This is about methods in general rather than handbell ringing, but I came to it while pondering Buckfastleigh as a possible method to ring next time we find ourselves with only four people. It's one of the methods from Chandler's 23-spliced, but isn't rung very often by itself. I've never rung it, although I did ring the 2nd place lead end version, which is called Silchester, years ago as it's in one of the every-lead-different variations of Smith's 23-spliced. Buckfastleigh is a double method with lots of pointy work which should be fun to ring.

Anyway, back to the topic. Something I occasionally find useful when conducting Bristol is that the odd place bells end up back in the same place at the half lead, while the even place bells move two places towards the front (except that 2nd place bell just moves to 1st place as it can't go any further). I started wondering whether it's possible in general to work out where place bells get to at the half lead. This makes most sense for methods with Plain Bob half leads, which includes all double methods (for example, Buckfastleigh), because the fact that the bells are in coursing order at the half lead means that the half lead place bells will all be related to each other in a sensible way.

What I realised is something quite neat that I don't remember ever being told or seeing written down before. Maybe it's well known, but it was new to me. It works for double methods.

Start with the place bell order, expressed as a number according to the system whereby Plain Bob place bell order is +1 and other place bell orders correspond to how many leads of Plain Bob they are equivalent to. So Superlative is +2, Bristol is -1, Buckfastleigh is +4 and so on.

We're interested in the half lead place bell order, so divide the place bell order by 2 because two half leads should be equivalent to one whole lead. If the place bell order is an odd number, first add 7 to it (in general, 1 less than the number of bells) to get an even number which is an equivalent way of expressing it. So Bristol is +6 instead of (actually, as well as) -1.

The twist is that the half lead place bells have to be interpreted as the distance from the treble, which is at the back, instead of as the distance from the front of the row.

For Superlative Major, the place bell order is +2 so the half lead place bell order is +1. Consider 2nd place bell. With place bell order +1, the next place bell is 4th, but we have to interpret it from the back of the row: 1st place from the back would actually be 8th place, which is where the treble is; 2nd place from the back would actually be 7th place; so 4th place from the back is actually 5th place. And 5th place is exactly where 2nd place bell finds itself at the half lead, starting 5-6 places.

This always seems to work, although I have not proved it in general. It's intuitively reasonable that the rotational symmetry of a double method means that the transition from the lead end (counting places from the front) to the half lead (counting places from the back) is the same as the transition from the half lead to the lead end. For another example, take Buckfastleigh, whose place bell order is +4. Dividing by 2 gives the half lead place bell order as +2. Consider 7th place bell. +2 means that it becomes 3rd place bell at the half lead, which counting from the back is actually 6th place. Checking the diagram, we can see that indeed 7th place bell ends up in 6th place at the half lead.

It works for the second half of the lead too. For example in Bristol Royal, the place bell order is +1, or equivalently -8 as an even number. Dividing by 2 gives the half lead place bell order as -4. Consider the 3rd, which leads at the half lead and is therefore 10th place bell at the half lead (as before, counting by distance from the treble). From 10th place bell, a place bell order of -4 (reverse alternate Cambridge) takes us to 2nd place bell, which is exactly where the 3rd finds itself at the next lead end.

Double methods are not the only ones with Plain Bob half leads - Yorkshire is another example. What I have described also works for Yorkshire, but changing the lead end place from 2nd to 8th spoils it. I think it works for Yorkshire just because the row at the half lead is the same as in Superlative. In general it's just a technique for double methods.

If anyone knew this already or knows where it's written down, let me know in the comments.