[r-t] All runs in a quarter, proper version

Alexander Holroyd holroyd at math.ubc.ca
Mon Feb 2 04:48:41 UTC 2015

With less conventional place notations (1234 etc) and more static music, 
one can do the same thing in a 1280:


Less good, in my opinion, although certainly ringable.  Considering the 
absolute theoretical minimum is 736, I find it pretty amazing that this is 

On Sat, 31 Jan 2015, Alexander Holroyd wrote:

> I'm pretty excited about how well this 1536 turned out!  I for one would be 
> keen to ring it.  Here is the whole thing written out, with the runs 
> highlighted:
> http://www.math.ubc.ca/~holroyd/runs.pdf
> I've put together a description of the process that went in to composing it. 
> In particular I hope this will answer Philip's questions, and perhaps 
> encourage others to explore this interesting new paradigm.
> I started out with the idea of trying to get all 480 front and back 4-runs 
> into a reasonably short length that someone might actually want to ring.
> How closely packed can the runs be?  Of course one can repeat a run 
> immediately with e.g. a 5678 place notation:
> 12345678
> 21435678
> but while that's a potentially interesting effect, I decided I wanted to try 
> to stick to more conventional local structure (e.g. maybe all triple and 
> quadruple changes).  Of course one can repeat runs every other row using 
> dodging:
> 12345678
> 21436587
> 24135678
> 42316587
> 43215678...
> but more than a triple dodge or so would get boring, and one must link to 
> other types of run.  In a few cases one can go directly to another type, e.g.
> 21435678
> 12346587
> or
> 24315678
> 23451768
> but there are not many such opportunities.  Many more linkages open up if one 
> allows things like the following, where two run-rows are separated by two 
> other rows, with both the run type and direction changing:
> 12834567
> 21384657
> 23148675
> 32418765
> Based on this, it might be reasonable to have no more than two other rows 
> between two run-rows in the entire touch.  However, I couldn't think of an 
> obvious way to impose enough structure to make this into something sensible 
> rather than a long random string of place notation.
> An obvious solution to the structure issue is to aim for a cyclic 8-part. It 
> is no longer reasonable to expect the property above, but instead one can 
> still hope for at most two other rows between potential run-rows, i.e. 
> run-rows plus things like -7812 that will become a run-row in other parts.
> My first attempts involved starting from simple basic blocks.  E.g. here is a 
> natural set of round blocks that together contain all the runs, if rung 8 
> times each according the cyclic part ends 23456781 etc.:
> 12345678: - *3
> 13245678: - *3
> 12345678: - *3
> 12345768: - *3
> The problem is that these are false against each other in multiple places. 
> E.g. rounds and 43215678 each occur in two blocks because they each contain 
> two runs, but so does 43216587 because it appears both as a 1234 run and as a 
> link row for 5678 runs.  This kind of thing seems inevitable whenever one 
> starts with simple repeating blocks.  Falseness is not necessarily a problem, 
> because one can eliminate it by local changes, and potentially even use it to 
> link blocks together.  However, the very nature of the shared runs issue 
> requires different solutions in different places. E.g. starting from the 
> above blocks, here is a true set of round blocks I found that contain all the 
> runs in a cyclic 8-part structure, retaining rotational symmetry (more on 
> that later):
> 43125678:
> 32145678: 4-4-4-3.6-5-5-
> 42135678: 56.1-5-5-
> 34215678: 1.34.5-5-5-36.1.36-5-5-5.34.1-4-4-
> 21438765: 4-4.5-5.4-4.5-5
> 43216587: 4-4.5-5.4-4.5-5
> 43215678: 4-4.5-5.4-4.5-5
> 43218765: 4-4.5-5.4-4.5-5
> These still need to be joined together, and the 8 parts need to be joined 
> somehow.  One could certainly do this, but it would probably end up not very 
> elegant, with little sign of the original structure.  Getting an exact 8-part 
> would be difficult without an ugly and perhaps run-free link section.
> I decided to abandon the method-based approach and simply look for an 8-part, 
> hopefully an exact one.  It turns out that there is still enough room for 
> manoeuver in the graph containing all potential run-rows plus links between 
> them involving at most 2 other rows.  I investigated the possibilities, but 
> the problem is that there is too much flexibility.  An arbitrary 1/8th 
> quarter peal length string of place notation is a lot to learn, and not very 
> satisfying.  I needed to impose more structure.
> One potential improvement is a 16-part, with the part ends being the cyclic 
> ones 23456781 etc. plus their reverses  (which form a dihedral group, 
> isomorphic to plain hunt).  I thought it might be possible to have a 
> symmetric 8-part, with a forward part end part followed by a reverse part end 
> one rung in reverse, but I can't see how to do this without asymmetric (and 
> perhaps run-free) links between them, which isn't good.
> There is another way to get more symmetry - rotational symmetry.  Suppose our 
> part starts something like
>   *12345678*
> 58  21435768
> x   12347586
> 16  13274568
> ...
> Then one can ring the same place notations reversed and in reverse order at 
> the end of the part:
> ...
>    13452768
> 38  31425678
> x   13246587
> 14 *12345678*
> 58  21435768
> x   12347686
> 16  13274568
> ...
> With the chosen part ends (and no others), any potential run-row in one half 
> will become a potential run-row in the other half, with runs at the front 
> becoming into runs at the back and vice versa.  And this will effectively 
> turn the touch into a 32-part, with only half as much place notation to be 
> learned and searched for.
> How does one avoid falseness in such a scheme?  Every row has a 'partner', 
> which is the one that we want to occur in the corresponding place in the 
> other half of the part.  E.g. the partner of 12345867 is 23145678.  And 
> taking partners respects the 16-part structure.  (Mathematically the partner 
> is the conjugate by reverse rounds R, and if P is a part end then R(PA)R = 
> P'(RAR) where P' = RPR which is another part end since R is in the normalizer 
> of the part end group).
> Here there is a delicate issue.  Some rows are their own partner. Fortunately 
> there are only two (modulo the 16 parts): rounds and 43215678. (In an 8-part 
> it would be rounds and backrounds).  Therefore we must use these as the 
> apices.  So what we want is a block starting from rounds and ending at 
> 43215678 (or its equivalent in some other part), visiting either each other 
> run-row or its partner (but not both).  Then we reverse this block both in 
> space and time to get back again and obtain one rotationally symmetric part. 
> An inevitable consequence of this is that the part end after one part must be 
> 43218765.  On the face of it this is not ideal, because then there will be 8 
> round blocks to be joined together somehow. But things will in fact turn out 
> well...
> After some experimentation with different place notation selections and 
> requiring a decent amount of traffic through 45 to avoid stagnation, I came 
> up with block B, which I was pretty happy with.  The different run types are 
> nicely mixed, with often 2 or 3 immediate repetitions of the same type using 
> dodges, but no more.   Only good place notations appear (unless you object to 
> 34 and 56, in which case I don't think there is much hope here). There are 
> some other options, but not that many, particularly as short as this.  As 
> described above, the block is rotationally symmetric, and 8 copies of the 
> round block BB give all the rows we want.
> That leaves the problem of joining the blocks.  Initially I was not 
> optimistic about doing this elegantly.  The rather frenetic place notation 
> means one should not expect that many Q-sets, especially without long places, 
> and there is the danger of getting e.g. to the part end 34567812, so that one 
> still has more joining to do.
> However, a miracle occurs.  There is a magical Q-set present that involves 
> all of the place notations 16,38,18,36 and preserves the rotational symmetry! 
> Replacing the original 38 with 18 shunts to the corresponding place in the 
> second half of the block, and we get back again by replacing 16 with 36, so 
> only 2 blocks are joined, and we don't get a parity problem.
> 12348657  12348576
> 21384567  13284567
> becomes
> 12348657  12348576
> 13284567  21384567
> and we never need to interrupt the run-fest.  Amazingly, the surrounding 
> place notations are such that we (just) avoid long places.  Furthermore, one 
> shunt gets to 45678123, so no other link types are required.  Many of us know 
> that these things rarely work out so nicely!
> Incidentally, I don't think I have ever seen this type of Q-set used before.
> At this point it's easy: we just successively link the blocks with 7 such 
> Q-sets.  The whole touch has exact rotational symmetry.  It is amusing that 
> the A and C parts have odd length, so alternate part ends come at handstroke. 
> I really find it amazing that all this works!
> What's the best way to describe this touch?  As I have tried to argue, I 
> think this should be entirely at the discretion of any band that wants to 
> ring it.  One-size-fits-all attempts to legislate simply get in the way of 
> new ideas like this.
> However, if one wants to think in terms of methods, one suggestion (among 
> other possibilities) is the following.  It has the advantage of just two 
> rotationally symmetric methods that don't look too daunting (a principle, and 
> a short 11222 differential), and lead-end calls.  The drawback is that the 
> lead-end orders are unintuitive, and the cyclic structure gets rather 
> severely hidden in this description.
> Frog Major:
> -4-
> 56-
> l.e. 57628413
> bob = 3
> Toad Differential Major:
> 4-4-
> l.e. 13254876
> bob = 6
>      12345678
>      --------
> (F)   35426781
> F     68751234
> F     13284567
> F_T_  64573218
> F_TT_ 31248765
> F_TT_ 86715432
> F_TT_ 53462187
> F_TT_ 28137654
> F_TT_ 75684321
> F_TT_ 42351876
> F_TT_ 17826543
> F_T_  46537812
> F     71862345
> F     24315678
> F     57648123
> F     82173456
> (F)   12345678
>      --------
> Start and finish at the half-lead of Frog: 56-4.56.1...
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