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

[Mark Davies]

Mark Simos writes,

> Mysteriously, each such swap (for example, the six swaps
 > taking one of the special sequences X or X' to its
 > six satellites in the other set) seems to seek out exactly
 > where a 4 would sneak in were one to do the swap and NOT
 > transpose the row.

Yes, I think this is expected, isn't it. It's because we are only 
considering swaps which generate another valid delta sequence. Valid 
delta sequences correspond to rows which do not have any repeating 
chords, so by definition our delta swap, if it causes just one chord to 
change, must select the only chord we were missing. Otherwise, we'd get 
a repeating chord. Since our missing chord is always the 4, that's the 
one we always swap with.

It's a direct consequence of the fact we are only selecting certain pair 
swaps between certain delta sequences - precisely the ones, in fact, 
which preserve the property of "six chords with no repeats".

> Listening to the effect of a delta swap, the obvious contour
 > repetition may be somewhat disguised by this effect. In a sense
 > we are really hearing a contour of 'root tones' rather than one
 > defined by voicing choices (bass lines).

I read the very interesting musical analogies, but am still confused! 
Does a delta swap (followed by a rotation to drop out the 4 again) sound 
good, then, or not?!

Turning to the dual pinwheels and their central "X"s:

> what is the order in which you are arranging these, by which the
 > rejected pair associations 'fall out' as bad neighbors?

Yes, there is a natural ordering. Taking just one set, {A..F} together 
with its central hub X', we have this pinwheel:

                   A
               F       B
                   X'
               E       C
                   D

X' can reach to any of A-F with a single pair-swap, so that provides the 
perfect symmetry of the pinwheel. But it also introduces relationships 
between the satellites A to F.

The pairs at opposite sides of the wheel (e.g. A,D or B,E) are produced 
by pair swaps at opposite parts of X'. Suppose X = abcdef, and A = 
(ba)cdef. Then D is abc(ed)f. The neighbours to a satellite (e.g. for A 
this is B and F) are generated by neighbouring pair swaps on X. Again if 
A = (ba)cdef, then B is a(cb)def and F is (af)bcde. Of course that is 
why neighbours cannot be reached by pair swaps: two overlapping pair 
swaps generate a triplet transposition.

Here's the complete ordering according to this scheme:

      X' = abcdef
      A  = bacdef
      B  = acbdef
      C  = abdcef
      D  = abcedf
      E  = abcdfe
      F  = fbcdea

This fits nicely, and is preserved in the chord/EIR world, as you can 
see if you use the "chord link" (the double pair-swap taking A->D' etc) 
to swap to and fro between the forward and reverse sets. Because of 
this, it's pretty easy to navigate between the two pinwheels using the 
valid pair-swap transitions we have. In fact, there's a nice way of 
doing it which has the X and the X' at opposing ends of the "extent", 
and only uses the chord link, the X and X' links, and one type of in-set 
delta link (the one taking A->C not A-D). I won't however spoil the fun 
of the discovery of this!

> Since my 'tour' needs to visit all 84 (14 EIRs with their 6 rotations)
 > I'm still struggling to understand the patterns of how various swap
 > and delta-swap linkages affect rotational positioning

Yes, in my analysis I'm not bothering to look at it like that. I'm 
considering all 6 rotations of an EIR (or of a delta sequence) to be one 
and the same. In my head, an EIR doesn't look like a linear sequence, 
but instead a bracelet, the beginning joined to the end in a circle. A 
round block, as we'd call it in ringing.

If, musically, you want to enumerate the rotations separately, then I 
agree this is more difficult, and I doubt whether you do have all the 
links you need (with just pair-swaps on EIRs or deltas). I will leave 
this as an exercise for the reader! However, to keep you sane, note that 
any rotation of an EIR maps to the same delta sequence, and vice versa. 
(This is obvious if you think about the EIRs - a rotation of an EIR 
doesn't change the order of the deltas).

> But why do most of these (4 of the 6) map directly to the deltas,
 > while 2 require lining up with a rotation? And, each such linkage
 > brings us to  an actual EIR rotated to a different starting chord.
 > Avoiding the 4, and avoiding a repetition of the 6 which starts the
 > special EIR, it feels like (can't say why!) there should be each
 > of the 5 starting tones available

Well, I suppose this is just a consequence of the fact we are starting 
with a particular rotation of the deltas. There is no real reason to 
consider every delta sequence from the +1. I'm sure if you considered 
every rotation of the deltas then it would all even out.

MBD