In change ringing, a row cannot contain any repeated bells and consecutive rows in a touch are related by swapping adjacent bells. We could extend this pattern to sequences of touches, which I will call "hyperbells"; a touch cannot contain any repeated rows and consecutive touches in a hyperbell are related by swapping adjacent rows. (Well, in common usage, a touch is a sequence of rows possibly containing multiple extents, but I'd like to reserve the term extent to say "extent of bells" or "extent of rows" or "extent of touches", and so on.).
Some row swaps are clearly illegal because they produce jump changes. For example, consider the following sequence of rows. (With abbreviated place notation.)
1234
x 2143
1 2413
x 4231
If we swap the middle two rows, we get
1234
? 2413
1 2143
? 4231
with the 1 and 4 bells performing jumps. There are only 6 distinct "methods" with four row "leads", in which swapping the middle two rows is valid. These 4-rows enumerate all the rows reachable with just odd-even swaps, so they must be followed with a hunt. I've given them nicknames below:
"Oxford" (O)
1234
x 2143
3 1243
x 2134
1 2314
"Kent" (K)
1234
3 2134
x 1243
3 2143
1 2413
"Canterbury (Place)" (C)
1234
2 1243
3 2143
2 2134
1 2314
As well as their reverses (rO = x2x1, rK = 2x2.1, rC = 3.2.3.1).
All the methods contain the same rows disregarding order, except for the lead head. Since this is the only difference, an extant would have a lead head for every possible pair of bells in the front two bells. That is to say, one lead head should have 1 and 2 in the front in any order, another should have 1 and 3, another 1 and 4, 2 and 4, 3 and 4. So there would be 6 "leads" in an extent. Below I tabulate the lead heads.
| LH | cyclic notation | methods |
|---|---|---|
| 2314 | (132) | O, C |
| 2413 | (1342) | K, rK |
| 1423 | (234) | rO, rC |
When you swap the middle two rows, you get the other method with the same lead head.
I've generated one extent so far, O rO O rO O rO, (which is equivalent to Double Bob), and if we swap all the middle-rows in the extent, we get C rC C rC C rC (which is equivalent to Double Canterbury Place Minimus). You cold also start with the reverse methods. Although these are equivalent to existing methods, I want rounds to be the 1st row, 4th row, or lead head of each lead (as w/rt the ordering above), because otherwise we would have to swap between consecutive "touches" (i.e. one touch will be one row short and another will be one row long). This could be considered a parallel to the rule of not swapping over the hand/backstroke. Ix think there may be other extents.
I've forgotten to account for one kind of hyperchange: neighboring swaps on rows. On higher stages this may be possible, but I think this does not work on minimus. For example, if a bell rings in the positions 1, 2, 3, 2, then swapping the first and second rows and the third and fourth rows produces the position-sequence 2, 1, 2, 3, which has no jumps. However, on minimus, the path of the other bells is limited by truth and because they are hunting, so in rows, we have
1234
21ab
2a1b
21ab
This is false because a .14. place is made twice.
Let me consider every possibility. Notated by relative position, on an infinite stage, the possible four row lines are: 1232, 1222, 1221, 1212, 1211, 1122, 1111; and their horizontal, vertical, and diagonal reflections.
Of these lines, in minimus the line can force falseness in minimus, and the cross hyperchange can force a jump in the lines of other bells. So by absolute position in minimus, only these lines are prima facie legal: 1221, 3443, 1212, 3434, 1122, 3344. However, even these lines are legal once you consider the row preceding and the row following the four rows. For example, consider the 1221 line. To enforce legal lines in the other bells and avoid falseness, the following is the only possible row sequence:
1234
21ab (a and b could be either 3 or 4... it is unimportant)
21ba
1243
CopyTo avoid falseness, this must be preceded and followed by a .14., producing:
1324
1234
21ab
21ba
1243
1423
When we perform the row swap, we get
1324
21ab
1234
1243
21ba
1423
Which has jumps in the 2 bell. The same observation can be made for the other lines. Also, the legal lines above do not cross between positions 2 and 3, so we cannot both join multiple legal lines and have a .14. place. The best one could do is have a 4-row "touch", where the rows behave like bells. However, one issue is that rounds could swap with another row, which would make the touch seem false. If you didn't permit rounds to swap, then we could have a 3-row touch on the remaining 3 rows.
How many rows is a hyperbell? With the first construction, there are 6 leads in each touch, and two possible methods for each lead. So there are 26 distinct touches in a hyperbell, and each touch is 4! = 24 rows long, for a total of 4! * 26 = 1536 rows, which is enough for a quarter peal. With the second construction, there are 4 or 3 rows per touch and 4! or 3! touches per hyperbell. We could apply the first construction with touches replacing bells, so there would be 26 hyperbells per hyperrow, which would be 1536 * 4 = 6144, enough for a peal. But there would only be 4 distinct rows in this peal.
Going further, one could try to make a hyperrow of hyperbells. A hyperbell cannot contain any repeated touches and consecutive hyperbells in a hyperrow are related by swapping adjacent touches. In the first construction, since any touch can go to any other touch (because each lead contains just one swap), hyperbells can be swapped freely, like bells and unlike rows. The size of a single hyperrow is the number of possible hyperbells, so there are 26! ≈ 1089 rows.
I should mention that a hyperbell is not an extent of touches, only of touches that can be reached by swapping rows. I think this is a consistent extension, as the bells constituting a row can vary, e.g. one can ring a method using nonconsecutive bells, especially on lower stages. Similarly, a hyperrow does not have to include every hyperbell, just as a peals on 8 or more bells don't generally make an attempt to be an extent. But I don't think anything would count as "music" when composing a hyperrow.
I made a Python script to run through all the possible touches. I had a bug with how the sympy library notates permutations, so I'm not confident that my results are correct, but here they are anyway:
as well as their horizontal reflections (replace all O with rO and vice versa), vertical reflections (reversing the order), combinations thereof, and also swapping out individual leads with the other method that shares the lead head, totalling 8 * 26 distinct touches. I have yet to analyze the music on each pair.
Actually, you can reduce it further to 2 touches: O K O rO K rO and O rO O rO O rO; and then generate the other 6 touches with just vertical translation (starting rounds at different leads). Additionally, you can start rounds at either the lead end or the lead head (where rounds wouldn't be affected by row swaps in the hyperbell), so there are 16 hyperbells (distinct w/rt the set of touches they contain), and 16 * 26 distinct touches.
Also see: