[r-t] FW: Krystals and connections with change ringing

Philip Earis Earisp at rsc.org
Tue May 10 11:29:54 UTC 2011

I've now had a reply from James - see below.

Following James' request I'll point him in the direction of further websites showing ringing polyhedra, eg:



Are there any other relevant links that I've missed?

If anyone has further ringing-related information for James, I suggest contacting him directly and cc'ing in this list.

From: James Ingram [mailto:j.ingram at netcologne.de]
Sent: 09 May 2011 10:49
To: Philip Earis
Subject: Re: Krystals and connections with change ringing

Dear Philip,

Thanks for the email! Fascinating indeed!. Great performance -- the audience obviously appreciated it!

I don't have any background in bell ringing, but I remember now that someone pointed out the connection when I first published the article in 1999.

The underlying connection has, I think, to do with Becket's frogs. These are like 3 free-swinging pendulums (=big bells).

I came at this in the early 1970s while grappling with a problem the early serialists (Webern etc.) had left us with: 12-tone music theory really depends on one being able to remember when each pitch last happened, but the notes don't have enough individuality to be clearly memorable. So I wanted to work with multiple occurrences of particular notes, because its then easier to remember them. Or something like that...
I'm currently working with notes (=events =bell rings) which have more "personality"... like bird tweets... and am currently rather excited about the possibility of building larger forms around this principle... rather than around things like melodies or "central pitches".

I was really really interested to see the graph at
and the corresponding polygons at
I also tried to work something like that out, but didn't really succeed. -- see the contour diagrams at the bottom of my article. I did A-level maths during the late 1960s, and investigated group theory during the 1970s. I can remember the notations for permutations in use at the time, but it didn't really get me anywhere...

However: knowing the proximity of contours is important for developing rules of counterpoint in polyphonic music. Bell ringing changes are homophonic, but there must be a way to create closely related polyphonic hand-bell music - in which chords and counterpoint are allowed... Groups of notes are more memorable than the ring of a single bell.
The above links show the proximities of domain 4 contours. Are there similar diagrams for domains 2, 3, 5, 6, and 7?

Greetings to your mailing list! Please forward this post to them if you like. :-)

Best wishes,

Am 08.05.2011 16:32, schrieb Philip Earis:
Dear James,

A musician friend has alerted me to your website, which I have found very interesting. I am not a musician, but am an active ringer and composer for church bells in the "change ringing" tradition, and based in Cambridge, England.

Part of your page on krystals has strongly provoked me interest.  Here you show two diagrams relating to linear fields and fields with foci distributed around a complete circle:

[cid:part1.01070407.09050108 at netcologne.de]

[cid:part2.07000703.05060803 at netcologne.de]

The language you use has a different vocabulary, but both these diagrams relate to change ringing in ways that are both fundamental and contemporary.  The graphical representation you have used in these two diagrams is very familiar to ringers, and so I wonder if you have experience of bellringing / change ringing yourself?  Either way, it is very exciting to see how fundamental ideas can have applications in very different forms of music.

If you are not familiar with change ringing, a reasonable explanation can be found at <http://en.wikipedia.org/wiki/Change_ringing>. Change ringing originated in the early 17th century, and continues to be widely performed and actively developed today. It takes place in towers where there are typically 6, 8, 10 or 12 bells, tuned to a diatonic major scale (though change ringing also takes place on handbells also, and I'm especially keen on this). In ringing notation, each bell is represented by a number: the core nature of change ringing is that sequences of permutations of all bells are rung to a constant, even rhythm accordingly to several well-defined constraints, including not repeating any permutations ("changes").  As such, composing tends to have a "mathematical" component (to meet the rigid formalisms, incorporate various symmetries etc) as well as an "artistic" component (trying to produce as an attractive arrangement of changes as possible within the constraints).  Lots of the area of mathematics known as "group theory" was independently discovered and formalised by change ringers in the 17th and 18th centuries before mathematicians got in on the act, but that's a separate story...

The diagram on your krystals page relating to the field with foci distributed around a complete circle is instantly recognisable to change ringers as the sequence we call "plain hunt".  This is the fundamental building block of all change ringing, as has been rung very frequently from the mid 17th century onwards.

The diagram you show of absolute hierarchies as characteristic for linear fields is especially exciting for me.  This represents a new strand of change ringing composition, called for complex reasons "mega tittums", which I have been actively developing in the past 5 years and which is becoming increasing popular. As an example, take a look at this video of change ringing on 24 handbells in this mega-tittums style, which I composed for a recent notable ringing event: <http://www.youtube.com/watch?v=4-fCRBNTNp0>.  (I'm ringing bells 5&6, ie the third ringer from the left of the picture).

Change ringing has a number of ways of representing compositions, but one is with what we call "grids". The 24 bell composition we rang can be seen in grid graphical representation at <http://www.changeringing.co.uk/wiki/images/7/78/Rwcentenary24belltouch.png>.  The first quarter is very similar to your diagram, except slightly expanded for stylistic reasons.  The middle section keeps the bells in the order for longer, before the conclusion returns them to their starting position.

On your website you also talk about geometric representations of the permutations of elements, pointing out that the "plain hunt" example is a helix projected into the plane of the page. This geometric representation is well known to people working in "ringing theory".  Indeed, many ringing sequences can be represented as various polyhedra, symmetry groups etc - for example take a look at http://plus.maths.org/content/os/issue53/features/polsteross/index and <http://www.ex-parrot.com/~richard/minimus/polyhedra/<http://www.ex-parrot.com/%7Erichard/minimus/polyhedra/>>.

I am excited about the connections between different forms of music, and how you have graphically represented your ideas. It would be good to hear your thoughts on possible links between our respective areas. I run the ringing-theory email list, and I'm sure many of the subscribers there would be very interested in this area also. I also have connections with other (non-ringing) musicians developing musical ideas in similar areas that may be of interest.

With best wishes,



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