# Change Ringing

The bells in the bell towers of churches are big and heavy. This limits how quickly they can be rung, making it impractical to use such bells for playing melodies.

Change ringing is a technique devised to permit a varying sequence of notes to still be played within these limitations.

Let's say one has three bells. One can begin by ringing them in order from the highest to the lowest: 1 2 3.

If one does that repeatedly, then each bell rings regularly in three beats. Given the limitation on how the bells can be moved, with effort, perhaps they could be sped up or slowed down to ring again after two or four beats instead, one less or one more.

Doing that, one can have a series of notes like this:

```1 2 3
X  |
2 1 3
|  X
2 3 1
X  |
3 2 1
|  X
3 1 2
X  |
1 3 2
```

This goes through all six possible orderings of the three bells.

One way to go through all twenty-four orderings of four bells is as follows:

```1) 1 2 3 4    9) 1 3 4 2   17) 1 4 2 3
X   X         X   X         X   X
2) 2 1 4 3   10) 3 1 2 4   18) 4 1 3 2
|  X  |       |  X  |       |  X  |
3) 2 4 1 3   11) 3 2 1 4   19) 4 3 1 2
X   X         X   X         X   X
4) 4 2 3 1   12) 2 3 4 1   20) 3 4 2 1
|  X  |       |  X  |       |  X  |
5) 4 3 2 1   13) 2 4 3 1   21) 3 2 4 1
X   X         X   X         X   X
6) 3 4 1 2   14) 4 2 1 3   22) 2 3 1 4
|  X  |       |  X  |       |  X  |
7) 3 1 4 2   15) 4 1 2 3   23) 2 1 3 4
X   X         X   X         X   X
8) 1 3 2 4   16) 1 4 3 2   24) 1 2 4 3
| |  X        | |  X
9) 1 3 4 2   17) 1 4 2 3
```

After each group of eight pulls, a change in the regular sequence, called a bob, takes place.

In this pattern, bell 1 follows what is called a plain hunting-course; it goes from sounding first to second, from second to third, and so on, and then back again, sounding twice in the same place only in first place and last place, which, being part of a plain hunting-course, is not termed place-making.

The other three bells approximate such a course, but must deviate from it at the time of the three bobs in the sequence (the third one being required at the end of the peal if one goes back to the start to repeat it again).

The first bob, between the eighth and ninth pulls, causes bells 2 and 4 to dodge, and bell 3 to engage in place-making;

``` 6) 3 4 1 2
|  X  |
7) 3 1 4 2
X   X
8) 1 3 2 4
| |  X
9) 1 3 4 2
X   X
10) 3 1 2 4
|  X  |
11) 3 2 1 4
```

the second bob, between the sixteenth and seventeenth pulls, causes bell 4 to make its place, and bells 3 and 2 to dodge.

Since bell 1 is the bell sounding the highest pitch, it is the lightest one in weight. Thus, if one only had three people present, one person being unavailable, and a stranger happened by, perhaps wishing to use the telephone as his car broke down, he could perhaps allow the other three to practice their change-ringing by taking on the easiest part by ringing this bell.

This is, of course, the opening situation in the book The Nine Tailors, by Dorothy L. Sayers, which brought change-ringing to the attention of the wider public, except that a peal (that is, a complete sequence of pulls) involving seven bells, rather than four, was involved which also had this characteristic.

There are several methods that will go through all 120 combinations of five bells. Let us start with the one called Plain Bob Doubles:

```1 2 3 4 5   1 3 5 2 4   1 5 4 3 2   1 4 2 5 3   1 4 2 3 5   1 2 5 4 3   1 5 3 2 4   1 3 4 5 2
X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |
2 1 4 3 5   3 1 2 5 4   5 1 3 4 2   4 1 5 2 3   4 1 3 2 5   2 1 4 5 3   5 1 2 3 4   3 1 5 4 2
|  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X
2 4 1 5 3   3 2 1 4 5   5 3 1 2 4   4 5 1 3 2   4 3 1 5 2   2 4 1 3 5   5 2 1 4 3   3 5 1 2 4
X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |
4 2 5 1 3   2 3 4 1 5   3 5 2 1 4   5 4 3 1 2   3 4 5 1 2   4 2 3 1 5   2 5 4 1 3   5 3 2 1 4
|  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X
4 5 2 3 1   2 4 3 5 1   3 2 5 4 1   5 3 4 2 1   3 5 4 2 1   4 3 2 5 1   2 4 5 3 1   5 2 3 4 1
X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |
5 4 3 2 1   4 2 5 3 1   2 3 4 5 1   3 5 2 4 1   5 3 2 4 1   3 4 5 2 1   4 2 3 5 1   2 5 4 3 1
|  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X
5 3 4 1 2   4 5 2 1 3   2 4 3 1 5   3 2 5 1 4   5 2 3 1 4   3 5 4 1 2   4 3 2 1 5   2 4 5 1 3
X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |
3 5 1 4 2   5 4 1 2 3   4 2 1 3 5   2 3 1 5 4   2 5 1 3 4   5 3 1 4 2   3 4 1 2 5   4 2 1 5 3
|  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X    |  X   X
3 1 5 2 4   5 1 4 3 2   4 1 2 5 3   2 1 3 4 5   2 1 5 4 3   5 1 3 2 4   3 1 4 5 2   4 1 2 3 5
X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |    X   X  |
1 3 2 5 4   1 5 3 4 2   1 4 5 2 3   1 2 4 3 5   1 2 4 5 3   1 5 2 3 4   1 3 5 4 2   1 4 3 2 5
| |  X  |   | |  X  |   | |  X  |   |  X  | |   | |  X  |   | |  X  |   | |  X  |   |  X  | |
1 3 5 2 4   1 5 4 3 2   1 4 2 5 3   1 4 2 3 5   1 2 5 4 3   1 5 3 2 4   1 3 4 5 2   1 3 4 2 5
```

After going through the first ten sequences of the five bells, each sequence called a pull, since each person ringing the bells will have pulled on his rope once, a change in the pattern, called a bob, takes place, to avoid returning to the initial sequence 1 2 3 4 5.

After the fourth ten, the same bob would result in the first forty pulls repeating, so a different variation is applied instead.

To keep the second forty pulls from repeating, the same alternate bob is used after the eightieth pull; the final forty pulls are not depicted above.