and is a way of expressing the cyclical result of the internal Bob routes replacing the internal Plain routes of the Qset.
Dickinson quotes this as the permutation (12345) – Rankin as the more general (123…n) – and both actually use the term transposition
to describe it. If the transformation lastquoted above is analysed, as a transfigure it is 24351 and as a transposition it is 23451. Its duty as a cyclic interchange is indeed as a transposition.
The effect of the 5fold cyclic transposition on the parity of the number of routes is explained by dissociating the transposition (12345) into four separate transposition steps (12), (13), (14), (15). With regard to the above example,
2 1 5 3 4 Swap the numbers in positions 1 and 2 1 2 5 3 4 Swap the numbers in positions 1 and 3 5 2 1 3 4 Swap the numbers in positions 1 and 4 3 2 1 5 4 Swap the numbers in positions 1 and 5 4 2 1 5 3
The graphical effects of these separate transposition are illustrated in Figure 4 on page 14, where the dotted internal lines are the routes to be substituted in the ensuing alteration. Each simple swap is in fact the alteration of a Qset of order 2.
As Dickinson explains, the transposition [swap] increases the number of cycles [routes] by one if p and q are in the same cycle, and diminishes the number of cycles by one if p and q are in different cycles
.
This is precisely the action of the Qset of order 2 (Figure 1 on page 14) and is easier to comprehend. As the number of swaps is four, an even number which is one less than the order of the Qset, the Parity Law follows.
It therefore transpires that as far as the Qset Parity Law of ringers is concerned, Rankin’s proof reduces to the relatively simple matter of dissociating the 5part cyclical transposition of the Grandsire Qset into four steps of altering Qsets of order 2. Indexes with their suffixes are not required, neither are permutations of chain segments. The reduced proof applies to Qsets of any order, including even orders in which parity is changed, and depends ultimately on the behaviour of the Qset of order 2, as does also Price’s proof which however uses mathematical induction.
The book on Stedman [18] explains at length the twinbob way of composing peals of Stedman Triples discovered by William Hudson in 1832. By far the majority of peals of Stedman Triples rung since have been on this plan.
The twinbob calls at S and at L are linked by Qset assemblies, as are the calls at H and Q.
The basic assembly is illustrated in Figure 5 on page 14.
This assembly is formed of two simple Qsets of order 2, linked for truth in the situation either – or
so that the parity of the number of round blocks is not disturbed, hence always even.
A peal must resort to singles for linkage.
This fact must have been quite evident to the early composers, but [18] p.265 makes only woolly references connecting it for truth with Thompson’s Qset paper on Grandsire Triples of 1886.
The sixty incourse leadheads of Plain Bob Minor may be placed on the facecorners of a dodecahedron (or its dual the icosahedron) symmetrically, so that the three leadheads of a Bblock form a clockwise rotation about a vertex, and the five of a Pblock are on a face, rotating anticlockwise as a pentagram. Each vertex has then a P/B Qset on it. (This is one of four different dispositions of all minor methods.) However, these sixty leads are false in pairs, each pair being diametrically opposite (antipodal) on the solid, their rows being mutually in reverse order. Hence in composing a true 360 with bobs only, one has to avoid falseness by selecting only 30 of the leads, one from each antipodal pair, and the Qsets are not fully used. (In the parallel case of Treble Bob Minor – including Surprise Minor – the full 720 results.)
However, the diagram on the dodecahedron may be (notionally) folded [19] to bring the two leadheads of each pair into coincidence. The simple surface of the dodecahedron becomes distorted into a multiplyconnected surface (has Henry Moore holes
in it), but of course it is not necessary to make a model of it – a flat diagram will suffice, with broken routes around its perimeter to be linked by letters to the ones diametrically opposite. All the Bblock, Pblocks, Qsets come together in mutual pairs, so that there are then just 10 Bblocks, 6 Pblocks, 10 Qsets in the diagram.
But the routes are no longer unidirectional because pairs of directed routes amalgamated were orientated in opposite directions.
In finding a round block by experimenting with the calling of the Qsets, one does not pay any regard as to which way round a lead is being used, until finally just one block is pricked out. And crucially, although there are an even number of Pblocks when all the Qsets are Plained (similarly for being Bobbed), it is possible to find a single round block.
The Qset Parity Law does not apply, because the pairedoff routes are not unidirectional.
White, in [8] pp.730–732, explains this folding starting with Plain Bob Doubles.
This folding removes the difficulty of avoiding false leadheads, as mutuallyfalse pairs become just one which may be used in either direction, and composition is now a simple matter of how to dispose the Qsets, each Plained or Bobbed.
The same argument applies to the composition of Original TriplesMinor with bobs only. With the incourse leads of Plain Bob Minor:
2 3 4 5 6  3 4 2 5 6  4 2 3 5 6  This is the usual 3part 360 of Plain Bob Minor, to be doubled to a 720 by two singles. The five underlined leads are all part of one plain course, but the isolated lead is reversed in its direction relative to the other four (this is true twice more cyclically). 
      
 2 3 5 6 4   3 4 5 6 2   4 2 5 6 3  
3 6 2 4 5  4 6 3 2 5  2 6 4 3 5  
6 4 3 5 2  6 2 4 5 3  6 3 2 5 4  
4 5 6 2 3  2 5 6 3 4  3 5 6 4 2  
 4 5 2 3 6   2 5 3 4 6   3 5 4 2 6  
 4 5 3 6 2   2 5 4 6 3   3 5 2 6 4  
5 6 4 2 3  5 6 2 3 4  5 6 3 4 2  
6 2 5 3 4  6 3 5 4 2  6 4 5 2 3  
2 3 6 4 5  3 4 6 2 5  4 2 6 3 5  
3 4 2 5 6  4 2 3 5 6  2 3 4 5 6  
     
[1]  Snowden, William (et al.): Grandsire:the method, its peals and history, 2^{nd} edn. London 1905. This edition contained a further paper on Grandsire Triples by Thompson, giving a number of partdiagrams with their Qsets for peal composition. The copy in the British Library was destroyed in World War 2. The 1^{st} edition 1888 was by the late Jasper W. Snowdon, edited by his brother William (a copy in the British Library). 
[2]  Mentioned in Andrew Hodges, Alan Turing, the Enigma (Vintage, 1992 edn.) p.445. Refer to a letter published in The Ringing World on 3 April 1953 p.219, from B.D.Price. 
[3]  Price, B.D.: Mathematical Groups in Campanology, Math. Gaz., Vol.53 (1969) pp.129133. 
[4]  Burnside, W.: Theory of Groups of Finite Order, (CUP 2^{nd} ed. 1911; reprinted Dover 1955) 
[5]  Fletcher, T.J.: Campanological Groups, American Mathematical Monthly, Vol.63 (1956) p.619. In the references, there is poor editing of Thompson, W. H. A Note on …(failure to discriminate by font between name and title) which has led White in [8] to style Thompson’s name erroneously as W.H.A. Thompson. 
[6]  A dual relationship in mathematics denotes a similarity between two entities or ideas, such that a comparison of the differences of one with the other reveals a correspondence between them. Another duality, concerned with graphical representation of permutations discussed below, is the duality of the icosahedron (a geometrical solid with twenty triangular faces) and the dodecahedron (twelve pentagonal faces). If the centres of the faces of either solid be joined to the centres of the adjoining faces, the result is the other. They have the same set of symmetry axes, and the 60 +ve changes on 5 bells (mathematically, the group A5) may be placed on either. W.H.Thompson must have used the dodecahedron to arrange the 360 +ve changes on six bells (leadheads of Grandsire Triples) symmetrically as his papers mention it, but exactly how he did it is apparently lost as there seem to be no known surviving models of his work. There is a sixfigure permutation Group isomorphic with the dodecehedron/icosahedron which Thompson might have used (the Group used by Hudson’s courses of Stedman Triples). 
[7]  Though I myself studied mathematics for a degree 19421945, there was no mention of Group theory (or, for that matter, statistics!) in my course, and my initiation was by studying smatterings of Burnside [4] as part of my ringing interests. Consequently I myself have never thought in terms of left or rightcosets. It is curious to realise in hindsight that while I was an undergraduate, the boffins at Bletchley Park were decoding German intelligence using much the same mathematics as in ringing composition; and when I was making a changeringing machine just after the war, the secondhand telephone relays I was using were very probably from dismantled Enigmadecoding equipment. 
[8]  White, Arthur T. (Western Michigan University): Ringing the Cosets; American Mathematical Monthly, Vol.94 No.8, October 1987 pp.721746. 
[9]  Thompson, W.H.:
A Note on Grandsire Triples, London 1886. Thompson’s proof of the Qset parity law for Grandsire Triples. There is a copy in the Central Council Library, but the British Library copy was destroyed in World War 2.
Thompson was born 26 September 1840. He gained admission to St. John’s College, Cambridge in 1858 but gained a scholarship to Gonville and Caius College in 1859. He studied mathematics and was 23^{rd} wrangler (first class honours) in 1863. He worked as an assistant magistrate for the East India Company at Dacca for five years, then studied at Inner Temple, London for the Bar and became a barrister in 1872. From 1872 he working in the Punjab, retiring in 1897/98. A batchelor, he retired to Brighton where he lived in lodgings, and died 10 July 1934 [10]. Obituary in The Times on 12 July. 
[10]  Alumni Cantabrigienses, British Library HLR 378.42659 (open shelves in Humanities 1). 
[11]  Davies, Rev. C.D.P.: Odds and Ends of Grandsire Triples, 1929. This booklet of pp.22 + folding enddiagram, restated [9] with an explanation of the 28 different Qset cases checked (pp.115 with enddiagram). Davies also made a revue of Thompson’s proof [9] in the two editions of [1]. 
[12]  Rankin, R.A.: A Campanological Problem in Group Theory, Proc. Camb. Phil. Soc. Vol.44, 1948, pp.1725 (received by the editor 20 November 1946). Robert Alexander Rankin was a foundation scholar of Clare College, Cambridge in 1934. In 1936 he was a wrangler (Mathematical Tripos Part II, 1^{st} Class) and in 1937 gained special credit in Part III. He was a Research Fellow of Clare 1939–47 and Official Fellow 1947–51 [13]. 
[13]  Harrison, W.J.: Notes on the Masters, Fellows, Scholars and Exhibitioners of Clare College, Cambridge, British Library ref. X.529/19550. 
[14]  Dickinson, D.J.: On Fletcher’s Paper Campanological Groups, American Mathematical Monthly, May 1957, pp.331332. 
[15]  The Ringing World leader
(half of the front page) of 30 April 1948, No.1934.
MORE ABOUT THE GROUP THEORY
… … …
His letter continues:
The final quote from B.D.P. asserts the word necessary, which Roger Bailey has, years later, demonstrated as untrue. In [16] I gave examples of peals of Bob Triples with partheads not forming a Group. A further case is provided in a tenpart peal of Erin Triples by Andrew Johnson, Erin peal No.8 in [17], which though based evidently on the Group [7.14] of ten partheads, conceals with a singledin block a twentypart structure whose partheads do not form a Group, but rather twenty rows of the fortypart Group [7.11]. The Group references such as [7.14] refer to my Catalogue of Permutation Groups on up to 7 Bells (1996) available as 
[16]  Price, B.D.: The Structure of Palindromic Peals (1998) available on www.ringing.info. 
[17]  Saddleton, P.A.B.: A Collection of Compositions of Stedman Triples and Erin Triples (Central Council publication 1999). 
[18]  Davies, Rev. Charles D.P. et al., Stedman (1^{st}. edition, Leeds 1903), Jasper Snowden Series. 
[19]  The term folddoes not necessarily mean a process like folding a bedsheet, or creasing and folding a piece of paper with an ink blot to produce an inkshape with an axis of bilateral symmetry. Here, the folding is the application of half the dodecahedrondiagram on to the other half, so that mutually false leads coincide; it can be thought of notionally in this case as a flattening of the 3dimensional diagram into two equal pancakes, slitting around the circumference, rotating one relative to the other through 180° then fusing together. This would bring together false pairs of leads which were at mutual antipodes of the original threedimensional diagram. As the underpancake became facedown, the routes on it were in the opposite direction to the fused routes of the upper pancake, making a nondirectional diagram. 
[20]  Price, B.D.: The Composition of Peals in Parts (1996) available on www.ringing.info. 
I apologise for dissociating text and diagrams. This is to keep down the cost of photocopying.
Page 13  Diagrams illustrating Thompson’s 28 kinds of Qset disposition. See page 4.  
Page 14  Miscellaneous diagrams  
Figure 1  The simple behaviour of a Qset of order 2, which is the basis of general proofs.
It illustrates a unidirectional Qset, and parity is altered, 1 ↔ 2.
It is logically the same as the intermediate switchof threeway (or more) light control.  
Figure 2  A Qset of order 2, not unidirectional. Parity here is not altered.  
Figures 3, 4  To illustrate Dickinson’s proof, on pages 7 and 8.  
Figure 5  The Qset assembly of Hudson’s twinbob courses of Stedman Triples. It contains two interlocked Qsets of order 2, unidirectional, hence parity is unaltered.  
Figure 6  A Qset of order 3, not unidirectional. Parity is altered.  
Figure 7  A third option enables a Qset of order 3 to be used atypically.  
Page 15  Reproduction (enlarged) of Thompson’s 28 cases of Grandsire Qsets, taken from the Rev. C.D.P.Davies’s folded endpaper of [11]. 
Thompson’s 28 kinds of Qset disposition 



Unfolding endpaper of Rev. C.D.P.Davies’s pamphlet Odds and Ends of Grandsire Triples [11] (enlarged by a factor of ×1.4)
HTML version by Andrew Johnson, October 2017
Arrived by search engine? Click here, for a comprehensive list of other changeringing links.