Impatient and nervous people often jingle coins in their pockets, but that is not what what we mean here by change-ringing.

The credit-card and the proposals for a "checkless society" threaten to make this irritating noise obsolete: besides which the new nickel-alloy-clad dimes and quarters with copper centers won't ring like the old ones.

No, the practice of 'ringing the changes' to be discussed here is at once an application of mathematics, a sport, a peculiarly-disciplined athletic exercise, a merry noise to promote community spirit, a custom, and the basis of a kind of comradeship in pursuing a unique avocation. These factors, all put together, create a tradition and a ritual over and above the existing traditions and rituals of the churches in whose towers the bells are rung.

While church-bells are undeniably musical instruments, opinion is divided as to whether the ringing of changes on a group of such bells is really music, whether melody or tune can legitimately be used of the effect that results when bells are rung in this special manner.

Change-ringing is a particularly English practice, as opposed to the playing of frankly musical compositions on sets of bells (carillons) in such countries as Belgium and the Netherlands.

So far as the mathematics involved is concerned, changes could be 'rung' on almost anything--the keys of a piano, the notes of a tin whistle, or for that matter the pots and pans in your kitchen!

Indeed, we will link up the theory of change-ringing with modern music of the kind that some people do not like because it is not 'tuneful' to their ears. Moreover, the mathematical principle of permutations which forms the basis of change-ringing also turns up in the most unexpected places in modern life, both in scientific or technical fields and in the activities of ordinary life, such as card-playing or the making and using of various decorative patterns.

New applications of the permutation theory to the use of computers are being made every day, so we will hear more of permutations and combinations as time goes on.

Traditional change-ringing thus represents a unique synthesis: full-sized and therefore heavy bells, usually hung in a church tower, entailing athletic exercise by the ringers, and also generating a set of rules and customs surprisingly like those of games and sports, this neatly tied together by the permutation principle and given community acceptance because of its association with churches that have peals of bells.

When your editor was about 8 years old, he lived near a church were change-ringing was practiced several times a week, so there are at least a few places in some U.S. cities where the custom has been imported from England.

Since American preferences usually is for the carillon, on which real tunes are played, it is expedient here to contrast the carillon with change-ringing. Sometimes both uses are made of a given set of bells.

First, the method of ringing: a single church bell, or a small group of them, is mounted in such a way that each bell can swing almost through a complete circle, say 300^o, from inverted mouth-up position through normal resting erect position to inverted position again. Locking or holding means are provided to hold the bell at one end of travel, and the ringer sets the bell in that inverted position by a special skill in manipulating the rope, which passes around a wheel, which in its turn rotates the frame holding the bell.

Since, as everyone knows, the clapper is free to swing in any direction, in this method of ringing it will not keep striking the bell in the same place--and this is very important, since a bell struck continually at one point will usually crack; remember the Liberty Bell and many other famous bells all the way from here to Russia. Even with good care, bells often crack and have to be recast. Bell history is full of instances where bells were damaged in transit.

In some instances, bells were cast in the yard of the church where they were to be used to avoid the transportation risks.

Despite the poetic allusions to silver bells and golden tones, gold and silver are not good things to make bells out of. Gold and silver plating have had to wait for electronic musical instruments to be useful, where they serve to make switch contacts more reliable. Bells are made chiefly of copper and tin, and the epithet 'tinny' is a shameful slander on a metal that contributes to the tone, not only of bells, but also of organ-pipes. Zinc goes into brass, used to make small bells and handbells. We will have to take up this subject some later time, lacking space here in this issue.

We said that change-ringing is done with freely-swinging bells, and this is in sharp contrast to the carillon that plays real tunes: since the carillon is usually played from a sort of giant keyboard, the hammers that strike being connected to it through a maze of wires and levers, and in this case the bells are mounted stationary, and they are struck from outside. To avoid cracking, the bells have to be turned every so often to present a different area to the hammer.

The tone of a single bell is very complex, and is quite unlike that of a stretched string. Also, it changes radically while it is sounding, and the sustained portion is very different from the 'strike-note.' This peculiar tone is part and parcel of the change-ringing syndrome, and helps explain why change-ringing of real bells is less boring than ringing the changes on anything else. Next best are handbells, which are used for practice.

The method of ringing is necessarily slower than a carillon can be played, or than any ordinary musical instrument can be played, or for that matter, than the just-mentioned handbells can be played.

This imposes a time-interval, making it difficult to sound the same note quickly twice in succession. In turn, this causes a change-ringing rule: a bell can't jump suddenly from any first to fourth place, but must proceed one step at a time. No given change may be repeated, except that the first change is also the last--'rounds' being the name for the normal obvious order, such as 1 2 3 4 5.

These and other rules and customs are of course incompatible with certain ordinary rules of music, such as the extensive use of repeated notes of melodies, and the almost inviolable rule in ordinary composition that there must be one or more main themes or motives, repeated again and again and returned to, and developed in various ways. Obvious example: Beethoven's Fifth Symphony.

Almost as if to contradict ordinary musical principles, and to make change-ringing harder for musicians to understand, the numbering of the bells is downward in pitch, not upward. 1 2 3 4 5 6 7 8 does not mean

thus antagonizing musicians of most any country. Do you suppose this has anything to do with the British custom of driving down the left, rather than the right, side of the road? Anyway, since this normal numerical order of the bells is a descending scale, the appropriateness of descending scale-passages in Christmas music such as Joy to the World will now be obvious. It will be equally obvious where the cliches written for movie-studio orchestras to play for films involving churches and/or bell-ringing, come from!

Bells for change-ringing are generally tuned to a portion of the major scale, but minor sometimes is used. Notes may be left out of the scale, and there is no theoretical objection to some other scales being used, although this has not been done so far as we know.

Now for the mathematical basis: permutations, or changes of order, are related to the factorial numbers. That is, if you want to know in how many different ways n things can be arranged, multiply together the ringers 1, 2, 3...

The result of this multiplication is known as 'factorial n' or sometimes 'n factorial.' The symbol used to be |n, but printers hated to set that in type, so n! is the usual modern version.

Like for instance: in how many ways--that is, how many different orders--can three things be arranged? 3!=1x2x3=6.

Smith, Brown and Jones. Smith, Jones and Brown. Brown, Smith, and Jones. Brown, Jones and Smith. Jones, Smith, and Brown. Jones, Brown, and Smith.

In how many ways can give things be arranged? 5!=1x2x3x4x5=120. Some piano teachers have students 'ring the changes' on the five fingers, going through the 120 possible different orders of five notes. Beyond 5!, the factorials quickly escalate to fantastically large numbers.

The ploughman homeward plods his weary way. Homeward the ploughman plods his weary way. The ploughman plods his weary homeward way. The ploughman plods his weary way homeward. And so on for quite a distance. In Spanish or Russian or Esperanto, more permutations would be permitted by their rules of grammar.

Seven chorus-girls could stand in a single line near the footlights in 5040 different ways.

The 52 cards in a standard deck can be shuffled into 52! different orders, but don't expect us to calculate that enormous number or write it out here! You will now realize why the exclamation-mark is an appropriate notation for factorials after all!

1x2x3x4x5x6x...x50x51x52; now let me see, that's *** well, of course, if you have no other use for your time...

Well, if 52! doesn't flabbergast you enough, how about (7!), i.e., 5040!? That is, if there are 5040 ways to ring 7 bells, then there must be factorial 5040 ways in which these changes could be taken in succession.

It this becomes evident why the rules as to which changes may follow and preceded which other changes, are so very strict. Even with these restrictions, a large number of change-ringing compositions are possible.

Since there are as many men ringing as there are bells, the memorizing of the sequence of the changes is far more difficult than it would be for a single performer to memorize a series of changes and to play them through on a piano or xylophone or flute or trumpet. Sometimes there is a conductor in the ringing-room who calls the changes, much as the caller at a dance-square meeting directs the dancers. Accordingly, certain maneuvers of the sequences of the bells have technical names: hunting, bob, dodging, slow work, quick work, ringing behind.

Since it is unlikely that many of you will take up change-ringing at some church, and since the present writer doesn't plan on such such a career either, these technical terms needn't detain us for long. Those who are more interested may read Wilson's Change-Ringing or consult the article on Change-Ringing in Grove's Dictionary of Music.

We can't go on further without calling your attention to Dorothy Sayers' mystery novel The Five Tailors, where the story is intimately woven with the facts and environment and atmosphere and customs of change-ringing.

The above table was swiped from Grove's Dictionary of Music and was probably compiled a century ago. Frankly, the estimated times for ringing all the possible changes, one of each permutation, seem unrealistically short for real bells equipped with rope and wheel--and no time for eating, sleeping, coffee-breaks, etc. Would the 96 strokes of the 4 bells be over inside one minute? The 5,040 changes on 7 bells involve 35,280 strokes which might not get all done and over with inside of 3.5 hours.

However, if one ran through the permutations on a piano or flute the changes would be over in far less time than indicated in the table, and of course a computer with sound-generator hooked up to it could give out the changes as a dizzy blur-whirr.

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That is, a contemporary composer who wanted to use permutations in his music would not be slowed down to the natural pace of real bells, and more importantly, would not have to follow all the change-ringers' rules, such as that forbidding sudden jumps like 52341 to 43125.

These rules, you have to remember, apply to a group of people, say eight men each holding the hope to one of a group of eight bells in a steeple high above them, and not enjoying too good a view of each other's rope-pulling, not being able to see much is any of the bell up there on the other end of the rope, and on top of that, a certain element of time-lag being involved. ALso, there is a continual confused jangle of after-sounds from all eight bells, containing at least one undertone (the hum-note) as well as several overtones, which overtones are not spaced anything like those of a string or wind instrument, but instead are grouped into a pungent clang that any jazz pianist would envy. Eight such after-sounds are going on at once, and if the strike-notes of the bells are in tune to a recognizable scale, that does not guarantee that all the after-sounds are in tune with themselves, let alone with those of the seven other bells!

If you were in the church's ringing-room pulling on one of those ropes, you would not want your place in a given change to jump erratically form the preceding change or to the next change. You would want the rules and restrictions, so that you could memorize the ringing methods and so that you wouldn't be embarassed by losing your place.

If you make a mistake, the whole team knows about it! Not knowing which ringer is pulling the rope to which bell, they blame the whole group for the lapse. Thus certain methods of ringing become traditional, and little room is left for really new change-ringing compositions, although there is still scope for variations, and for many selections from a series too long to be practical, as for instance 5040 out of 3,628,800.

Such shorter selections from a long series of possible permutations may be made by joining otherwise distant steps according to the rules of the particular method. This can be done, usually, at a number of points, and working out such selections becomes a sort of homework or mathematical puzzle. Since the methods of ringing determine in advance a whole series of changes, it is not necessary to write out five-thousand-and-some individual changes before the ringing can be done.

A long series of changes, according to some authorities, 5,000 or more, is called a peal. But peal is also the name for a group of bells used for change-ringing. A short series of changes, such as 200 or so is called a touch.

Note that major means 8 bells undergoing changes, and has nothing to do with the musician's major mode: whether the peal of bells is tuned to a major, minor, or some freaky way-out scale would make no difference. Similarly, minor involves six working bells, even if they are tuned to a major scale and whether there are more bells in that steeple or not.

The treble is the highest-pitched bell of the group being rung at the time, and the tenor is the lowest. But remember that the pitches are numbered downward, not upward, so that the treble is No. 1 and the tenor will be 4 or 7 or 9 or whatever number is in use. Also, in many methods the tenor is rung behind; she does not enter into the changes the other bells make in those methods, but always comes last in each change, thus punctuating the change like a bar-line in regular music.

Raven's The Bells of England gives the following example of a 'plain hunt' on 4 bells:

Now get out your red pencil and rule a thin line through all the diagonal rows of numerals 1 in the list of changes just given. The diagonal lines of 1's thus thus revealed will tell you what 'hunt' means much better than any page of soggy verbiage we could cook up.

Now visualize a large English church, and inside it, the ringing-room with ropes and a fairly athletic and patient man at each, and assume you are one of these fellows and it's your turn to pull your rope and you do so and the bell sounds somewhere up there, and the after-jingle is also going on. It's a day or two after Christmas, 1829 or such a matter, and they haven't invented television yet, and the street-traffic outside does not reverberate with the sound or the stench of automobile exhaust or motorcycle bells--and certainly not electric doorbells! This is the context in which to think about hunting up and hunting down, and is also the context and atmosphere in which to bring to life the tables of numbers given here.

That is, do not think of these numbers as being originated in our age of automation and coming out on a cash-register tape or an adding-machine tape or a computer printout or a deck of IBM punchcards.

Today's methods of using permutations and combinations, even the musical ways, are quite different, and we will allude to this before closing.

We are so used to visualizing history as wars and rumors of wars, turmoil, treachery, disorder, cruelty, injustice, revolutions, intrigue, surprise attacks, etc., that it should be a real relief to you to realize that there are some islands of tranquility, tidiness, order, proper functioning of rules, and consistency in the world, after all, and one of these is the practice of change-ringing of bells, which seems to have started in 17th-century ENgland and was promoted by such experts as Fabian Stedman, a printer who printed up sheets with the arrays of numbers and published a book called Tintinnalogia, and after whom the Stedman method is named.

Note that this is only a small part of the long history of bells themselves, both small and large, so that we will have to return to bells in a later issue and tell how ancient gongs and metal bars evolved into crude riveted-together cowbells and such, and later into the graceful outline now thought of as 'bell-shaped,' and then after change-ringing, the more refined system of tuning bells to themselves and to one another, developed for the carillon on which actual tunes are played: and because carillon-playing involves sounding several bells simultaneously or almost so in chords, tuning must be more careful than that required for change-ringing. In change-ringing, it would be a mistake if two or more bells were struck simultaneously.

This has created a curious paradox in more recent years: English foundries cast and tune fine bells for carillons, but nearly all of the finished carillons are shipped away to Europe or the United States, the English churches preferring to continue with the changes of Kent Treble Bob Major, Stedman's, Surprise Major, and Grandsire Triples.

Actually, there is no reason why the same set of bells cannot be both part of a carillon and also be used for change-ringing. Of course, this might involve some mechanical complications, but it is far from impossible. ALso, there is an important and very useful intermediate case: the Canterbury and Westminster clock-chime tunes, copied on tower-clocks and mantel-clocks and grandfather-clocks virtually around the world.

Now to wind things up, let's turn to musical applications of the mathematical principle of permutations and combinations outside change-ringing. Take the familiar Westminster Chime tune, for a starter: there is an approximation to the ringing of changes, but there is the freedom of repeating a note. Immediately it becomes more interesting and less monotonous.

Then we may progress to bugle calls, where there are only four notes, but this time rhythm is introduced. Four small bells could sound bugle calls, and so could auto horns.

Quite a number of finger exercises for piano, violin, and other instruments involve permutations and combinations. The musical form called Theme and Variations may apply permutation in some instances. Once rhythm is admitted (the change-ringing bell-strokes are supposed to be of equal duration) rests, i.e., silences, may enter into permutation with notes.

If permutation, preferably in a more sophisticated form than ringing the changes, were substituted for the monotonous exact repetitions, such as the Alberti bass and rigid ostinati, everyone would gain. The uninspired oom-pah-pah of the traditional waltz was one of the tortures of our boyhood piano practice, and we found out that other budding pianists agreed with us. How simple it would have been if composers permuted or varied the bass-accompaniment-pattern! Who could possibly lose by this creative introduction of a little variety?

Similar remarks apply to marches, as well as the various dance-forms. Popular music has frequently committed the gravest excesses in the accompaniment department, the most stultifying boring incessant monotony, reminding this listener of nothing so much as a Linotype machine, a printing-press, a windshield-wiper, a stamping-mill, or a pump.

The monotonous rhythms an accompaniment formulas of much popular music reduce the performers to mere machines: a machine could do that stuff much better! They mock their own humanity; they degrade themselves.

We won't describe What Ought To Be Done About It here because the trouble is precisely that there has been too much prescription and not enough freedom. If everybody applies the same remedies to escape the musical straitjacket, the situation will be just as bad as before. Individuality implies difference.

Which brings us to another application of the permutation principle: the tone-row of twelve-tone serialism. Change-ringing with its harnessing of factorials and suggestion that the changes themselves can be permuted, curiously anticipates Schoenberg's atonality and its consequences by just over 200 years.

Arnold Schoenberg, realizing that the ordinary musical key-system reigning supreme in the 19th-century Romantic Period would come to thwart progress in certain directions, constructed a new compositional system in which the ideas of tonic, dominant, subdominant, major, minor, modulation, and--to prevent anarchy--a new kind of order was imposed, with its new set of rules and principles.

His followers unfortunately often lack his genius, so that contemporary serialism, the modern consequence of Schoenberg's twelve-tone system, often sounds boring...and worse yet, the followers' compositions are far too much alike, although Schoenberg did not fall into his own trap like that.

The attempt to exploit the whole-tone scale and augmented triad beyond where he left off gives rise to the most trivial, banal, annoying, and futile of soap-opera cliches, and as for the diminished-seventh chord, dividing the 12-tone octave into equal quarters, that had already been exploited to the point of Ugh!

This is said in respect of Debussy and Schoenberg and Bach and some other composers, not to detract from their achievements but to point out that the attempt to copy them slavishly and to repeat them is an act of disrespect and an admission of lack of creative inspiration and originality.

The names C-sharp and D-flat applied to the second the 12 equal notes of the tonal system, or their French equivalents Re diese and Re bemel, are thus misleading when used to discuss the works of the successors of Schoenberg, because they imply Notes Numbers 2, 4, 7, 8, and 11 are in some way inferior to or subordinate to, those derived from Notes 1, 3, 5, 6, 8, 10 and 12 of the atonal dodecaphonic scale. We could use the last twelve letters of the alphabet opqrstuvwxyz or the first twelve greek letters alpha beta gamma delta epsilon zeta eta theta iota kai lambda mu or the duodecimal numerals 0 1 2 3 4 5 6 7 8 9 X E, or one or another of the sets of twelve syllables that have been invented for this purpose.

Twelfth-inch cross-section graph-paper is available for notation of much music. Player-piano rolls also offer an impartial notation which furthermore causes the performance of the music.

Why twelve-tone serialism rather than 13 or some other number? The crassest and crudest of economics and expediency and taking-for-granted, we believe. Pianos were handy; pianists and piano-tuners were available as highly-trained standardized products; it would have cost too much to go to 11 or 13 tones per octave and really get away from the old tonal key-system.

This composer does not follow the serialist teaching in his own compositions, but that is not a repudiation of permutation principles as a possible musical resource. Rather it is the consequence of being mainly busy with the exploration of an expanded tonal system applied to such systems as the 17, 19, 22, 24, 31, 34, 41, and other equal and unequal divisions of the octave now economically feasible because of electronics and automation.

Here's an idea for you: why don't you compose on a greatly-liberalized change-ringing permutation basis, using percussion instruments of indefinite pitch or frank noisemakers, instead of bells? By adding rhythmic variety as the bugle-calls do, interest can be sustained.

It is not too likely that the change-ringing custom will spread around the world much more that the already-obtaining sporadic instances of emigrating Englishmen bringing it with them to overseas churches, but in its homeland it is a sufficiently firmly-established tradition that there can be no danger of it dying out. It is sufficiently athletic and tiring, especially in the long peals of over 5000 changes, that maybe it will eventually be accepted for the Olympics!

Our only listening to it was in about 1926 and again in 1931, when we lived a block from the Trinity Church in Portland, Oregon, where change-ringing went on about two or three times a week.

This brings up a question: why does the city of Los Angeles have such a prejudice against church bells? Is there a law against ringing them, or a tacit agreement not to ring them very loudly? In almost every city we have visited, far more bells were heard.

It can't be claimed that modern auto-horns and pneumatic riveters and motor-cycles and backfiring and freight trains and other city noises, all of which are tolerated--and now jet aircraft--are more pleasant than bells, so it probably has nothing to do with anti-noise laws.

This being the December issue, we thought that this article would be appropriate: the change-ringing tradition surely explains the numerous bells on Christmas-cards and holiday decorations, and the proverbial expressions Ring the changes on something or other and "Ring out the old year, ring in the New" will surely have more meaning for you.

The basic mathematical idea had its appropriate expression in 17th-century England, and then, as we saw, gave rise to atonal serialism as the 19th century gave way to the 20th. The latest thing is 'aleatory' or 'chance' music, which has more to do with random numbers than permutations and combinations, but is not too remote from the latter.

Now let us speculate: what would Edgar Allen Poe's famous poem, The Bells, have been like, if he had heard streetcar bells, cash-register bells, typewriter bells, electric doorbells, and--oops! The telephone bell is ringing.