# Method ringing

Method ringing (also known as scientific ringing) is a form of change ringing (the practice of ringing a series of mathematical permutations on tuned bells, rather than a melody). In method ringing, the ringers are guided from permutation to permutation by following a specific algorithm, called a method. Where the treble bell follows exactly the same pattern as the other bells it is called a principle. The underlying mathematical basis is intimately linked to group theory.

The practice originated in England and remains most popular there today; in addition to bells in church towers, it is also often performed on handbells.

## Introduction

Understanding the conventions of change ringing is necessary to discuss method ringing. Among these are ways of referring to bells: the one highest in pitch is known as the treble and the lowest the tenor. The majority of belltowers have the ring of bells (or ropes) going clockwise from the treble. For convenience, the bells are referred to by number, with the treble being number 1 and the other bells numbered by their pitch (2, 3, 4, etc.) sequentially down the scale. (This system often seems counterintuitive to musicians, who are used to a numbering which ascends along with pitch.) The bells are usually tuned to a diatonic major scale, with the tenor bell being the tonic (or key) note of the scale.

The simplest way to use a set of bells is ringing rounds, which is sounding the bells repeatedly in sequence from treble to tenor: 1, 2, 3, etc.. (Musicians will recognise this as a portion of a descending scale.) Ringers typically start with rounds and then begin to vary the bells' order, moving on to a series of distinct rows. Each row (or change) is a specific permutation of the bells (for example 123456 or 531246)—that is to say, it includes each bell rung once and only once, the difference from row to row being the order in which the bells follow one another.

Since permutations are involved, it is natural that for some people the ultimate theoretical goal of change ringing is to ring the bells in every possible permutation; this is called an extent (in the past this was sometimes referred to as a full peal). For a method on ${\displaystyle n}$ bells, there are ${\displaystyle n!}$ (read factorial) possible permutations, a number which quickly grows as ${\displaystyle n}$ increases. For example, while on six bells there are 720 permutations, on 8 bells there are 40,320; furthermore, 10! = 3,628,800, and 12! = 479,001,600.

Estimating two seconds for each change (a reasonable pace), we find that while an extent on 6 bells can be accomplished in half an hour, a full peal on 8 bells should take nearly twenty-two and a half hours and one on 12 bells would take over thirty years! Naturally, then, except in towers with only a few bells, ringers typically can only ring a subset of the available permutations. But the key stricture of an extent, uniqueness (any row may only be rung once), is considered essential.

This is called truth; to repeat any row would make the performance false. Another key limitation keeps a given bell from moving up or back more than a single place from row to row; if it rings (for instance) fourth in one row, in the next row it can only ring third, fourth, or fifth. Thus from row to row each bell either keeps its place or swaps places with one of its neighbours.[1] This rule has its origins in the physical reality of tower bells: a bell, swinging through a complete revolution with every row, has considerable inertia and the ringer has only a limited ability to accelerate or retard its cycle.

A third key rule mandates rounds as the start and end of all ringing. So to summarize: any performance must start out from rounds, visit a number of other rows (whether all possible permutations or just a subset thereof) but only once each, and then return safely to rounds, all the while making only small neighbour-swaps from row to row. These rules dramatically limit the options open to a method-maker. For example, consider a tower with four bells. An extent includes 4! = 24 changes and there are, naturally, 24! possible orders in which to ring each change once, which is about 6.2 × 1023. But once we limit ourselves to neighbour-swaps and to starting and ending with rounds, only 10,792 possible extents remain.[2]

It is to navigate this complex terrain that various methods have been developed; they allow the ringers to plot their course ahead of time without needing to memorize it all (an impossible task) or to read it off a numbingly repetitive list of numbers. Instead, by combining a pattern short and simple enough for ringers to memorize with a few regular breaking points where simple variations can be introduced, a robust algorithm is formed. This is the essence of method ringing.

## Method ringing in action

The "Blue Line" of Plain Bob Minor. This has 60 changes. Note that, for clarity, the row at the bottom of each column is repeated at the top of the next. Note that in this copy of the diagram the "Blue Line" is actually coloured red. The course of the treble is coloured blue.

### The plain course

This defines the changes over a relatively short sequence, ranging from a mere handful up to a few hundred changes at most. To learn this, a ringer must memorize the course taken by his or her bell during this span, called the plain course. To help learn it, ringers often use a diagram in which a plain course is written out, row by row, and a particular bell's course is given visual shape, being traced by a blue line.

A method's plain course begins and ends in rounds and thus can be considered as a performance in its own right, albeit a brief one.

A lead is part of the plain course. It commences when the method starts and lasts until the treble gets back to the same place. In the diagram of Plain Bob Minor shown, the lead starts when the treble rings in seconds place and lasts until the treble has rung twice at lead. It is common practice in diagrams to draw a line under the lead end to assist in understanding the method. Most methods have a plain course consisting of a number of leads where the pattern is the same, but different bells are in differing places. In the diagram given, the number 4 bell rings the same pattern as the number 2, but one lead earlier.

In principles (where the treble does the same work as other bells and is affected by calls) the definition of a lead can become more complex.

### Calls and compositions

In a longer peal, a conductor makes a call directing the ringers to make a slight variation in the course. (The most common calls are called bobs and singles.) These variations usually last only one change, but cause two or more ringers to swap their paths, whereupon they continue with the normal pattern. By introducing such calls appropriately, repetition can be avoided, with the peal remaining true over a large number of changes. For example, an extent in a minor method is 720 (6!) changes, so would require 12 repetitions of the plain course shown.

To know when to make calls and which ones to make, a conductor follows a plan called a composition which he or someone else devised; if properly constructed it will ensure a true performance of the desired length. Today computers make checking a composition's truth easy; but the process once involved a mix of mathematics and laborious row-by-row checking.

Probably the greatest composer [3] of the 20th century was Albert J Pitman, who composed over a hundred [4] peals between 1910 and 1965, entirely by hand. None of his compositions was then, nor since, discovered to be false.

### Place Notation (shorthand)

As well as writing out the changes longhand (as in the accompanying illustration of Plain Bob Minor) there is a shorthand called Place Notation.[5] For each row in which all bells change place, such as the first change, use an "x" or a "-". In rows where one or more bells stay in place write down the place numbers which do not change, so that the second row is written "16". Plain Bob Minor is therefore x16x16x16x16x16x12.

Many methods are symmetrical, and so only the first half lead is given, along with possibly the lead end. Plain Bob Minor is thus: x16x16x16 le:12. Where two changes consisting of numbers follow each other, use a dot to separate them. Plain Bob Doubles (i.e. on 5 bells) is: 5.1.5.1.5 le:125, or if written at full length 5.1.5.1.5.1.5.1.5.125.

## Method names

Methods are generally referred to by an official name assigned to them by the Central Council of Church Bell Ringers; such names have three standard parts: the method's name proper, its class, and its stage.

The name proper is the method's personal name. The oldest methods have long-established names; but new methods are constantly being devised and rung, and the Central Council generally allows each to be named by the band which first rings a peal in it. Most often these methods end up with a place name, such as the band's village; but people's names and still more fanciful inventions are not uncommon.

The class describes the method, putting it in some established category of methods that work in similar ways. Methods in the simplest category omit this second name and use a simple two-part name.

The stage indicates the number of bells, using unique terminology:

As can be seen, there are different naming systems for even- and odd-bell stages. The odd-bell stage names refer to the number of possible swaps that can be made from row to row; in caters and cinques can be seen the French numbers quatre and cinq while the stage name for three-bell ringing is indeed "singles". Higher odd-bell stages follow the same pattern (sextuples, septuples, etc.) while higher even-bell stages have more prosaic names: fourteen, sixteen, etc.).

Note that the names refer to the number of bells being permuted, which is not necessarily the same as the number being rung: for it is typical to ring triples methods not on seven bells but on eight, with the tenor covering: only the seven highest bells permute; the eighth and lowest bell is simply rung last in every row. So likewise with caters, usually rung on ten bells, and other higher odd-bell stages.

Put together, this system gives method names sound that is evocative, musical, and quaint: Kent Treble Bob Major, Grandsire Caters, Erin Triples, Chartres Delight Royal, Percy's Tea Strainer Treble Place Major, Titanic Cinques and so forth.

### Performances

A short composition, lasting perhaps only a few hundred changes, is called a touch; but many ringers look forward to the greater challenge of a quarter peal (about 1,250 changes) or a peal, (about 5,000 changes). This number derives from the great 17th-century quest to ring a full extent on seven bells; 7 factorial is 5,040. Sturdier bellframes and more clearly understood methods make the task easier today; but a peal still needs about 3 hours of labour and concentration, something any ringer can be proud to complete.

Most ringers follow the definition of a peal as regulated by the Central Council. This requires a minimum of only 5,000 changes where major or a higher stage is being rung; but demands at least the full 5,040 changes on lower stages. For triples, this ensures at least a full extent; for lower stages a full extent falls well short of the goal and ringers must complete several full extents to reach 5,040 (working out mathematically to at least 7 extents on six bells, at least 42 on five, or at least 210 on four; three-bell peals are not recognised by the Central Council).[6]

To qualify as a peal, the ringing must meet a number of other key criteria. Among other things, each bell must be rung continuously by the same person; bands cannot swap in an extra person to give ringers the occasional break. Likewise the ringing must be done entirely from memory; ringers cannot consult the method's blue line nor can the conductor (who must be one of the ringers) have a written reminder of the composition.

More commonly rung is the quarter peal, typically consisting of 1,260 changes and typically taking 45 minutes to ring. Half peals are more rarely rung, but have been known. One example is in Buckfast Abbey in Devon, where there are two half peal boards.