[r-t] New mirror principles

[Richard Pullin]

 Methods with mirror or horizontal symmetry are still quite rare. These
involve pairs of bells ringing mirror blue lines of each other in perfect
unison. To achieve this you must use horizontally symmetric place
notations, which in Minor are limited to 16, 34, 1256, X. The plain course
of Kidderminster Minor sweeps up all 48 of the mirror rows on six bells -
the mirror pairs being bells 16, 25, 34 - and to produce a 720 you simply
need to visit all 15 of the different possible pairings.

I think principles make the most elegant mirror methods as treble-dominated
methods require the treble and tenor to both be hunt bells, which seems a
bit bizarre.

On eight bells there are 384 mirror rows, the permitted place notations
being 18, 36, 1458, 1278, 3456, 123678, X. The nicest of these are 18, 36,
X, but it seems you can only attain 192 rows using just these place
notations, in much the same way that you can only ring 60 rows in Doubles
without using single changes.

I produced what is probably the simplest principle to use these three place
notations, and called it Horizontal Major:
https://complib.org/method/40781?accessKey=97137a1a6d26fa1ec7463700b5fb092c394d4856

Compositeur generated some more principles, one of which I selected and
called Mirror Major as it has palindromic symmetry in addition to the
horizontal symmetry (this is what distinguishes horizontal from mirror
symmetry):
https://complib.org/method/40782?accessKey=4bd6cbe546d92baa55773f4afc982889190a13df

Building on Horizontal Major, I added two 1458 PNs in each lead to produce
a principle containing all 384 mirror rows, and called it Double Factorial
Major:
https://complib.org/method/40789?accessKey=32b4c8bd2455dfb1bab2ea528d4d5a021a9284f7
Again this does not have palindromic symmetry, but I don't think that's a
shortcoming as the whole point of the exercise was to appreciate
alternative types of symmetry.

Bobs-only 720s of Kidderminster Minor are possible because an odd number of
courses is required for an extent, meaning that you simply need to add
normal q-sets of 3 bobs. This is an incredibly rare feature. Fortunately a
bobs-only 40320 of Double Factorial Major is possible for the same reason,
an extent being 105 courses. I set to work producing one and the result is
below. I imagine this might be the first bobs-only extent of Major (using
'normal' bobs that affect three bells, at any rate) and probably the first
where adding q-sets of 3 bobs was the sole technique for constructing the
extent. Frustratingly there were two remaining courses I couldn't q-set in
and it took me a few more days to shuffle things round and finish the
composition. It was very interesting having to deal with courses that were
defined by pairings of bells rather than courses defined by a unique course
head.

40320 Double Factorial Major
1  2  3  4  5  6  7  8  12345678
--------------------------------
      2        -     3  72145386
3              -        82145367
         2        -     82751463
   -        3           38751462
   2              -     82351764
         2              82513764
   -        3           48513762
   2     -        -     82451367
         -  3  -     3  62145378
      -              -  12356478
                  2     12456873
   -              -  -  31264578
      2              3  51364278
      -              -  31245678
--------------------------------
3-part.
Bob = 34
Double Factorial Major:
18.36-36-18-36-36-36-18-36-36-36-1458.36.18.36.18-18.36.18.36.18-18.36.18.36.18-18.36.18.36.18-1458-36
Link to composition: https://complib.org/composition/76406

The method is named after the double factorial number sequence in
mathematics, which is what determines how many mirror rows there are for
each number of bells.
This is like the normal factorial sequence but uses only odd or even
numbers. So 6!! is 2 x 4 x 6 = 48, which is why there are 48 mirror rows on
six bells. 7!! is 1 x 3 x 5 x 7 = 105, which is why an extent of Double
Factorial Major has 105 courses, and why extents for other even-bell
methods that contain all mirror rows in their plain courses always require
an odd number of courses in their extents.
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