'Tis the season for bells and churches; and bells and churches immediately bring to mind one of the great musical arts -- change-ringing. It's certainly the noisiest of the musical arts, and one of the most purely mathematical.
Change-ringing is governed by the fundamental musical quirk of those huge church-bells in the church-tower: they have such a huge momentum, they aren't suitable for melodies. What you can do, when you have several bells, is ring them in different orders; and that's precisely what change-ringing is: the exemplification of a mathematical permutation by changes of bells. For instance, suppose you have a church with four bells. The bells are numbered from the highest-sounding bell (the treble) to the deepest-sounding bell (the tenor):
1 2 3 4
This particular order of bells is called ringing rounds. We can then change the rounds. And the series of numbers we list to keep track of the bells is the simple, straightforward musical notation of change-ringing. For instance:
1 2 3 4
2 1 4 3
2 4 1 3
4 2 3 1
4 3 2 1
3 4 1 2
3 1 4 2
1 3 2 4
1 2 3 4
Do you see a pattern in the bells? If you don't, pick a bell and trace out where in the order it is found. This type of pattern, which guarantees that you go from rounds to rounds without repeating a row, is called ringing according to method, or ringing by method (the particular method found above is usually called the Plain Hunt). And, obviously, the more bells you have, the more changes you have in method ringing. When you ring through 5040 changes, without break and without repeating a row, you have one peal. (The number is chosen because it's the number of total possible permutations -- called an 'extent' -- you have with seven bells.) A peal lasts several hours; a quarter peal lasts about forty minutes. From what I understand, there was once an extent on eight bells -- ringing all eight bells each change, without repeating, from round to round; it lasted twenty hours. Obviously there are many extents that have never been rung; an extent with sixteen bells would take well over a million years.
Although change-ringing is designed for tower bells, you can ring changes and peals on handbells, as well, and, from what I understand, change-ringers who work with tower bells usually practice their patterns on handbells (for the obvious reason that practicing with a tower full of very loud bells tends to annoy the neighbors).
Now, it may not seem likely that just ringing bells in different permutations would leave much room for creative composition, but there are so many different ways you can ring bells, and some of them sound so much better than others, that this isn't so. For instance, you can have any number of different bells, so there are terms used in the names of compositions that indicate the number of bells:
(You may wonder why five bells gives you doubles and seven bells gives you triples; nine and eleven have names of the same sort, although it's not so obvious for nine. The reason is that it's the number of bells that can change position in the row at every change.) A plain course occurs when you just ring a pattern, but you can also, during the ringing, call on particular bells to make changes in the pattern. The rigid rules are still in place -- you still can't repeat rows, and you still ring from rounds to rounds. But not every pattern goes through the full extent of the bells you have; so you can add a variation or divergence from the plain course. This divergence from a plain course is called a touch; the commands for bells to change have different names depending on what is required. In addition, the patterns themselves can vary somewhat. I gave the Plain Hunt above, in which the bell continually 'hunts', i.e., goes straight from back to front or front to back; but you can add a Dodge. In the following pattern 3 is dodging with 5:
4 1 3 2 5
1 4 3 2 5
1 2 4 5 3
2 1 4 3 5
2 1 4 5 3
4 2 1 5 3
Further, many important changes have their own names. This is a hagdyke, for instance: 12563478.
The following websites are interesting resources for change-ringing:
* Thanks the North American Guild of Change-Ringers, you can hear and see some changes rung.
* You can also listen to change-ringing with handbells (some of it very beautiful) at this website.
* The Glossary at The Change Ringers Web Directory is the single most useful resource on the web for those who are lost when it comes to change-ringing terms.
* Minor Strikerz is designed for young ringers
* Want to do a little virtual change-ringing? Kees van den Doel's Bells Applet is a Java program you can use online to try out different methods. I recommend setting the controls to maximum irregularity (if you have a quick computer; if not, try more regularity -- use maximum regularity if you want to get the clearest sense of the ordering) and then toying with some of the others on the left-hand side (you can change tempo, mute bells). Then play with some of the right-hand controls.
For my part, I'd like someday to be able to ring a touch or two in person. Like most people, my acquaintance with change-ringing came through Dorothy Sayers's Lord Peter mystery, The Nine Tailors (one of Sayers's all-time best, which is saying something). 'Nine tailors' is the pattern rung when a man in the parish has died; hence the expression, 'Nine tailors make a man'.
There are other musical arts related to bells, the most famous of which is that of the carillon; a carillon is an instrument involving (at least) 23 bells of a particular sort; it's a sort of bell-organ, so it admits of more melody. You can find out more about it through the Guild of Carilloneurs of North America, and there are a few samples of its music here and here.