Braiding Rainbows

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Quite a few technical terms and jargon words are introduced here. The first time each is mentioned or defined, it is highlighted like this, and in addition is described by an external link to the excellent glossary at

The Basics: Rounds

Because church bells are relatively heavy, and are rung by swinging rather than being hit with a hammer, they can only be made to sound about once every two seconds or so. They are swung in a complete circle from the balance point one way to the balance point the other way, and the sound is made when the freely-swinging clapper catches up with the swinging bell and comes to rest on the metal of the bell just as the bell slows at the top of its swing. This means that, although the speed of ringing cannot be varied very much, the exact moment that the bell sounds may be controlled fairly precisely, the result is that skilled bell ringers can ring sequences of bells to a very regular beat. Traditionally, this starts with a descending scale known as "Rounds" after which the order of bells may be varied while keeping the tempo the same. Also traditionally, the highest note of bell (known as the "Treble") has the number 1, and the descending scale of Rounds on eight bells from Treble to Tenor (the heaviest bell) is written out thus:

1 2 3 4 5 6 7 8


Ringing all the available bells once (in any particular order) is known as a "Row". The order of bells in the row can be varied according to a pattern or "Method", which is applied repeatedly until Rounds is reached again. Because of the weight of a large metal bell, and because it is swinging through a complete circle, from any one sequence of bells (Rounds, for instance), the place that a bell takes in the sequence may only be changed by sounding it one place sooner or one place later in the sequence, or indeed staying in the same position in the row (this latter is called "Making a Place"). If, on each subsequent row, a bell continues to go earlier or later each time, so as to end up at the front or the back of the sequence, this is known as "Hunting", and if it alternately goes before and then after another bell, it is known as "Dodging". According to bell ringing conventions, a bell is not allowed to stay in the same place in the row for very long. Apart from occasionally "making" a single place, it will mostly be continuously Hunting or Dodging. Changing the order of bells by one place (either by hunting or dodging) or making a place are all that it is possible to do in English church bell ringing, but by using these simple moves, complex mathematical methods can be devised to ring long sequences of rows without ever repeating one. The different ways of starting and ending with Rounds and producing sequences of non-repeating changes are thus the basis of English bell ringing, and a source of much of the pleasure people derive from it.

Here is a list of the changes on four bells in which the bells all "hunt" at the same time straight from one end of the row and back again with no dodging, known as a "Plain Hunt":

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

A bell ringing Method consists of a short set of rules by which the order of the bells is varied. Once the set of rules has been applied once, it may repeated many times, and assuming it was started with the bells in Rounds, and if the set of rules is repeated often enough, sooner or later the bells will always get back to Rounds again. Once through the set of rules is called a "Lead", and the same rules repeated enough times to get back to Rounds is known as a "Plain Course". This will normally be a lot shorter than the maximum possible number of combinations on that particular number of bells. In the example above there are eight different rows, even though there are twenty four possible different rows on four bells. In order to ring more rows than occur in a Plain Course, the bell ringing conductor will make "Calls" at particular moments during ringing. A Call (usually "Bob!" or "Single!") causes the ringers to vary the rules of the Method in such a way as to start a different set of rows. By judicious use of calls, it is possible either to ring every possible combination on that number of bells, or to ring a "Touch" of any desired length, in each case without repeating any of the sequences of bells, and also in each case returning to Rounds again before stopping.

The Grid

For a bell ringer to be able to learn a new method, it needs to be in a written form, and there are a number of ways of doing this. One way of making information about a method available is by use of a "Grid". The Grid for a method is produced by applying the rules of the particular method successively to produce a list of rows in the same way as for the Plain Hunt above, and then joining all the same numbers with lines. Here is the grid for one Lead of the method known as Cambridge Surprise on eight bells:

You may notice that the Treble bell (1) ends at the same place that it started but that none of the other bells do. This is no accident. The Treble follows a regular pattern that will repeat itself every Lead, and is marked here with a red line. Each of the other bells follows a different path and ends at the position started from by another bell, forming a cycle through all the non-Treble bells 2 to 8, although not in that order.

The Blue Line

Another way of writing out and learning a method is known as the "Blue Line", which consists essentially of a complete Plain Course written out as separate Leads with a different bell line from the Grid picked out on each Lead:

[Diagram from Blueline software at]

In the diagram above, the path taken by the eighth bell is highlighted in blue and each lead follows on from the previous one. Because each bell ends a lead at a different place from which it started, every bell will end back in Rounds having followed each possible line exactly once.

Learning the Blue Line for each bell is in fact the most usual way for a bell ringer to choose to learn a new method, and this is normally done from a diagram similar to the above.

Place Notation

There is yet another way of describing a method on a particular number of bells. It is very succinct and efficient, and is the way in which Methods are officially specified. Starting with the Grid for Cambridge Surprise again, and ignoring all Hunting and Dodging, we are left with just the "Places Made":

If we write the position of any Places Made in a column next to the grid, and an "x" against each row where all the bells swap with their neighbours, we get a concise way of describing a method. If we add the places made at the "Half Lead" and "Lead End" , and omit the second half (which is a reversed copy of the first half), we have enough information to describe the whole method on that number of bells:

So, one form (it can vary) of this "Place Notation" for Cambridge Surprise on eight bells as above is:

& x 38 x 14 x 1258 x 36 x 14 x 58 x 16 x 78 + 12

where the "&" means that the following notation is repeated in reverse order apart from the last item, and the "+" means that the notation following the plus sign is not reversed. It should be remembered that different Place Notation is required to describe the same method on a different number of bells.

A convenient form of Place Notation, particularly useful for computer input, is to exclude "External Places", in other words not to include whenever the first or last bells make a place, as these can easily be worked out. It is also sometimes convenient to include the full notation and not to compress it with "reflecting" sections. Cambridge Surprise in this form for eight bells would look like this:

x 3 x 4 x 25 x 36 x 4 x 5 x 6 x 7 x 6 x 5 x 4 x 36 x 25 x 4 x 3 x 2

This is particularly interesting because it allows the pattern of "places made" near the Treble bell to be seen quite easily:

x 3 x 4 x 25 x 36 x 4 x 5 x 6 x 7 x 6 x 5 x 4 x 36 x 25 x 4 x 3 x 2

Place Notation for twelve bells uses the symbols "9", "0", "E", and "T" for the bells numbered nine, ten, eleven, and twelve respectively. Again by removing "external" places as above, we can also see a similar pattern for the same method on twelve bells:

x3 x4 x25 x36 x47 x58 x69 x70 x8 x9 x0 xE x0 x9 x8 x70 x69 x58 x47 x36 x25 x4 x3 x2

It should be noted that although some methods lend themselves well to this kind of analysis, a great many do not.