[r-t] Compositions of the Decade: Part 2 - Doubles
Philip Earis
Earisp at rsc.org
Thu Nov 19 15:37:03 UTC 2009
Doubles is the base from which change ringing really developed. It is a paradox that doubles has been both well-studied and much overlooked over the centuries.
The golden age for doubles was in the 17th Century, when a wide variety of methods were developed. Tintinnalogia (freely available online at http://www.gutenberg.org/etext/18567) remains a fresh and fascinating read. However, plenty of new ideas continue to abound today.
Infinite possibilities
Ringing on five is of course based around ringing 120-change extents - small enough to make things manageable, both from a ringing and composing point of view. Indeed, many problems can easily be exhaustively searched using a computer.
Because of the constraints, the boundaries between doubles compositions and methods can be rather arbitrary - the two concepts become intertwined.
However, the beauty is that rearranging five bells in different ways still allows massive possibilities. A single grain of sand contains around 7.8*10^19 (78 billion billion) atoms. The entire universe is believed to contain around 10^79 atoms. There are 6.7*10^198 possible ways of arranging the extent on five bells. In other words, there remains an eternity of new methods available. Doubles really retains its ability to interest, delight and surprise.
Declining numbers
Whilst many ringers' first introduction to change ringing is with doubles, ringers often seem keen to move away from five bell methods as quickly as possible.
There has been an alarming decline in doubles in recent decades, at least as far as peals are concerned - at the beginning of the decade peal numbers had fairly consistently been averaging about 200 a year (about 3% of all peals rung). By 2008 numbers had dropped to a record low of 123 peals (just 1.8% of the total). A further steep decline looks likely in 2009.
Even more worrying is that just one of the peals of doubles rung in the whole of 2008 contained methods which weren't either plain hunt based or Stedman. Now there is nothing wrong with plain doubles methods per se, but this illustrates even more quite how unexplored the field of doubles ringing is.
It is frustrating to hear people say contemptuously that there's nothing worthwhile that can be done on five bells. This disdain is snobbery borne out of ignorance. A ringer who shuns lower numbers is usually running away from a challenge. It's easy to formulate a peal of doubles that is vastly more complex than the most "advanced" spliced maximus that is rung.
A further paradox is that despite declining peal numbers and negative attitudes, the last decade (especially recent years) has seen great innovation resulting in excellent new extents of doubles. Building on new ideas from the 1990s, which for example saw many differential doubles methods rung, doubles is one of the big growth areas in ringing theory.
Recently, the main thrust of this development has come from Professor Alexander Holroyd, working out of his Vancouver lair. The Professor (one of the few ringers to have a mathematical constant named after him) has used his group theory expertise and innovative experimentation with different symmetries to great effect, as we shall see.
Themes over the decade
It is interesting how some of the new doubles developments have close parallels with the way early ringing pioneers worked in the 1600s. As in much of ringing, an effective way to finding a solution to a problem is by solving a simpler related problem.
With doubles, the key to finding interesting extents has often been to produce an in-course half extent - ie all 60 changes obtained only using double-changes (place notations 1, 3 and 5) - and then use a single to obtain the whole extent.
The most common extents of double rung, accounting for the vast majority of rung doubles, are Grandsire, Stedman, and Plain Bob. All of them elegantly produce extents based on in-course half-extents (with Plain Bob the argument is admittedly a bit more stretched and requires stitching together 10-change in-course blocks).
As we'll see, the theme of in-course half extents will appear in my choices below, along with different symmetries and the difficulties in classifying some doubles extents.
Without further ado, here are my chosen doubles compositions.
1) Jump Stedman - Ander Holroyd - First rung September 2008
The first "composition of the decade" preserves the in-course half-extent beauty of Stedman, and miraculously converts it to a wonderful plain course extent, which is conceptually extremely satisfying, and great fun to ring
Just like in conventional Stedman, the method is divided into sixes, which have hunting on the front three bells whist the back two double dodge. Here there are four types of six, rung in the order (quick -> jump down -> slow -> jump up)
See the previous description on this list at:
<http://bellringers.net/pipermail/ringing-theory_bellringers.net/2008-September/002748.html>
And although not a new composition, Robert Johnson's 2006 proof of how an in-course half extent (like conventional Stedman doubles) can always be expanded into a full extent (with Stedman, the resulting method is Crambo) deserves an honourable mention here.
2) Multi-spliced doubles - Philip Saddleton - c2003-2009 (Unrung and unpublished)
The past decade has seen progress in multi-splicing more conventional, treble-hunting doubles methods as well. Following his achievements in the realm of spliced minor compositions in the previous decade, Philip Saddleton has turned his hand to doubles.
He has managed to include all 220 symmetrical single-hunt plain methods in 42 extents, using 2-lead, 3-lead, 4-lead and combination splices to fit everything in.
The extents will be published as part of the new doubles collection - hopefully appearing soon.
I hope Philip won't mind me reproducing one extent here - a combination splice - as a sample of his work.
2345 96S
2453 94S
2534 88D
3245 158T
3524 148E
4352 44D
5423 125T
5342 127T
5234 117E
4523 55S
4235 48D
3452 150E
2345
I suspect Matthew Frye deserves credit for giving ideas for some of the extents.
3) Banana Doubles - Ander Holroyd (building on Richard Smith) - First rung March 2009
Another theme for the decade (on all stages) has been using different kinds of symmetry, rather than just the "conventional" palindromic symmetry.
One neat form of symmetry is "glide" symmetry, where the changes in the second half-lead are the reverses of those in the first. Whilst this has been used before (Double Eastern Bob Major, first rung in 1752, glides merrily along), it was employed to great effect in my second doubles composition of the decade:
Banana is a marvellous principle. There are some similarities to Stedman, with six consecutive changes of hunting on three, but the glide symmetry gives it a super fluidity. It combines a superficial simplicity with inspirational delight wonder when rung.
120 Banana Doubles
Alexander E. Holroyd
% 1 % 2 % 3 12345
------------------
- - 54213
------------------
5 part
Method: 3.2.3.2.3.4.3.4
bob = 2; hl bob = 4
The so-called "plain course" of Grandsire doubles can be considered a reverse-engineering of a neat in-course half-extent. In the same way, Banana Doubles can be considered the "pick of the bunch" of the exhaustive list of 101 Doubles methods that Richard Smith published in 2006, with the following properties
- Principles
- Plain course generates the extent
- No more than two consecutive blows in one place
Richard's full list can be seen at: http://ex-parrot.com/~richard/doubles/extents/principles-2-blows.txt - it is a subset of the 52,227,975 methods he found that aren't restricted to 2 consecutive blows in one place. It was pleasing to see a band ringing 42 different doubles principle plain-course extent methods in a peal in 2008.
4) Magic block doubles - Philip Saddleton - September 2008 (unrung)
It's always possible to argue about whether something really is a reverse-engineer of something else. A notable and even more extreme example which highlights the problem of how to classify something was published by Philip Saddleton.
The father of "magic blocks" spliced, which had a big impact on minor ringing in the decade, PABS has here produced an extent containing seven different overworks and eight different underworks. It's possibly the ringing equivalent of a bonsai tree.
5 bells
touch=+3.1,"B1",
&5.3.5,"F1",
&1.5.2,"B2",
&5.3.2,"F2",
&1.34.2,"B3",
&25.3.34,"F3",
&1.3.23,"B4",
&2.23.34,"F4",
&1.5.2,"B2",
&5.3.5,"F1",
&1.3.2,"B1",
&5.23.5,"F5",
&1.34.23,"B5",
&5.23.5,"F5",
&1.3.2,"B1",
&5.3.34,"F6",
&4.5.23,"B6",
&5.3.2,"F2",
&4.5.23,"B6",
&2.3.5,"F7",
&4.3.23,"B7",
&2.3.5,"F7",
&4.5.23,"B6",
&5.23.34,"F8",
+1.3.2,"B1"
5) Hybrid doubles (15 change divisions) - Ander Holroyd - November 2008
Few methods have been rung with an odd number of changes per division. Red Square Hybrid Doubles puts Ander's group theory knowledge to innovative use, dividing the extent into 8 leads of 15 changes (with the treble of course ringing 3 blows in each place per lead) that form a group.
+125.145.3.123.1.345.125.1.345.123.1.3.125.145.3
Extent: pppsppps; single = 1 for last 145
http://www.bellringers.org/pipermail/ringing-theory_bellringers.net/2008-November/002756.html
6) In-course 120 - Andrew Johnson - October 2006
Responding to a challenge on this list, Andrew produced a very neat example of an in-course 120 of doubles, where each row occurs once at handstroke and backstroke.
+3.1.3.5.1.3.5.1.3.5.3.1.3.1.3.5.1.3.5.1.3.5.3.5
A 240 containing each row twice can trivially be obtained with a pair of singles.
7) Dixonoid doubles -Philip Earis and Andrew Tibbetts - autumn 2001
Continuing the theme of things being difficulty to classify, the long established idea of "dixonoids" or rule based constructions made an appearance in the early years of the decade.
Here, the place notation is defined "on the fly" based on which bells are leading. In the plain bob version, all bells plain hunt, with 2nds made when the treble leads (as in bob doubles), but with 4ths additionally made at the backstroke whenever 2 or 4 lead:
120 Dixon's Bob Doubles
2345
- 5342 1
- 4235 2
- 4352 3
- 5432 2
- 3425 2
- 2345 2
- = 145 at treble's backstroke lead
In the Grandsire version, a 240 containing each row once at each stroke, the bells plain hunt, with thirds made the handstroke after the treble leads (as in normal Grandsire), and again with 2nds made when the treble leads (as in bob doubles), but with 4ths additionally made at the backstroke whenever 2 or 4 lead:
240 Dixon's Grandsire Doubles
2345
s 4325 1
s 3425 6
s 2354 1
s 3254 6
s 3524 3
s 5324 6
p 2345
s=123 at treble's backstroke lead only
8) Ocean Finance Doubles - Ander Holroyd - First rung March 2008
+3.5.123.1.3.123
Extent: TppTppTppTppTpAppppA
T = 345 (instead of 123) at division end A = 145 (instead of 123) at division end
This is a clever asymmetric principle with six changes per division. Extents usually consist of an assembly of mutually true courses. This one doesn't, relying instead on a composition consisting of two distinct blocks. The blocks permute in the same order, neatly providing the complementary rows for their analogue so the extent is obtained.
Reviewing the selected compositions above, it does seem to have been a bit of a CUG-fest. This is not intentional - please do tell me what I've missed.
Coming soon-ish - part 3 - a minor earthquake...
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