Place notation

All change ringing is concerned with taking a certain ordering of bells, called a row, and generating a new row from it according to rules. A method can be thought of as a set of these rules applied in sequence.

In practice there are many different ways that methods can be written down. Most frequently they are represented visually as a blue line diagram, which can aid in memorisation. Some methods can be learned according to a rule such as "first, treble-bob, last, near, full, far..." which expresses the blue line concisely as a short phrase.

However, in order to refer to a specific method uniquely, it is necessary to have a compact way of expressing what exactly every bell does while ringing it. Place notation is a tool invented in order to do just that.

The basic concept

First, let us recall some definitions:

Row

An ordering of the bells 1 to N, where N is the total number of bells being rung. A row is written in ringing order left to right. Each bell rings once and only once within a row.

For example:

• 1 3 5 2 4 6 is a row;
• 9 8 7 6 5 4 3 2 1 is a row;
• 1 1 3 4 7 8 is not a row (why?).

Change

The action of moving from one row to another by altering the order of the bells; or, the relationship between two rows in terms of how they differ. (NB: A change can also be another name for a row, which can be a little confusing, though the meaning is usually clear in context.) In order to qualify as a change, the following condition must be satisfied:

• The two rows must have the same number of bells.

For most of change ringing history, changes were also required to satisfy a second condition:

• In going from one row to another, no bell should move more than one position forward or backward within the row.

Changes which do not satisfy this latter condition are known as jump changes.

Methods using jump changes are still extremely rare, and so the explanation that follows will assume for the moment that we are dealing with changes which satisfy both of the conditions above.

Suppose that we have two rows related by a change:

A: 1 4 2 5 3 6

B: 4 1 2 3 5 6

How can we express the change succinctly?

We might start by making a note of which bells do not change position between the two rows. In the above example, bells 2 and 6 are the only ones which remain fixed across the change. We could try simply writing down the numbers of the bells, but that would not be a sensible solution. To see why, look carefully at this other pair of rows related by a change:

C: 6 5 4 3 2 1

D: 5 6 4 2 3 1

With a bit of scrutiny it should be apparent that the relationship between rows C and D is the same as that between rows A and B: if we were to relabel the bells in row C so as to match row A, then the row D obtained under the same relabelling would match row B.

In other words, the change between rows A and B is the same as the change between rows C and D, even though the rows themselves are different. Our notation needs to express this fact in a fashion which is independent of the specific rows. Therefore, rather than look at the numbers of the bells which do not move between rows, we should pay heed to the position of those bells instead.

Let's look again at the first example, this time making explicit note of how the bells move:

A: 1 4 2 5 3 6
    x  |  x  |
B: 4 1 2 3 5 6

As noted before, bells 2 and 6 are the only ones which stay in the same place, those places being 3rd and 6th within the row. Similarly, going from row C to row D, the bells in positions 3 and 6 are the only ones which don't move:

C: 6 5 4 3 2 1
    x  |  x  |
D: 5 6 4 2 3 1

In change ringing, a bell which stays in the same position between rows is said to make a place in that position. Place notation is so named because it expresses changes exclusively in terms of which places are made between rows.

The fact that the bells which make places uniquely define a change is due to the constraints we placed upon changes to begin with. When changing between rows, there are only two possible behaviours allowed of any bell: it can either remain where it is, or it can swap places with a bell next to it. Therefore, all bells which do not make places must swap with their neighbours, and so this information is in a sense already accounted for by simply noting the bells which do make places.

Place notation elements and strings

To represent a change, we list the positions of the bells which make places between two rows in numerical order. The change in the examples above would be written as 36. This is an element of place notation (sometimes also called a unit): a single change represented in place notational form.

The reason place notation is so useful is because changes on arbitrary numbers of bells can be uniquely defined using single elements, which are very compact and easy to work with. Furthermore, representing changes as strings of numbers allows them to be communicated to a computer.

Here are some more example changes and the place notation elements which represent them:

1 3 2 5 4 6
| |  x   x
1 3 5 2 6 4

Place notation: 12

1 3 5 7 2 4 6 8
|  x  | |  x  |
1 5 3 7 2 6 4 8

Place notation: 1458

1 2 3 4 5 6 7 8 9 0
 x  |  x   x  | | |    
2 1 3 5 4 7 6 8 9 0

Place notation: 3890

There is a special type of change possible on even numbers of bells in which no bell makes a place between rows. For example:

1 2 3 4 5 6
 x   x   x
2 1 4 3 6 5

This change is sometimes called the cross change for obvious reasons, and is notated using the special element - (or x).

To perform a method, several changes must be made one after the other in sequence. To use place notation to write methods, we need a way of concatenating place notation elements together.

A string of place notation is a series of elements written in sequence. To avoid ambiguity about the boundaries between elements, dots . are added between any two elements which are not the cross change. For example, the sequence of changes in which change 14 is followed by change 36, which is followed by change 1458, would be written 14.36.1458.

It is not necessary to use dots to mark the bounds of the cross change: replacing 36 with the cross change in the former example would give 14-1458.

This simple notation is already powerful enough to uniquely express every "conventional" method that has ever been rung. For example, here are some place notation strings corresponding to a single lead of some well known methods:

-16-16-16-16-16-12

Plain Bob Minor

-36-14-12-36-14-56-14-36-12-14-36-12

Cambridge Surprise Minor

-58-14.58-58.36.14-14.58-14-18-14-58.14-14.36.58-58.14-58-18

Bristol Surprise Major

Condensed place notation

Since methods can have leads which are (in theory) any finite length, place notation strings can become rather long and cumbersome when written out in full. However, for the vast majority of methods we may leverage their inherent symmetry to produce a condensed form of the place notation.

The very significant majority of named methods are palindromic, which means that their sequence of changes is the same when reversed. If we examine the place notation of the three methods above we can see that, if we ignore the final elements in each case, the remaining strings are all symmetrical about the element at the halfway point. For example, take Cambridge Surprise Minor:

-36-14-12-36-14-56-14-36-12-14-36-(12)
                ^^

The lead has a symmetry point at the change 56 (and also at the final change 12, if you want to be picky). This means that this place notation string is about twice as long as it needs to be, because all the changes in the second half are already expressed in the first half in reverse order.

We can now introduce the convention that a comma , within a place notation string separates any palindromic substrings. With this convention, the place notation for Cambridge becomes

-36-14-12-36-14-56,12

The full place notation can be obtained from the condensed form by way of the following steps:

1. List the elements before the comma from left to right;
2. List the elements before the comma from right to left;
3. Join the strings obtained from steps 1 and 2 together at the repeated element;
4. Repeat steps 1 and 2 for the section after the comma;
5. Concatenate the strings obtained from steps 3 and 5.

Writing these steps out for our example, we have:

Step 1: List the elements before the comma from left to right.

-36-14-12-36-14-56

Step 2: List the elements before the comma from right to left.

56-14-36-12-14-36-

Step 3: Join the strings obtained from steps 1 and 2 together at the repeated element.

-36-14-12-36-14-56-14-36-12-14-36-

Step 4: Repeat steps 1 and 2 for the section after the comma.

12

Step 5: Concatenate the strings obtained from steps 3 and 5.

-36-14-12-36-14-56-14-36-12-14-36-12

And we are done.

Step 4 seems somewhat trivial in this example because the substring after the comma in the condensed place notation has only a single element. However, it is crucial that both substrings be treated as palindromic for this convention to work. This is because there are many palindromic methods which have their points of symmetry in a different place.

The best example of this is Grandsire, in which the point of palindromic symmetry is offset from the midpoint of the lead. The full place notation for a lead of Grandsire Doubles is

3.1.5.1.5.1.5.1.5.1

which condenses to

3,1.5.1.5.1

Alternative notational conventions

When creating a custom method on Complib, you may use either the full or condensed form of the notation as specified above. You can use either - or x (or a mixture of both!) to represent the cross change. Complib will automatically condense the notation when validating the method, provided that the place notation itself is valid.

There are other conventions for writing place notation, many of which are still very common. For the sake of convenience, here are a few you are most likely to encounter.

Ampersand notation

An ampersand & at the start of a place notation string is used to indicate that the string which follows is palindromic about its last element. This is similar to the comma notation.

For example, &-16-14 represents the same string as -16-14,.

Leadhead change

A place notation element prefixed lh or le, or sometimes simply separated from the main string by a space, is used to denote the leadhead (or leadend) change. This is most often used in conjunction with ampersand notation.

Sometimes the actual leadhead row will be given rather than the place notation element. In order to obtain the terminal element, you will need to work out what change must be made at the end of the lead to get to the specified row.

Leadhead codes

Many commonly rung methods have Plain Bob leadheads, each of which has a special alphanumerical code at a given stage (see Leadhead codes). Some conventions allow the use of a leadhead code instead of a terminal element of place notation. This is almost always used in conjunction with ampersand notation.

For example, the place notation of Cambridge Surprise Minor is sometimes condensed as

b &-36-14-12-36-14-56

which means the full string is palindromic about the element 56, and the leadend change is the element which corresponds to the code b, which is 12. This convention requires knowledge of the leadhead codes and which elements they represent, and so can be a little trickier to use. To learn about leadhead codes and how to use them, see the dedicated section.

Jump changes