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

Mark Davies mark at snowtiger.net
Sat Jun 19 13:38:09 UTC 2010

Mark wrote,

> Interesting - hadn't thought of defining the swaps on the deltas!
 > I'm not quite clear what that would mean either mechanically (as
 > an actual ringing pattern) or musically (in terms of motivic
 > transformation).

Well a single pair-swap of a delta will in theory change only one chord, 
however this will always have the result of bringing in the forbidden 4, 
so in order to preserve the original set of chords a rotation of the 
starting chord will be required, affecting the whole sequence.

For example, suppose you had a delta sequence starting +1, +3 giving 
initial chords 1 2 5. Swap these deltas to give +3, +1 and the chords 
become 1 4 5. Only one has changed, but you've got the forbidden 4, so 
need to pick a new starting chord. Using the "4 avoidance rule" you 
might end up with something like 2 5 6.

I don't know whether pair-swaps of chords or their deltas will really 
give you the musical effects you're looking for, but I guess there's no 
harm trying!

By the way, it turns out you can't get between all the delta sequences 
using pair-swaps only. The deltas, just like the EIRs, split into two 
sets of six sequences, with each sequence in the first set having a 
reverse in the second; additionally there are the two "special" 
sequences which are also reversals of each other. Let's call the first 
set {A, B, C, D, E, F} and the reversals {A'...F'}, with the special 
sequences labelled X and X'.

Looking at the EIRs first, each sequence in the first set can be linked 
by pair swaps to one and only one other sequence, which is in the 
reversals set but at the "opposite end" to the actual reverse; the 
special EIRs can't be linked to anything. In other words, the only pair 
swaps are between:

       A, D'
       B, E'
       C, F'
       D, A'
       E, B'
       F, C'

This is because the chord sequences A-F are all related to X by a 
triplet rotation, and similarly for the reversals. In other words, if X 
is abcdef, then we can get to A by rotating abc in one direction only: 
bcadef. But a triplet rotation plus a pair swap is enough to completely 
reverse a sequence of three: abc -> bca -> cba. So if we start at the 
reversal of X, fedcba, which is X', we can rotate the other three chords 
to give D', edfcba, and then A and D' must only be two independent 
pair-swaps away from each other:

     X = abcdef
     A = bcadef  (rotated abc)
         cbaedf  (swapped bc, ed)
     D'= edfcba  (same sequence, starting at e)
     X'= fedcba  (rotated def back)

Turning to the deltas, it turns out that a rotation abc->bca always 
corresponds to a reversal of four deltas, e.g. abcd->dcba. This looks 
much more symmetrical in the delta world! And it means the pair swaps 
link completely different sets.

Now the special sequences X and X' directly link to all the sequences in 
the opposite set, i.e. there is a single pair-swap taking X to any of 
A'-F'. That's because, if you start at X', fedcba, and reverse any set 
of four, e.g. cdefba, you get one of the A'-F' sequences, but are only 
one pair-swap away from reversing X' completely to get X. So we can link 
to X just by swapping the remaining pair:

     X'= fedcba
     A = cdefba  (reversed cdef)
         cdefab  (swapped ab)
     X = abcdef  (same sequence, starting at a)

It also means that, within a set, there are many sequences which are 
joined by double pair-swaps. For instance, A links to C, D and E by pair 
swaps. Only its neighbours B and F are unavailable (because the pair 
swaps would overlap, creating a triplet rotation).

To summarise, in the EIR world we have just six connections, from one 
each of the {A..F} set to one in the {A'...F'} set. In the delta world, 
we have connections within the sets, and from X' to the {A..F} set and X 
to the {A'..F'} set, but no connection between the two sets.

The obvious thing to do is to put both types of connection together, 
i.e. use both EIR and delta swaps, and you can roam freely across all 
fourteen sequences!


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