[r-t] FW: A compositional question I am hoping a change-ringing theorist could help with! (re-sending)

Mark Davies mark at snowtiger.net
Wed Jun 9 23:17:42 UTC 2010

This is interesting. There is a relationship amongst pairs of numbers 
here. There seems (to my mind) to be two special rows in Mark's list 
which have the natural ordering for the set of numbers, based on the 
following circle:

           2     3
           5     6

If you go one way round the circle, you get 136752, which is the second 
row in Mark's left-hand column, and if you go the other way you get 
125763, which is the second row in the right-hand column, i.e. the 
reverse of the other. Presumably the reason for this is that some chord 
pairs (e.g. 1 and 3, 3 and 6 etc) have a natural affinity for each other.

The remaining twelve rows can all be produced by taking any neighbouring 
triplet in the circle (of which there are six: 136, 367, 675, 752, 521, 
213) and rotating it one way. For instance, take 675 and rotate to give 
756. Then you can read off the two new rows (in each direction round the 
circle) to give 137562 and 126573, which form a pair in Mark's list.

In fact, if you rotate any of the triplets abc -> bca, where we are 
reading clockwise round the circle above, you get a pair of rows in the 
list. Add the six pairs produced by these six rotations of triplets, to 
the one pair produced by the natural order, and you get the seven pairs 
he describes.

But the rotations the other way, for instance 675 to 567, don't form 
pairs of rows in his list. Why is this, when his rows seem otherwise 
symmetrical under reversal? Has he just missed rows such as 135672, or 
is there some underlying musical reason for the symmetry violation?


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