Қоңыраулармен математикалық өрнектердің үнемі өзгеруі

Sounding continually changing mathematical permutations on bells
Method ringing (also known as scientific ringing) is a form of change ringing in which the ringers commit to memory the rules for generating each change of sequence, and pairs of bells are affected. This creates a form of bell music which is continually changing, but which cannot be discerned as a conventional melody. It is a way of sounding continually changing mathematical permutations. It is distinct from call changes, where the ringers are instructed on how to generate each new change by calls from a conductor, and strictly, only two adjacent bells swap their position at each change. In method ringing, the ringers are guided from permutation to permutation by following the rules of a method. Ringers typically learn a particular method by studying its "blue line", a diagram which shows its structure. The underlying mathematical basis of method ringing is intimately linked to group theory. The basic building block of method ringing is plain hunt. The first method, Grandsire, was designed around 1650, probably by Robert Roan who became master of the College Youths change ringing society in 1652. Details of the method on five bells appeared in print in 1668 in Tintinnalogia (Fabian Stedman with Richard Duckworth) and Campanalogia (1677 – written solely by Stedman), which are the first two publications on the subject. 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.

There are thousands of different methods, a few of which are the below.

Plain hunt is the simplest form of generating changing permutations continuously, and is a fundamental building block of change ringing methods. It can be extended to any number of bells. It consists of a plain undeviating course of a bell between the first and last places in the striking order, with two strikes in the first and last position to enable a turn around. Thus each bell moves one position at each succeeding change, unless it reaches the first or last position, where it will remain for two changes before proceeding to the other end of the sequence.

The highest bell in pitch is known as the treble and the lowest the tenor. The majority of bell towers 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. 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. Plain hunt is the simplest way of creating bell permutations, or changes.

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 bells, there are (read factorial) possible permutations, a number which quickly grows as 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. 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 decelerate 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.

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.

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 second 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.

To obtain more changes than available in the plain course, 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 of the 20th century was Albert J Pitman, who composed over a hundred peals between 1910 and 1965, entirely by hand. None of his compositions was then, nor since, discovered to be false.

As well as writing out the changes longhand (as in the accompanying illustration of Plain Bob Minor) there is a shorthand called Place Notation. 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.