Change ringing

Change ringing is the art of ringing a set of tuned bells in a controlled manner to produce variations in their sounding order.

Change ringing originated following the invention of English full-circle tower bell ringing in the early 17th century, when it was found that ringing a bell through a large arc of swing gave more control over the time between successive strikes of the clapper. This culminated in ringing bells through a full circle which allowed ringers to easily produce different striking sequences; either called out by one of the ringers, known as "Call changes", or continuously changing sequences following mathematical rules, known as "Method ringing".

The considerable weights of bells employed in full-circle ringing means that they cannot be easily halted, and the change of speed between successive strikes is limited. Therefore in change ringing each bell strikes once in each sequence, and only moves one place in the striking order of successive sequences. These constraints give change ringing its basic rules for generating sequences, which are known as "changes". Control is exerted by the ringer when each bell is mouth upwards and moving slowly near the balance point, and the rope management involved means that each bell nearly always requires its own ringer. The weight of a tower bell means it can only strike once about every two seconds—the time for a full rotation in one direction.

Although change ringing is practised worldwide, it is by far the most prevalent on church bells in English churches, where it was first developed. Such a church's ring of bells requires each bell to swing freely for slightly more than one revolution in alternate directions. Change ringing is also performed on handbells, and on carillons (where the bells are fixed and struck with hammers) though both of these instruments are more commonly used to play conventional melodies and employ harmonies.

Tower bells

Main article: Ring of bells
Bell ringing at Stoke Gabriel parish church, Devon, England. This is in the "ringing chamber".
The bells of St Bees Priory shown in the "down" position, in which they are normally left between ringing sessions. This is in the "bell chamber".
The bells of St Bees Priory shown in the "up" position. When being rung they swing through a full circle from mouth upwards round to mouth upwards, and then back again.

Today, some towers have as many as sixteen bells that can be rung together, though six or eight bells are more common. The highest pitch bell is known as the treble, and the lowest is the tenor. 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 counter-intuitive to musicians, who are used to a numbering that ascends 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. Some towers contain additional bells so that different subsets of the full number can be rung, still to a diatonic scale.[1] For instance, many 12-bell towers have a flat sixth, which if rung instead of the normal number 6 bell allows 2 to 9 to be rung as light diatonic octave; other variations are also possible.[2]

Mechanism of a bell hung for English full-circle ringing.

The bells in a tower reside in the bell chamber or belfry usually with louvred windows to enable the sound to escape.

The bells are mounted within a bellframe of steel or wood. Each bell is suspended from a headstock fitted on trunnions (plain or non-friction bearings) mounted to the belfry framework so that the bell assembly may rotate. When stationary in the down position, the centre of mass of the bell and clapper is appreciably below the centreline of the trunnion supports, thus giving a pendulous effect to the assembly, and this dynamic is controlled by the ringer's rope. The headstock is fitted with a wooden stay, which, in conjunction with a slider, limits maximum rotational movement to a little less than 370 degrees. To the headstock a large wooden wheel is fitted and to which a rope is attached. The rope wraps and unwraps as the bell rotates backwards and forwards. This is full circle ringing and quite different from fixed or limited motion bells, which chime. Within the bell the clapper is constrained to swing in the direction that the bell swings. The clapper is a rigid steel or wrought iron bar with a large ball to strike the bell. The thickest part of the mouth of bell is called the soundbow and it is against this that the ball strikes. Beyond the ball is a flight, which controls the speed of the clapper. In very small bells this can be nearly as long as the rest of the clapper.

Below the bell chamber there may be one or more sound chambers, (one of which is likely to house the clock mechanism if the church has one) and through which the rope passes before it drops into the ringing chamber or room. Typically, the rope's length is such that it falls close to or on to the floor of the ringing chamber. About 5 feet (1.5 m) from the floor, the rope has a woolen grip called the sally (usually around 4 feet (1.2 m) long) while the lower end of the rope is doubled over to form an easily held tail-end.

Unattended bells are normally left hanging in the normal ("down") position but prior to being rung, the bells are rung up. In the down position, the bells are safe if a person touches them or pulls a rope. A bell that is up is dangerous to be near, and only expert ringers should ever contemplate entering a bell chamber or touching a rope when the bells are up. The ringer pulls on the rope and starts the bell swinging. Each time the bell swings the ringer adds a little more energy to the system, similar to pushing a child's swing. Eventually there is enough energy for the bell to swing right up and be left over-centre just beyond the balance point with the stay resting against the slider.

Bellringers typically stand in a circle around the ringing chamber, each managing one rope. Bells and their attendant ropes are so mounted that the ropes are pulled in a circular sequence, usually clockwise, starting with the lightest (treble) bell and descending to the heaviest (tenor). To ring the bell, the ringer first pulls the sally towards the floor, upsetting the bell's balance and swinging it on its bearings. As the bell swings downwards the rope unwinds from the wheel and the ringer adds enough pull to counteract friction and air resistance. The bell winds the rope back onto the other side of the wheel as it rises and the ringer can slow (or check) the rise of the bell if required.

The rope is attached to one side of the wheel so that a different amount of rope is wound on and off as it swings to and fro. The first stroke is the handstroke with a small amount of rope on the wheel. The ringer pulls on the sally and when the bell swings up it draws up more rope onto the wheel and the sally rises to, or beyond, the ceiling. The ringer keeps hold of the tail-end of the rope to control the bell. After a controlled pause with the bell, on or close to its balancing point, the ringer rings the backstroke by pulling the tail-end, causing the bell to swing back towards its starting position. As the sally rises, the ringer catches it to pause the bell at its balance position.

In English style ringing the bell is rung up such that the clapper is resting on the lower edge of the bell when the bell is on the stay. During each swing, the clapper travels faster than the bell, eventually striking the soundbow and making the bell sound. The bell speaks roughly when horizontal as it rises, thus projecting the sound outwards. The clapper rebounds very slightly, allowing the bell to ring. At the balance point, the clapper passes over the top and rests against the soundbow.

In change ringing where the order the bells are struck in is constantly altered, it is necessary to time the swing so that this strike occurs with precise positioning within the overall pattern. Precision of striking is important at all times. To ring quickly, the bell must not complete the full 360 degrees before swinging back in the opposite direction, while ringing slowly the ringer waits with the bell held at the balance, before allowing it to swing back. To achieve this, the ringer must work with the bell's momentum, applying just the right amount of effort during the pull that the bell swings as far as required and no further. This allows two adjacent bells to reverse positions; the quicker bell passing the slower bell to establish a new pattern. Although ringing up certainly involves some physical exertion, actual ringing should rely more on practised skill than mere brute force. Even the smallest bell in a tower is much heavier than the person ringing it. The heaviest bell hung for full-circle ringing is in Liverpool Cathedral and weighs 82 long cwt 0 qr 11 lb (9,195 lb or 4,171 kg).[3][4] Despite this colossal weight, it can be safely rung by one (experienced) ringer.

(Whilst heavier bells exist—for example Big Ben—they are generally only chimed, either by swinging the bell slightly or having the bell hung dead and using a mechanical hammer.)

Handbells

Main article: Handbell

Change ringing is performed on handbells, and is quite popular in its own right. Many record-length peals, including the longest peal ever rung, are by handbell ringers.

Normally each ringer has a bell in each hand and sit or stand in a circle (like tower ringers). The tower bell terms of handstroke and backstroke are retained, referring to an upwards and downwards ring of the bell respectively; and as in towers, the ringing proceeds in alternate rows of handstroke and backstroke.

Occasionally, a technique called lapping, or cross and stretch is used. Ringers stand or sit in a straight line at a single convenient table on which the bells are placed. They pick up a bell each time they ring it, and then put it down. As the bell sequence changes, however, the ringers physically swap the bells accordingly—so the bells move up and down the table and each row is rung in strict sequence from right to left. Ringers in cross and stretch thus do not have responsibility for their own personal bell, but handle each as it comes.

Some handbell change ringers practice a hybrid of these two methods, known as body ringing: ringers standing in a line each hold one bell, exchanging places in the line so that the changes sound correctly when the bells are rung in sequence from right to left.

Ringing the changes

The simplest way to sound a ring of bells is by ringing rounds. This is a repeated sequence of bells descending from the highest to lowest note, which is from the lightest to the heaviest bell. This was the original sequence used before change ringing was developed, and change ringing always starts and ends with this sequence.

After starting rounds, at a given command, the ringers vary the bells' order, to produce a series of distinct sequences known as rows or changes. In this way permutation of the bells' striking order proceeds. For example 123456 can become 214365 in the next sequence. There are two distinct ways that the ringers know what to do. In call change ringing each row is specifically called for by one ringer (the conductor) who tells the others how to change their bells' places from row to row. In method ringing, however, the ringers receive little instruction during ringing as they have learned a "method" of changing the bells' positions in the sequence which results in a series of continuously varying changes. A conductor's intervention is needed only periodically, when a slight variation in the pattern is necessary, or to correct errors.

The physical constraint of the mass of the bells means they can only be slightly delayed or advanced in the striking order, so they cannot be omitted from a sequence, and can only be made to change by one position in successive sequences.

Call change ringing

Most ringers begin their ringing career with call change ringing; they can thus concentrate on learning the physical skills needed to handle their bells without needing to worry about methods. There are also many towers where experienced ringers practise call change ringing as an art in its own right (and even exclusively), particularly in the English county of Devon.

Calls are made with spoken commands such as "X to Y" or "X and Y" or "X after Y", in which X and Y refer to two of the bells by their numbers (not by their positions in the row); such a call signifies that after the call, a pair of bells will have swapped, resulting in X following Y. However, there are several different ways of representing any given change. By far the most common two are known as calling up and calling down; each has its merits and inconveniences, but generally any given tower consistently uses one system in preference to the other.

As an example, consider the following sequence of rows, and the calls a conductor would use to evoke them:

Row Conductor's intent Call, if calling Up Call, if calling Down
1,2,3,4,5,6 to swap bells 2 and 3 "2 to 3" "3 to Treble"
1,3,2,4,5,6 to swap bells 4 and 5 "4 to 5" "5 to 2"
1,3,2,5,4,6 to swap bells 2 and 5 "2 to 5" "5 to 3"
1,3,5,2,4,6 to swap bells 1 and 3 "1 to 3" "3 to lead"
3,1,5,2,4,6

Thus it can be seen how these two ways of calling differ:

In both cases, the ringer of the bell immediately above (behind) the swapping pair must also be alert, as this bell follows a new bell after the swap. Rarer forms of change calling may name just one of the moving bells, call the moving bell by position rather than number, or call out the full change.

Method ringing

The "Blue Line" of Plain Bob Minor, shown in red. Note that, for clarity, the row at the bottom of each column is repeated at the top of the next.
Main article: Method ringing

Method ringing is the continuously changing form of change ringing, and gets its name from the use of a method to generate the changes. There are thousands of different methods, and Plain Bob Minor, a method on six bells, is shown in the accompanying diagram.

The method is committed to memory by each ringer, so that only a few commands are given by the ringer in charge (the conductor). This does not consist of memorising the individual sequences, but using a variety of techniques such as:

  1. Memorising the path of the bell, shown as a blue line in the diagram.
  2. Looking for visual signposts such as when a particular bells crosses with another.

In the example shown, the pattern starts and ends in rounds. To extend this to the full unique 720 changes possible (factorial 7, which is 1×2×3×4×5×6×7 = 720 changes) from time to time the conductor calls out the need for another variation by calling "bob" or "single".

Thanks to it, ringers can spend hours ringing thousands upon thousands of unique changes with no outside direction or coordination. They do not have to memorize impossible quantities of data, nor do they attempt to read it all off a lengthy sheet of numbers. Rather, they are all following a method, a relatively simple pattern they have learned to direct them from row to row. Where the treble bell follows exactly the same pattern as the other bells it is called a principle.

For some people, the ultimate goal of this system is to ring all the permutations, to ring a tower's bells in every possible order without repeating – what is called an extent (or sometimes, formerly, a full peal). The feasibility of this depends on how many bells are involved: if a tower has n bells, they have n! (read factorial) possible permutations, a number that becomes quite large as n grows. For example, while six bells have 720 permutations, eight bells have 40,320; furthermore, 10! = 3,628,800, and 12! = 479,001,600. Estimating two seconds for each change (a reasonable pace), one finds that while an extent on six bells can be accomplished in half an hour, a full peal on eight bells should take nearly twenty-two and a half hours. (When in 1963 ringers in Loughborough became the only band in history to achieve this feat on tower bells, it took them just under 18 hours.[5]) An extent on 12 bells would take over thirty years.

Since extents are obviously not always practicable, ringers more often undertake shorter performances. Such ringing starts and ends with rounds, having meanwhile visited only a subset of the available permutations; but truth is still considered essential — no row can ever be repeated; to do so would make the ringing false. A peal is an extended performance; it must last at least 5000 changes on eight or more bells and at least 5040 on seven or fewer bells (5040 being 7!, the length of a full extent on seven). A performance of 1250 (on 8 or more) or 1260 (on 7 or fewer) changes likewise makes a quarter peal (quarter for short); a peal or a quarter tends to last about three hours or 45 minutes, respectively.
Plain Bob Minor
Plain Bob Minor played using synthesised bell sounds. The bells start ringing rounds followed by a plain course of Plain Bob Minor (60 of the 720 changes that are possible on six bells) and finish in rounds again.
Problems playing this file? See media help.

Half-muffling

For commemorative services such as funerals, memorial services and Remembrance Sunday, the bells are rung half-muffled. To ring half-muffled a thick leather pad called a muffle is strapped to one side of each bell's clapper. This deadens the sound of alternate strokes of the bells, the muffled stroke (usually the backstroke) sounding similar to an echo of the unmuffled stroke. In some areas the tenor bell (the lowest note) is left unmuffled to heighten the contrast.[6]

Very rarely, normally only for the death of a Sovereign, are the bells rung fully muffled.[6] One notable exception to this tradition was when nurse Edith Cavell's remains were repatriated after the First World War. As the ship bearing the coffin arrived in Dover, a full peal of Grandsire Triples (5040 Changes, Parker's Twelve-Part) was rung on the bells of the parish church. The peal record shows:

Rung with the bells deeply muffled with the exception of the Tenor which was open at back stroke, in token of respect to Nurse Cavell, whose body arrived at Dover during the ringing and rested in the town till the following morning. The ringers of 1-2-3-4-5-6 are ex-soldiers, F. Elliot having been eight months Prisoner of War in Germany.[7]

Other common change-ringing terms

History and modern culture

Change ringing as we know it today emerged in England in the 17th century. To that era we can trace the origins of the earliest ringing societies, such as the Lincoln Cathedral Guild, which claims to date to 1612[9] or the Antient [sic] Society of Ringers of St Stephen in Bristol which was founded in 1620 and lasted as a ringing society until the late 19th century.[10] The recreation began to flourish in earnest in the Restoration era; an important milestone in the development of method ringing as a careful science was the 1668 publication by Richard Duckworth and Fabian Stedman of their book Tintinnalogia, which promised in its subtitle to lay down "plain and easie Rules for Ringing all sorts of Plain Changes." Stedman followed this in 1677 with another famous early guide, Campanalogia.

Throughout the years since, the group theoretical underpinnings of change ringing have been pursued by mathematicians.[11] Bells have been installed in towers around the world and many rings in the British Isles have been augmented to ten, twelve, fourteen, or even sixteen bells. Today change ringing is, particularly in England, a popular and commonplace sound, often issuing from a church tower before or after a service or wedding. While on these everyday occasions the ringers must usually content themselves with shorter "touches," each lasting a few minutes, for special occasions they often attempt a quarter-peal or peal, lasting approximately 45 minutes or three hours respectively. If a peal attempt succeeds, towers sometimes mark the occasion with a peal board mounted on the wall of the ringing chamber; at St Peter Mancroft in Norwich there is one documenting what is generally considered to have been the first true peal: 5040 changes of Plain Bob Triples (a method still popular today), rung 2 May 1715.[12] Today over 4000 peals are rung each year.

Organisation and extent

The Central Council of Church Bell Ringers, founded in 1891, is dedicated to representing change ringers around the world. Most regional and local ringing guilds are affiliated with the council. Its journal, The Ringing World,[13] has been published weekly since 1911; in addition to news and features relating to bellringing and the bellringing community, it publishes records of achievements such as peals and quarter-peals. Ringers generally adhere to the Council's rules and definitions governing change ringing.

The Central Council, by means of its peal records, also keeps track of record length peals, both on tower bells and handbells. (The record for tower bells remains the 1963 Loughborough extent of Plain Bob Major (40,320 changes); for handbells it was set in 2007 in Willingham, Cambridgeshire, with 72,000 changes of 100 different Treble Dodging Minor methods, taking just over 24 hours to ring[14]) More importantly, perhaps, along with keeping track of the first peal ever rung in a method, the Central Council controls the naming of new methods: it generally allows the first band to ring a method to name it.

Much ringing is carried out by bands of ringers meeting at their local tower to ring its bells. For the sake of variety, though, many ringers like to take occasional trips to make a tower grab ringing the bells of a less familiar tower. The setting, the church architecture, the chance to ring more bells than usual, the bells' unique tone, their ease or difficulty of ringing, and sometimes even the unusual means of accessing the ringing chamber can all be part of the attraction. The traditional means of finding bell towers, and still the most popular way today, is the book (and now internet database) Dove's Guide for Church Bell Ringers.

As of 30 September 2015 there are 7,140 English style rings. The Netherlands, Pakistan, India, and Spain have one each. The Windward Isles and the Isle of Man have 2 each. Canada and New Zealand 8 each. The Channel Isles 10. Africa as a continent has 13. Scotland 24, Ireland 37, USA 48, Australia 59 and Wales 227. The remaining 6,798 (95.2%) are in England (including three mobile rings).[15]

Number of bells

Methods of change ringing are named for the number of working bells, or those which switch order within the change. It takes a pair to switch, and commonly the largest bell (the tenor) does not change place. For example, there may be six bells, only five of which are working, allowing for only two pairs. A method of ringing for these bells would be called doubles. Doubles is the most common group of methods rung in the United Kingdom, since 90% of parish churches with bell towers in the UK are fitted with only six bells.

"Plain Bob Doubles" is a method rung on five bells whereas "Plain Bob Triples" is the same method rung on seven working bells.

There are two separate ways are used to refer to the number of bells. One way is used for even numbers, the other for an odd number.

Even numbers of bells
Number of bells Name
4 Minimus
6 Minor
8 Major
10 Royal
12 Maximus
Odd numbers of bells
Number of bells Name
3 Singles
5 Doubles
7 Triples
9 Caters
11 Cinques

The name for 9 bells is pronounced "kate-ers" and comes from the French "quatres". The name for 11 bells also comes from the French and is pronounced "sinks" c.f. Cinque Ports.

The names refer to the number of bells which change places in each row. With three bells only one pair can change, and so it is singles. With seven bells there are clearly three pairs with the one left over not moving this row.

Named changes

Mathematical abstraction though each row may be, some rows do have a musical or melodic meaning to the listener. Over the years, a number of these have acquired names — they are named changes. Both the conductors directing call-change ringing and the composers coming up with plans for a bout of method ringing sometimes like to work their favourite named changes in. The table below lists some popular named changes on eight bells; many of these names are also applicable by extension on more or fewer bells.

Change Name
12345678 ( listen ) Rounds
87654321 ( listen ) Back rounds or Reverse Rounds[16]
13572468 ( listen ) Queens (an apocryphal story says it appealed to Elizabeth I)
15263748 ( listen ) Tittums (so named because of the ti-tum ti-tum sound it makes)

Such names are often humorous; for example, the sequence 14235 on five bells is called weasels because it is the tune of the refrain to the children's song Pop Goes the Weasel.[17]

Called changes are listed at MAW Call Change Collection

Striking

Although neither call change nor method ringing produces conventional tunes, it is still the aim of the ringers to produce a pleasant sound. One of the most important aspects of this is good striking — not only should the bells never clash by sounding at the same moment, the bells should sound to a perfect rhythm, tapping out a steady beat.

It is the custom to leave a pause of one beat after every alternate row, i.e., after the ringing of each ‘backstroke’ row. This is called 'open handstroke' ringing (or open handstroke leading). In Devon, Cornwall and parts of Yorkshire, this custom is not followed when call-change ringing; instead the bells strike steadily without the pause. This latter custom is known as the closed-hand or cartwheel arrangement. However for method ringing the universal practice is to ring with open handstrokes, even in the South West of England.

Striking competitions are held where various bands of ringers attempt to ring with their best striking. They are judged on their number of faults (striking errors); the band with the fewest faults wins. These competitions are organized on regional and national levels, being particularly popular among the call-change ringers of Devon where it is customary to include the quality of the rise and lower of the bells as part of the judging criteria. Competitions for method ringers usually start "off the stay"—i.e., the bells are rung up before the competition begins. At the annual National 12 Bell Striking Contest the bands are ringing methods and producing a different change approximately every 2.5 seconds, with a gap between bells of 0.21 seconds. To an expert ringer's ear at this level of competition a variation of a tenth of this would be discernible as a striking fault.

Splicing bell ropes

Rope splicing plays an important role in English style ringing. Judicious splicing, and a better understanding of ropes, can help prolong the life of this most important component.[18]

Sport

In 2016 readers of The Ringing World magazine wrote to insist that bell ringing was "an art and a sport", as demonstrated by regular "striking competitions." It was suggested that classification of change ringing as a sport by Sport England could save it from becoming obsolete. But the Central Council of Church Bell Ringers opposed the move, suggesting that it would jeopardise its relationship with church bodies, since bell ringing should be seen as part of Christian worship, not exercise. The council's president, Chris Mew, said: "Where is the glamour of the sports field and where are the David Beckhams of the belfry?"[19]

In literature and television

The mystery novel The Nine Tailors by Dorothy L. Sayers (1934) contains a great deal of information on change-ringing. Her fictional detective, Lord Peter Wimsey, demonstrates his skill at ringing, and the solution to the central puzzle of the book rests in part upon his knowledge of the patterns of change ringing.

Connie Willis, who frequently and overtly references Sayers in To Say Nothing of the Dog (1997), features bell ringers in her earlier novel Doomsday Book (1992); a group of American women led by a Mrs. Taylor frequently appears practising for or ringing both handbells and changes.

The British television series Midsomer Murders aired an episode in the fifth season on a series of murders within a bell-ringing team, in "Ring Out Your Dead".

In the science-fiction novel Anathem by Neal Stephenson (2008) changes are rung in a cloistered monastery for mathematicians to signal different ceremonies.

See also

Notes

  1. ↑ Bryant, David. "The History and Use of Semitone Bells". Retrieved 11 June 2014.
  2. ↑ Dove entry for Worcester Cathedral, a twelve bell tower with 3 semi-tone bells. Search on Dove's Guide for towers with 13 or more bells, note that Christ Church Cathedral, Dublin is listed as having 16 bells, with additional sharp 6, 9 and 13; Swan Bells, Perth, Western Australia, is again a 16 but with additional flat 3 and 10. Retrieved 28 May 2008.
  3. ↑ "history". Whitechapel Bell Foundry. Retrieved 23 June 2009.
  4. ↑ "Liverpool:Cathedral Church of Christ". Dove's Guide for Church Bell Ringers. Central Council for Church Bell Ringers. Retrieved 2 May 2015.
  5. ↑ Online peal board, from the Central Council records committee Archived June 13, 2008, at the Wayback Machine.
  6. 1 2 "Commemorative Ringing". The Central Council of Church Bell Ringers official website. The Central Council of Church Bell Ringers. 13 March 2004. Retrieved 31 May 2011.
  7. ↑ Peal record Retrieved 21 October 2014
  8. ↑ Harrison, John. "Glossary of ringing terms". Retrieved 26 March 2016.
  9. ↑ "Company of Ringers". Lincoln Cathedral website. 2006. Archived from the original on October 29, 2006. Retrieved 18 April 2007.
  10. ↑ "THE ANTIENT SOCIETY OF RINGERS". Website of St Stephen's, the parish church for the City of Bristol, England. Archived from the original on 3 July 2007. Retrieved 18 April 2007.
  11. ↑ "Changes" can be viewed as permutations; sets of permutations constitute mathematical groups, which in turn can be depicted via so-called Cayley graphs, which in turn can be mapped onto polyhedra. Web pages: Some literature on the mathematics of change ringing:
    • Ian Stewart, Another Fine Math You've Got Me Into (New York, New York : W.H. Freeman, 1992), Chapter 13 (pages 199-219).
    • F.J. Budden, The Fascination of Groups, (Cambridge, England: Cambridge University Press, 1972), Chapter 24: "Ringing the changes: groups and campanology," pages 451-479.
    • Arthur White and Robin Wilson (March 1995) "The hunting group," The Mathematical Gazette, vol. 79, no. 484, pages 5-16.
    • Arthur T. White (1987) "Ringing the cosets," American Mathematical Monthly, vol. 94 , pages 721-746.
    See also the Steinhaus–Johnson–Trotter algorithm.
  12. ↑ Bells and Bellringing, from the CCCBR. There is some evidence there may have been an earlier peal (also Plain Bob Triples), rung January 7, 1690 at St Sepulchre-without-Newgate in the City of London by the Ancient Society of College Youths; see Landmarks in the History of the Society, from the ASCY.
  13. ↑ The Ringing World
  14. ↑ "72000 Treble Dodging Minor (100m)". peals.co.uk. The Ringing World.
  15. ↑ County Lists from Dove's Guide for Church Bell Ringers, Central Council of Church Bell Ringers, retrieved 12 October 2015
  16. ↑ Some sources (e.g., ) define back rounds slightly differently, as 76543218.
  17. ↑ This is particularly effective at the end of ringing down. The bells are in order, and so if not chimed leave a pause, the sequence becomes: 1..4..23.5 where a dot indicates a pause.
  18. ↑ Beech, Frank (2005). Splicing Bell Ropes Illustrated (first ed.). Central Council of Church Bell Ringers. pp. 1–32. ISBN 0-900271-82-5.
  19. ↑ Jamieson, Sophie (19 February 2014). "Ding-dong for bell ringers as row breaks out over bid to be classed as a sport". The Daily Telegraph. Retrieved 19 February 2016.

References

External links

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