[r-t] A new category of splice?

Sun Jul 1 21:35:37 UTC 2012

In recent days MBD and I have been discussing and making salivating progress on a compositional project. Mark has stumbled across exciting new methods which are "halflead splices", ie where a given halflead in each of the methods contains the same rows. I'm not aware of any previous examples of halflead splices, and will try to introduce what's been found.

Previously I had thought that the most simple / fundamental splice between two methods was the lead splice, ie where a given lead of two different methods contains the same rows.

The most commonly rung lead splice methods have a few rows reordered in the same short section, typically by having adjacent places made in different positions.  Eg Beverley and Surfleet surprise minor are only different when the treble is in 5-6, with Beverley having places made in 3-4 and Surfleet with places made in 1-2.

This is not the only way to get a lead-splice, however.  Another way is to change the notation to "jump" to a different place in the lead, with a jump back at a corresponding suitable place later in the lead. Consider eg Bourne Surprise Minor (&-3-4-2-3-34-3,2). When the treble is in 3-4, in the first lead we have the rows:

   261435

x  624153

12 621435

x  264153

...and at the halflead we have the rows:

x  264531

36 624351

ie the pairs of bells (2&6, 3&5) come together when the treble dodges in 3-4.  Because these same pairs cross at the half-lead, they will come back together in the corresponding place (ie when the treble dodges 3-4 down) in the second half lead.

So this means that we can get a lead splice of Bourne by swapping a different pair of bells when the treble is dodging in 3-4. The resulting lead splice method is Wath Surprise (&x3x4x5x3x34x3,2)

So far so standard. However, when developing a new eight bell composition (more on this soon), the following methods have dropped out of infinite assemblies of changes, screaming "ring me!":

H = &-16-58-14-18-14-18-58-38.56.38-14-18-14-1458-36-1458-58-18-58-16.34.16-14-18-58-18-58-14-38-58, +14

J = &-16.58-58-18-58-16.34.16-14-18-58-18-58-14.58-36-14.58-14-18-14-18-58-38.56.38-14-18-14-14.38-58, +14

Both methods are differentials (with a 3-cycle and a 5-cycle), with the very attractive "Double Helix" property that in the 3-cycle the bells (here 1,2,3) ring in all 8*7*6 = 336 possible relative orders. I call such methods "helixoids".

Here, our helixoid methods H and J also have the attractive bonus features of double symmetry, of having the "natural" coursing order, and of having a pleasing line (neither too static nor too fluid / featureless).

It's how the methods splice together, though, which is very interesting. As featured at the top of this message, they are magical mystical halflead splices. Clearly the vastness of the helixoid methods (with 112 change leads) gives possibilities beyond short treble-dodging methods.

The principle in which this works is essentially the same as the Bourne / Wath example, though here with two pairs of related rows (A and A', B and B') to jump between. With methods H and J, pairs of bells come together at changes 3 (A) and 26 (B), with corresponding pairs coming back at changes 30 (A') and 53 (B').

Given the method symmetry, as these places A and B are both in the same quarter-lead we can change the bells that swap here to jump around a couple of times to get the half-lead spliced property.

The net effect is that within each half-lead, a block of 23 changes are basically rung the opposite way round, and linked using different changes.

A risk with this approach is that the two methods might feel very similar. However, I think they have a really rather different character. Indeed, method H is right place (for a suitable definition of right place), whilst method J has many substantial wrong-place features.  Take a look...

H: <http://ringing.org/main/pages/method?notation=-16-58-14-18-14-18-58-38.56.38-14-18-14-1458-36-1458-58-18-58-16.34.16-14-18-58-18-58-14-38-58&pn-query=Display&stage=8&le=14>

J: <http://ringing.org/main/pages/method?notation=-16.58-58-18-58-16.34.16-14-18-58-18-58-14.58-36-14.58-14-18-14-18-58-38.56.38-14-18-14-14.38-58&pn-query=Display&stage=8&le=14>

I'll come to how the methods were found and what they are used for in a separate message shortly.

I do like this half-lead spliced concept.  Has it been documented before?  Are there rung methods which have this feature? I imagine Richard Smith may dig something out. I'm looking forward to stimulating discussions.

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