20 parts | Ringing

Grandsire Triples

20-part peals

In Stedman Triples, 20-part peal compositions are well established and have been popular for over 150 years. The most famous example is Thurstans' Four Part, which is in fact a 20-part divided into four distinct quarters. Five of the part ends for this are Rounds and the cyclic the rotations of Rounds on the front five, i.e: 5123467, 4512367, 3451267, 2345167, and 1234567 (note that these aren't the actual six-ends.) The remaining part ends are the rotations for Queens, Tittums, and Back Rounds.

I remember thinking in 2010 that such a group of part ends should be technically divisible for Grandsire Triples, each part being 252 changes long. A peal based on B-Blocks could work. The difficulty would be squeezing in five members of a Q-set of omits into each part - to use the excellent Philip Saddleton expression, it might be difficult to find room to manoeuvre.

In February 2011 I got to work with putting this together. I chose the same group of 20 part ends as for Thurstans' peal (rotations of Rounds, Back Rounds, Queens and Tittums), with the 7th as the observation bell, and here they are, again in four quarters:

a1 23456 c1 35246

a2 34562 c2 52463

a3 45623 c3 24635

a4 56234 c4 46352

a5 62345 c5 63524

b1 54326 d1 42536

b2 43265 d2 25364

b3 32654 d3 53642

b4 26543 d4 36425

b5 65432 d5 64253

It is worth bearing in mind that the 'b' figures relate to one another in the same way that the 'a' figures relate to one another, and the 'c' figures relate to one another in the same way that the 'a' figures relate to one another, etc. This is a mathematical group, and the 20 part ends are intrinsically linked to one another in the way described by J. W. Parker in his turn of the century pamphlet on Stedman Triples.

After a bit of searching I was surprised to find a block based on BPB x 5 that was mutually true to the above groups a, b, c and d. As always with Grandsire Triples, I then had to find out what the missing B-Block was, and where it could be singled in to the main composition (all of this was done with pen and paper, as I don't know how to programme a computer for such searches!) This done, I then had to link the 20 parts together.

First, I came up with no. 12a, where the 20 parts are divided into an exact five part.

Then I came up with no. 12b, where the 20 parts are divided into a near four-part, rather similar to Dexter's no. 2 variation of Stedman Triples. This is much more logical and easy to call than it might look at first glance. I think that no. 12b is neater, as you get to ring most of the 20 parts 'unbroken,' whereas no. 12a effectively has every part broken up.

No. 12b was the first of the two to be rung, conducted by myself at Dodderhill on 25th June 2011 - this was possibly the first time that a 20 part peal of Grandsire Triples had ever been rung. The composition has subsequently been conducted by Ben Constant a couple of times, and Graham Hayward was the first person to conduct no. 12a, which was rung at Belper in December 2013. My thanks to both of them for their interest.

It still amazes me that nobody else came up with a peal composition of Grandsire Triples on this plan before myself. I continue waiting to be informed that someone has. Previous composers must have known that this group was a possibility for Grandsire Triples. Perhaps J. J. Parker, E Bankes James, etc, were only interested in part systems that have more regularity for the conductor. Sometimes one comes across '20 part peals' which are in fact another part plan broken up into 20 notational sections, but are strictly not 20-part peals at all. For me, a 'part' has to relate mathematically to the other parts by group theory.

Strangely enough, Ander Holroyd came up with his own 20 part and posted it on the ringing theory mailing list on 5th May 2011, just a couple of months after I had composed my own. Ander's composition was identical to my no. 12b, so I guiltily had to tell him that I'd beaten him by a few months!

Not long afterwards, I came up with another 20 part that has lots of calls, but one lead missing from each part. This is one of the 'shunt' leads where the 7th is in the hunt, so there are effectively four plain courses missing from the peal. I did manage to break these into a rather inelegant irregular four part division, but the peal is very inferior to no. 12.

Update: Having recently (2019) looked through my original composition pad I was reminded that before trying the 20-part group above, I first tried the "extended Middleton's group" (45326, 32546, 54236, 35264 x5), but the three part-blocks that I looked at were all false. Probably a true version would be possible, with Rounds inside the SBBS block and this would no doubt be a variation of my no. 12. Later in 2011, I came up with a third version of my 20-part: no. 12c. For some reason I have never given much attention to this variation, despite the fact that it is of considerable merit, and the linking points are closer than they are in 12a and 12b. The main part is called exactly the same.

5040 Grandsire Triples

by RBP (no. 12c)

234567

752634 1

347265 2

523647 1

345276 3

763524 2*

s 547263 1

325647 1

763425 1

s 457263 1

634725 2

576234 1#

345627 2

20 part

In part 3, call a single at # to produce * three times in succession before proceeding to part 4.

In parts 8, 13 and 18, call a single at * to produce #.

Update: In May 2019 I discovered that 20-part peals are possible on the 3-lead course plan. However, I think of 3-lead course peals as really being 120-parts (particularly the examples with strictly only two calls in each course), with my 20-part peals on this plan consequently having a "sort of" status. My peal has the 7th observation at the quarterly part ends, though this rotation with the 2nd observation looks better on paper. I haven't seen 3-lead course peals like this before, and believe them to be original.

Update: For the sake of completeness, I recently (November 2019) turned my no. 12c into an exact 5-part. The resulting no. 12d is 12c's equivalent of 12b's 12a (are you with me so far?)

Half a 20-part

If you have another look at the table of 20 part ends above, it is interesting to see how it divides into other part plans for peal compositions. For example, there are plenty of five-part peal compositions based on the group 'a' (though many five-part peals have different actual part ends) and scores of ten-part peal compositions based on the combined groups 'a' and 'b' (though many ten-part peals have different actual part ends.) These ten-part peals divide into two half peals - each half peal being a five-part - and the two halves are joined together irregularly by using singles to 'teleport' from one half of the peal to another point in the other half of the peal, and then back again 2520 changes later. This is a well-known trick, and example of such a ten-part can be found here.

However, in 2012 I observed that a 10-part based on the combined groups 'a' and 'c', or 'a' and 'd', never seems to have been published, or at least not that I've come come across. But such a peal seemed feasible, as the part ends would be a sub-group of this functioning mathematical group. Interestingly, this 'half 20-part' would be half in-course and half out-of-course (like Parker's 12-part), unlike the other kinds of ten-parts. I got to work looking for a mutually true part block, but didn't have any success, for reasons that I can't remember.

I returned to the idea afresh in late 2016 or early 2017, and became quite excited when I thought I had come up with a mutually true part block of 504 changes which could form a peal. However, when I checked it on the computer, it was false! Very strangely, it was almost as if the 'half 20-part' was trying to 'force iself' into an actual 20-part, and part ends from the two groups that I had left out appeared in the composition. It was also very strange how there didn't seem to be a clear hint on paper beforehand that this would happen. Very interesting, but why doesn't this happen for other peals based on sub-groups? What is different about this particular sub-group? If you know, I'd love to hear from you. For more about group theory, see this excellent paper by the late Brian D Price.

Update: I have looked into half-20 parts a few times more recently, and now have a more scientific explanation! The problem is that the 10 part ends in question are not a subgroup of the 20, and are therefore not a group in their own right. This is what lead to the falseness problems in the aborted Grandsire peals. I have come up with a 10-part of Bob Triples based on this idea, again showing the great versatility of that method.

A 10-part peal of Grandsire might just be possible on this plan, but it certainly seems very unlikely. The B-blocks have to be chosen and scrutinized very carefully indeed. The way of joining the two half-peals also has to be done carefully (as was the case with the Bob Triples peal.) In Grandsire you have precious little control over which B-blocks get joined together, due to the cumbersome q-sets of 5 elements.

The Bob Triples peal I suppose counts technically as a non-group 10-part. A similar example is this peal of Erin Triples. The 10 raw part ends are indeed a group, but each notational part is in fact 2-parts joined together by singles, making for 20 parts in total. However, the 20 parts do not form a group as they are half of the part ends from the 40-part group in Price's paper. The peal is therefore 'half a 40-part,' or a 'non-group 20-part.'