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

Philip Earis Earisp at rsc.org
Tue Jun 15 07:54:42 UTC 2010

Hi Mark,

Great - I'm forwarding your new message to the theory list, as requested. 

The thing known mysteriously as "MBD" is actually a chap called Mark Bucephalus Davies, a fine stallion who escaped from Oxford University and now resides in Gloucestershire...


-----Original Message-----
From: Mark Simos [mailto:msimos at berklee.edu] 
Sent: 14 June 2010 20:06
To: Philip Earis; Chris Kippin; Ian Partridge
Subject: Re: [r-t] FW: A compositional question I am hoping a change-ringing theorist could help with! (re-sending)

[Philip - feel free to repost this to the theory list if you feel it is appropriate and not too long...]

Dear Bell Ringer Theorists:

Sorry for my tardy reply - swamped with a project applying the very modal beasties we've been discussing. (Also, I'm not on the theory list, so please cc me directly on any comments, otherwise kind Mr. Earis will need to keep playing switching station!)

I am truly grateful for your attention to my little problem. Many thanks for all the helpful suggestions, especially "MBD" (I'd like a name so I can credit him/her if and when I write this up) who succeeded (see excerpt below) in exactly what I was hoping for: perceiving a symmetry in the relationships of the rows I'd provided that I was not able to see.

Several people have inquired about how I derived the 'interesting subset' of permutations I posed. I had not intended to be coy about the musical derivation of this set - it's all new discoveries for me and I'm not sure whether the result will be compositions to be copyrighted, a compositional method to be authored about, or what. But the arithmetic principle behind the set is fairly simple. Read on if you are interested...

The musical reference is that the numbers represent roots of triads diatonic to a given major-scale mode (skipping the diminished chord) - the example set given (sans major or minor indications) is for roots of a Lydian sequence.  First off, I'm interested in progressions which use all and only these roots, each exactly once. At first cut this suggests 6! or 720 'root tone rows.' (You can think of this so far almost like a diatonic, even modal, version of a serialist's tone row.)

If one thinks of the root tones of chords in a progression as a kind of melody unto themselves, we can look for progressions with certain qualities of intervallic flow and contour. The first subset I explored were sequences where the same interval is never repeated consecutively. Here I treat the 'interval' between two succeeding tones in the sequence by the closest interval = hence 2-5 would be considered an ascending 4th, 5-2 a descending 4th rather than an ascending 5th. Thus there are 6 such intervals available: 2nds, 3rds, and 4ths in ascending and descending forms. So this restriction means: no cascades or runs such as 1-2-3, and no 'arpeggiations' such as 1-3-5 (though inversions like 3-5-1 can occur). Curiously, these disallowed figures are mentioned in an article Philip Earis forwarded to me about perceived musical qualities in ringing changes. Apparently, 'cascades' of scalar sequences or arpeggiated triadic runs suggesting the outlines of chords are perceived by many ringers as particularly  'musical' in effect. So clearly the aesthetic I am after is not to capture conventional intuitions of musicality in a tonal sense. However, my restrictions also disallow movement by fourths (such as 2-5-1), which are less obviously tonal in effect. In any case, I counted 38 such sequences out of 720.

A further subset is of sequences where each of these 6 intervallic transitions shows up exactly once. (This is a proper subset of the 38 above, if you stick to rows that are proper permutations - so, for example, an ascending 2nd cannot be followed by descending 2nd, which would repeat a root). This is where I wound up with 7 sequences and their retrograde forms. For example:

1 3 6 7 5 2 / 1 ...	(illustrated in the reply below) has transitions: (U - up/ascending; D - down/descending)

 U3 U4 U2 D3 D4 / D2

Let's call this latter property the 'exhaustive interval row' property or EIR. (Not to be confused with rows that only appear to have this property, and thus are EIR apparent... sorry)

Since I obtained the set by brute-force enumeration and checking I had no confidence that I had the 7 basic forms in a meaningful order. However, since the EIR property would clearly be invariant under retrograde, I provided the retrogrades as well.

As it happens I mis-stated my original problem slightly. I consider each row as a cycle, so that the interval between last and first tone in the row must obey the same properties, and in fact is needed to complete the exhaustive set of intervals used. Actually the EIR property also remains invariant under rotation (since these are cyclical patterns). So counting retrogrades combined with rotations there are 7*2*6 or 84 EIR sequences. Every row of my seven exists in 12 forms (6 rotations, direct and retrograde forms). I'm pretty sure these are the only 84 EIR-y sequences of the 720 permutations--but I've been wrong already several times! 

I didn't mention rotation in my original post because I was interested in relationships among the 14 sequences that I hadn't seen. That is the very insight which the mysterious MBD brought. To find the configuration he/she described, MBD had to pick from both sides (direct and retrograde) of my original set, and had to employ some rotations as well. My holding the "1" in first position was a nod to modal implications of the rows (a longer discussion). But it turns out this was a blind spot.

This is testimony to the power of finding the right visual metaphor or abstraction for thinking about your problem. I'm not sure why the sequence that MBD placed in the center of the mandala belongs there, why it has special status. But it certainly does!

This configuration has opened my eyes to many new possibilities with the sets. I am now working with a slightly extended notion of a 'jump-tolerant' change between rows. It appears you can't visit the 84 configurations using only one-place swaps (but I'm not sure yet). But because of the 'pinwheel' pattern MBD describes below, you can move between the rows via transitions which preserve 3 of the 6 tones in their positions. This requires using retrograde and rotated forms. So far I've found several ways of doing this 'tour' but so far haven't seen if there is a way to visit all 84 of the forms with one such 'method' (taking the term in its broadened sense here). That's what I'm still mulling on.

Hope this helps explain the set I'm working with and why - and perhaps will lead to some further insights. It would be cool if this all resulted in a ringing method that was actually playable on a set of diatonic bells!


> To: ringing-theory at bellringers.net <ringing-theory at bellringers.net>
> Sent: Thu Jun 10 00:17:42 2010
> Subject: Re: [r-t] FW: A compositional question I am hoping a change-ringing theorist could help with! (re-sending)
> 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:
>              1
>           2     3
>           5     6
>              7
> 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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