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          Stedman Triples

                60-course plans

A plain course of Stedman Triples is 84 changes long. Therefore, 60 plain courses (P-Blocks) are required to form an extent of 5040 changes - the same number, incidentally, as Plain Bob Triples.

But unlike Plain Bob Triples, in Stedman there isn't a set of 60 P-Blocks that contain between them all of the 5040 changes. Three ingenious composers called John Carter, Joseph W Parker, and Joseph J Parker all independently proved that 40 P-Blocks is the greatest number of mutually true P-Blocks possible, and their results were all published in the same issue of the Bell News in answer to an enquiry. John Carter and Joseph W Parker worked out the same set of 40 P-Blocks - Joseph J Parker's was a different set.

 

So the challenge for composers was to use bobs and singles to invent a set of 60 courses that do contain all 5040 changes. The courses use bobs (and sometimes singles) internally, as much a fixed aspect of the course as the plain sixes are. This must be so, as we have seen that 60 P-Blocks joined together externally by bobs is not possible.

 

                                                   Hudson's Group

In 1832 the Sheffield composer William Hudson used a set of 60 courses to produce a new peal of Stedman Triples. Whether he was the first person to discover this mathematical group of 60, or simply the first person to apply it to Stedman Triples, I don't know. Here are the 60 courses (bearing in mind that 231456 is usually the Rounds course end in Stedman, rather than 123456, due to the traditional start not being at the six end):

 

231456      342516     453126     514236     125346

324156      435216     541326     152436     213546

321465      243615     462135     614325     136245

234165      426315     641235     163425     312645

361524      653214     526134     215364     132654

635124      562314     251634     123564     316254

416523      154263     521643     265413     642153

145623      512463     256143     624513     461253

415632      164352     631542     356412     543162

146532      613452     365142     534612     451362

625431      246351     432561     354621     563241

264531      423651     345261     536421     652341

 

These 60 course ends form a mathematical group in the sense that any one of them transposed by any of the others will produce another member of the same group. This special group relationship must exist between all of the course ends, otherwise the set of courses would be useless for the task of finding a true peal.

 

How might Hudson have worked out this group of 60 courses? He must have known that the basic falseness in Stedman depends on the three bells working together on the front in each six. Because of this, any of the course ends above cannot relate to each other via a three-bell transposition or transfigure. For example, we have the course end 614325 which means that we cannot have course ends 612435 or 613245. These other two relate to 614325 in that bells 2,3,4 have been rotated. But this would dictate that repetition occurs somewhere in the peal in some of the sixes where bells 2,3,4 meet each other on the front. To prevent this happening, none of the course ends can relate to any of the others via a three-bell rotation. This is exactly what happens in Hudson's group, and was possibly the reason he discovered and used it.

 

However, as a set of mutually true 60 P-Blocks is not possible in Stedman Triples, Hudson's Group still contains repeated changes. But the falseness is pretty minimal. The crucial sixes in the plain course are 3,5,7,12. Hudson discovered that by calling bobs at sixes 3,4,5,6 in every course eliminates all of the falseness, but also produces one of the 60 course ends from the above table as the first course end. Calling a bob at 3 gets rid of the false six and introduces one of the missing sixes that needs to be found. Calling a bob at 4 magically leads us back into one of the other 60 Hudson courses. The process is repeated by calling bobs at 5,6. So by calling bobs at 3,4,5,6 for every one of the course ends above produces a true 5040. This is exactly what Hudson did in his peal:

5040 Stedman Triples

by William Hudson (1832)

1 3 4 5 6

  - - - - 356412

D - - - - 521643

D - - - - 234165

  - - - - 365142

* - - - - 642153

  - - - - 453126

10-part, calling D at * in parts 2 and 7.

'D' stands for Double - a special call that swaps two pairs of bells over and keeps the peal in-course. Calling bobs at 3,4,5,6 produces a 5-part transposition at the course end, so this course is a useful building block for the peal. Another building block with the same effect would be bobs called at 5,6,7,8 in every course, which is simply this one rung backwards. You can also call bobs at 3,4,12,13 in every course, though this comes round after two courses rather than five, so is a smaller building block. It's reversal is 7,8,12,13. I used this to produce a peal on 6th March 2012, and a similar peal can be found on John Warboys' website:

5040 Stedman Triples

RBP (no. 1)

  2  S  L  Q  14
  s               346125
        x  x   s (321465)
     x     x      641235
     x     x      321465
  s               245136
 (s               416523
     x     x   s)(421536)
        x  x   s (435216)
        x  x      125346

10 part, replacing the parenthesis block with s2,S,Q,S,Q,s14 in part 1, and omitting it completely from part 6.

 

S = Bobs at 3,4

H = Bobs at 5,6

L = Bobs at 7,8

Q = Bobs at 12,13

s2 = Single in six two, immediately bringing up the course end

s14 = Single in the last six of the course, prevnting the course end from coming up

s9 = Single that immediately brings up six 11

s11 = Single that takes you two sixes back to six 9

 

Hudson's great legacy was that he created the twin-bob plan, which remains even to this day by far the most popular type of composition for peals and quarters of Stedman Triples (we now notate these compositions as per the table above.) To make sure that there is no falseness, we must always have exactly 240 bobs in the twin-bob peals no mtter how they're arranged. It is not necessary to stick to the building blocks of SH, or HL, or SQ, or LQ in every course, but every pair of bobs removed from one of these bocks must always be replaced by a complementary pair elsewhere in the peal to prevent repeated changes. There is a huge range of different peals possible on the twin-bob plan, including 5-parts, 10-parts, exact bicycle 3-parts, 20-parts, 6-parts, 2-parts and 1-parts. At least 2 singles have to be used, and the greatest number of singles possible in a twin-bob peal is 198 as can be seen in this example.

It must have been very exciting when Hudson discovered this plan. Before then composers had been striving to reduce the number of calls in a 5040, and 240 bobs was a major breakthrough. Only recently has a peal of Stedman Triples been composed with few than 242 calls (238, by Alan Burbidge.) And another huge advantage of the twin-bob plan is that it is easy to ring: having the bobs come in pairs means that you always go in Quick and Slow alternately.

 

                                                     Carter's Courses

John Carter is one of the all-time great minds in Change Ringing composition. He produced many twin-bob peals, but in February 1898 he composed a peal of Stedman Triples on an entirely new plan - the first such innovation in the method since 1832. In so doing, he demonstrated to the world that twin-bob peals weren't the end of the story. There are other plans available, and Carter opened the door to those possibilities. The following course is the main building block for this plan:

  2314567 six no

  3426175   1

- 3461275   2

  4137652   3

  4175326   4

  1542763   5

- 1527463   6

  5716234   7

- 5762134   8

- 7251634   9

- 7216534   10

  2673145   11

- 2631745   12

  6124357   13

- 6143257   14

 

This calling is mutually true to the same 60 course ends used by Hudson. (In other words, by starting from each of the 60 Hudson course ends and calling the above course, you would get a true 5040.) To join up the 60 courses, Carter adds in pairs of bobs (Extras) at sixes 4 and 5, and takes away pairs of bobs (Omits) at sixes 8 and 9. This technique – invented by William Shipway – is also used in some twin-bob peals. When singles are used in Carter’s plan, the courses have to be rung in reverse until the next single.

 

Carter’s calling was radically different to the twin-bob plan. In twin-bob peals the 7th is unaffected by all bobs and only affected at the singles. In Carter’s plan, the 7th is affected three times in the course. In twin-bob peals the bobs come in pairs. In Carter’s plan, the bobs come in odd numbers, hence the nickname Carter’s Odd-Bob for his first peal on the plan.

Here is an example of a composition on this plan, with only two singles. This is a great feature, as it shows that ths system is in no way 'inferior' to the peals with two-singles-only that composers had been striving for in the years up to 1846 when Thurstans produced his famous peal. All sorts of peals can be had in different numbers of parts on Carter's plan, such as this five-part. A number of other composers went on to produce peals using Carter's courses.

                                              How does it work?

As outlined above, Hudson’s 60 course ends form a mathematical group with every one of the course ends relating to the other 59. Using this framework, the composer can work out the 14 different types of sixes (technically known as cosets) which will form a true course within the group of 60. Perhaps using a table of the 14 cosets to tick off as he/she goes along, the composer then uses bobs (and maybe singles) to devise a 14-six course through each coset once and once only, the 14th six linking back to the starting point (the 14th course end either being Rounds or one of the 60 course ends from the group.) 14 sixes isn’t a great length, so this can be done without computer assistance.

 

Whatever method Carter used, this is certainly one way of constructing the courses he used. By using bobs to affect the 7th, Carter managed to create something very original. In each course the 7th is affected in three different ways, so it still occupies all of the same positions that it does in twin-bob peals – Quick, Slow, and all of the dodging. A very clever way of producing an original calling. Having the 7th regularly affected also creates an entirely different rhythm, producing effects like consecutive sixes ending in 748 or 678, which aren’t possible in twin-bob peals.

                                 60-part peals - the 'long-and-short course' plan

As described above, there are various ways of joining together the 60 courses in a peal. One way is to use pairs of Extras and Omits. Another way is to use singles. In both the twin-bob plan and Carter's plan, singles change the direction of the ringing, and the standard courses have to be rung in reverse until the next single is called. This means that the conductor must learn the reverse courses along with the normal courses.

But there exist special kinds of peals based on 60 courses, where the single takes you to another part of the course that is still being rung in a forwards direction. It isn't necessary to learn a revrse course, because all of the courses are rung in the same direction. And it isn't necessary to use Extras or Omits in these peals, because the singles can link up all the courses. These peals are special indeed. They are amongst the easiest odd-bob peals to learn and call. I like to think of them as '60-parts' because every one of the 60 courses is effectively called the same, plus the joinings. A twin-bob course or a Carter course theoretically multiplies 60 times to form a true 5040, but it is necessary in actual ringing to join up those courses with Extras, Omits, and backwards courses rung between pairs of singles. Not so with the 'long-and-short course' peals, a term coined by Alan Burbidge. The standard course is 14 sixes long. The joinings are as follows: call a standard course but then call s14 to stop it coming round, which takes you to six 2 of the standard course; continue from there to the course end. This is the long course (24 sixes). The short course is s2.s4 (4 sixes.)

 

The great J. W. Parker appears to be the one who discovered these peals. The other advantage of these joinings is that for 10-part peals they shunt the bells into a part-end order, so that the touch doesn't come round at the first part end. This technique incorporates the missing course at the same time, effectively killing two birds with one stone. Very elegant! Parker's peal shows how neat the plan is:

5040 Stedman Triples

by J W Parker

 

Standard course (A):  s2.5.6.10.s11.13 (14)

Long course (L):  s2.5.6.10.11.13.s14.15.16.20.s21.23 (24)

Short course (X): s2.s4 (4)

 

Calling:

 

231456

215364 A

256143 A

264531 A

543162 X

136245 L

164352 A

10 part, calling L in place of first A in 5th part, and X in place of first A in 10th part.

 

This can of course be rotated so that the 6th is the secondary observation bell - arguably a more intuitive way of ringing it:

 

231456

215364 A

256143 A*

264531 A

243615 A

631542 X

514236 L

10 part, calling L in place of A* in first part, and X in place of A* in sixth part.

(Halfway part end: 324156)

Other part plans are possible, including 3-parts. The same course can be rotated, with the 7th still the observation bell. Here is an example by Alan Burbidge:

5040 Stedman Triples

by Alan S Burbidge


2314567
-------
21354   [t a]a a t l a
32415      a a a t l a
43521      a a a t l a
54132      a a a t l a
15243      a a a t l a
-------
Repeat, calling s2.s4 [4] for [t a].

a = 3.4.8.s9.11.s14
t = 3.4.8.s9.11.s12 [12] (next six is position 3 of next course)
l = 3.4.8.s9.11.s14.s16.s18 [18]

This course and its variants aren't the only ones that can produce long-and-short course peals. Another well-known course is the 'Triangular Single Course,' which is called s1.s3.s6.s10.13 (14). The sixes in which singles occur are the Triangular Numbers from mathematics, hence why Alan Burbidge chose this name. A. J. Pitman composed a peal based on the reverse variation of this course:

5040 Stedman Triples

by A J Pitman

Standard course (A): s1.s5.s8.s10.12 (14)

Short course (X): s1.s5.s8.s9.10 (12)

Long course (L): s1.s5.s8.s10.s11.s12.14 (16)

Calling: A A L X A A ten times, but in 3rd part call X A in place of last A, and in 8th part call L in place of final two As.

To see the course ends, click here.

Though this course uses many more singles than Parker's course, it is arguably advantaged by the fact that the linking points are closer together.

Why are these long-and-short course peals possible?

Well, the long and short of it is that the courses are mixed parity, containing singles internally as part of the courses' makeup (even though all of the 60 course ends are themselves still in-course.) This means that when a single is added externally to link one block to another, the linkage point can be in one of the out-of-course sections of the course and effectively join together two in-course blocks. But this isn't possible in twin-bob peals (except in the 20-parts) or Carter's courses, because those courses are made up from bobs-only blocks, so a single - which changes the direction of the ringing without choice - forces the newly introduced course to be rung in reverse (a good way of thinking about in-course and out-of-course in Triples and Major is that one is forwards ringing and the other is backwards ringing. This is not so in Doubles, Minor, Caters, or Royal, where in-course and out-of-course truly are two separate states that are locked off from each other.)

 

                                 John Lancashire and the s1.s10 courses

The Leicester ringer and architect John Lancashire discovered an interesting new plan: call a single whenever the observation bell (the 7th in this case) is lying behind in 7ths place. So the basis of these courses are s1.s10. This plan does not lend itself so readily to a set of 60 identical mutually true courses, so Lancashire had to use all sorts of intricate joinings, extras and q-sets across the courses. Consequently, if we got a computer to generate all of the different possible 60-course plans, Lancashire's calling would not be among them.

 

There are other interesting and meritous features of this plan, and Lancashire's compositions in particular. As the singles in s1.s10 leave the observation bell unaffected, it is possible to produce compositions in which this bell is entirely unaffected throughout the peal and only rings plain courses. These appear to be the only 5040s of Stedman Triples using normal calls in which a bell is completely unaffected for the entire peal. Another astonishing feature of his peals is that some of them have no more than two consecutive calls. His first compostion was rung in 1898. It was actually rung and published before John Carter's new peal. However, Carter sent a letter of congratulation to Lancashire, published in Bell News, in which he also mentions his own peals that had been composed several months earlier. It appears, then, that Carter's plan was discovered first. Carter described Lancashire's 5040 as the best yet produced in the method. Here is Lancashire's peal of 1898 (note that it does not have a normal start):

 

5040 Stedman Triples (Start with a full slow six)

by John O Lancashire

1 3 4 5 6 7 8 10 12 13 123456

s   s -        s  -  - 341265

s -   s -   -  s  -  - 613245

s   - s        s  -  - 326154

s -   - s   -  s       514623

s       s -    s  -  - 623514

s -   - s   -  s  -  - 136524

s       -   -  s       256314

s   - s        s  -  - 632514

This is called five times, but the first course of the 2nd, 3rd, 4th and 5th parts is instead called: s1.3.5.s10.12.13 to produce:

1 3 4 5 6 7 8 10 12 13 621435

s - s   s -    s       413526

s -   -        s  -  - 253146

s -   s -   -  s  -  - 432156

s       -   -  s       512346

This is called five times, but the first course of the 7th, 8th, 9th and 10th parts is instead called: s1.s6.7.s10.

The 7th is unaffected throughout the peal, and there are never more than 2 consecutive calls, with only 82 of these.

The composition is a five-part that has to be split up into two sections to prevent the first part end from being Rounds. An irregular q-set can be used to ring some of the part backwards and so produce an exact 5-part, but this results in more than two consecutive calls.

Lancashire continued working on the plan, producing a remarkable 10-part peal in 1901. This reduced the number of consecutive calls even further, with only 21 instances of two consecutive calls and one instance of three consecutive calls (the peal was published with the 4th as observation bell rather than the 7th, so that this set of three calls was broken up across the beginning and end of the peal.) There are 222 singles and only 90 bobs. The feature of having the observation bell completely unaffected throughout is compromised, but the s1.s10 structure remains. As can be seen, he used the Triangular Single Course (described above) as a good basis for reducing consecutive calls. Here are the figures for the 7th observation version:

5040 Stedman Triples

by John O Lancashire

A = s1.s3.s6.s10.13.14.s17.s20.s22.24 (26)

B = s1.s5.s8.s10.11.s13.s16.s20.23 (24)

C = 2.5.s7.s9.s12.s16.19 (20)

D = s1.s3.s6.s10.13 (14)

E = s1.s2.4.7 (8)

F = s1.s5.s8.s10.11.s13.s16.s20.23.s24.s25.s28.s32.35 (36)

231456

561342  A B C D

421635  A B C D

351264  A B C D

641523  A B C D

321546  A B C E D D

461253  A B C D

531624  A B C D

241365  A B C D

651432  A B C D

231456  A F

Other composers have written about how complex and intricate this peal is vis-a-vis the ways that the right sixes are joined together to preverse truth. As mentioned, one course on this plan doesn't cover all of the 14 different types of six within the group of 60, so the ways that he skips from one block to another with reverse courses (such as course B which starts off as the reversal of course A, which starts off as the Triangular Single Course) and other unusual joinings is highly advanced and must have taken hours of work. It moved the brilliant composer and theorist, J W Parker, to describe Lancashire as a 'genius,' and the above peal as "the most wonderful peal of Stedman Triples yet discovered." Fuller in-depth descriptions of how these courses and six-types were joined together have been written elsewhere.

A number of high-ranking composers have produced further peals on Lancashire's s1.s10 plan, including Brian D Price, and A J Pitman. There is a huge number of possibilities, and variations of courses.

 

   

 

 

   

     

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