Gödel, Escher, Bach: An Eternal Golden Braid (58 page)

Achilles: Well, in the mathematics of acoustico-retrieval, there arise rr questions which have to do with the number of solutions of cer Diophantine equations. Now Mr. T has been for years trying to fit way of reconstructing the sounds of Bach playing his harpsichord, which took place over two hundred years ago, from calculations in% ing the motions of all the molecules in the atmosphere at the pre time.

Anteater: Surely that is impossible! They are irretrievably gone, g forever!

Achilles: Thus think the nave ... But Mr. T has devoted many year this problem, and came to the realization that the whole thing hinged on the number of solutions to the equation

an +bn=cn

in positive integers, with n > 2.

Tortoise: I could explain, of course, just how this equation arises, but I’m sure it would bore you.

Achilles: It turned out that acoustico-retrieval theory predicts that Bach sounds can be retrieved from the motion of all the molecule the atmosphere, provided that EITHER

there exists at least one solution to the equation

Crab: Amazing!

Anteater: Fantastic!

Tortoise: Who would have thought!

Achilles: I was about to say, "provided that there exists EITHER such a solution OR a proof that there are tic) solutions!" And therefore, Mr. T, in careful fashion, set about working at both ends of the problem, simultaneously. As it turns out, the discovery of the counterexample was the key ingredient to finding the proof, so the one led directly to the other.

Crab: How could that be?

Tortoise: Well, you see, I had shown that the structural layout of any pr of Fermat's Last Theorem-if one existed-could be described by elegant formula, which, it so happened, depended on the values ( solution to a certain equation. When I found this second equation my surprise it turned out to be the Fermat equation. An amusing accidental relationship between form and content. So when I found the counterexample, all I needed to do was to use those numbers blueprint for constructing my proof that there were no solutions to equation. Remarkably simple, when you think about it. I can't imagine why no one had ever found the result before.

Achilles: As a result of this unanticipatedly rich mathematical success, Mr. T was able to carry out the acoustico-retrieval which he had long dreamed of. And Mr. Crab's present here represents a palpable realization of all this abstract work.

Crab: Don't tell me it's a recording of Bach playing his own works for harpsichord!

Achilles: I'm sorry, but I have to, for that is indeed just what it is! This is a set of two records of Johann Sebastian Bach playing all of his
Well Tempered Clavier
. Each record contains one of the two volumes of the
Well-Tempered Clavier
; that is to say, each record contains 24 preludes and fugues-one in each major and minor key.

Crab: Well, we must absolutely put one of these priceless records on, immediately! And how can I ever thank the two of you?

Tortoise: You have already thanked us plentifully, with this delicious tea which you have prepared.

(The Crab slides one of the records out of its jacket, and puts it on. The sound of
an incredibly masterful harpsichordist fills the room, in the highest imaginable
fidelity. One even hears-or is it one's imagination?-the soft sounds of Bach singing
to himself as he plays ...)

Crab: Would any of you like to follow along in the score? I happen to have a unique edition of the
Well-Tempered Clavier
, specially illuminated by a teacher of mine who happens also to be an unusually fine calligrapher. Tortoise: I would very much enjoy that.

(The Crab goes to his elegant glass-enclosed wooden bookcase, opens the doors, and
draws out two large volumes.)

Crab: Here you are, Mr. Tortoise. I've never really gotten to know all the beautiful illustrations in this edition. Perhaps your gift will provide the needed impetus for me to do so.

Tortoise: I do hope so.

Anteater: Have you ever noticed how in these pieces the prelude always sets the mood perfectly for the following fugue?

Crab: Yes. Although it may be hard to put it into words, there is always some subtle relation between the two. Even if the prelude and fugue do not have a common melodic subject, there is nevertheless always some intangible abstract quality which underlies both of them, binding them together very strongly.

Tortoise: And there is something very dramatic about the few moments of silent suspense hanging between prelude and fugue-that moment where the the theme of the fugue is about to ring out, in single tones, and then to join with itself in ever-increasingly complex levels of weird, exquisite harmony.

Achilles: I know just what you mean. There are so many preludes and fugues which I haven't yet gotten to know, and for me that fleeting interlude of silence is very exciting; it's a time when I try to second-guess old Bach. For example, I always wonder what the fugue's tempo will be: allegro, or adagio? Will it be in 6/8, or 4/4?

Will it have three voices, or five-or four? And then, the first voice starts ... Such an exquisite moment.

Crab: Ah, yes, well do I remember those long-gone days of my youth, days when I thrilled to each new prelude and fugue, filled with excitement of their novelty and beauty and the many unexpected' surprises which they conceal.

Achilles: And now? Is that thrill all gone?

Crab: It's been supplanted by familiarity, as thrills always will be. But that familiarity there is also a kind of depth, which has its own compensations. For instance, I find that there are always new surprises whit hadn't noticed before.

Achilles: Occurrences of the theme which you had overlooked?

Crab: Perhaps-especially when it is inverted and hidden among several other voices, or where it seems to come rushing up from the dept out of nowhere. But there are also amazing modulations which ii marvelous to listen to over and over again, and wonder how old B2 dreamt them up.

Achilles: I am very glad to hear that there is something to look forward after I have been through the first flush of infatuation with the
Well Tempered Clavier
-although it also makes me sad that this stage cot not last forever and ever.

Crab: Oh, you needn't fear that your infatuation will totally die. One the nice things about that sort of youthful thrill is that it can always resuscitated, just when you thought it was finally dead. It just takes the right kind of triggering from the outside.

Achilles: Oh, really? Such as what?

Crab: Such as hearing it through the ears, so to speak, of someone whom it is a totally new experience-someone such as you, Achilles. Somehow the excitement transmits itself, and I can feel thrilled again.

Achilles: That is intriguing. The thrill has remained dormant somewhere inside you, but by yourself, you aren't able to fish it up out of your subconscious.

Crab: Exactly. The potential of reliving the thrill is "coded", in sot unknown way, in the structure of my brain, but I don't have the power to summon it up at will; I have to wait for chance circumstance trigger it.

Achilles: I have a question about fugues which I feel a little embarrass about asking, but as I am just a novice at fugue-listening, I was wondering if perhaps one of you seasoned fugue-listeners might help me learning .. .

Tortoise: I'd certainly like to offer my own meager knowledge, if it might prove of' some assistance.

Achilles: Oh, thank you. Let me come at the question from an angle. Are you familiar with the print called
Cube with Magic Ribbons
, by M. Escher?

Tortoise: In which there are circular bands having bubble-like distortions which, as soon as you've decided that they are bumps, seem to turn it dents-and vice versa?

FIGURE 56. Cube with Magic Ribbons, by
M. C. Escher (lithograph, 1957
).

Achilles: Exactly.

Crab: I remember that picture. Those little bubbles always seem to flip back and forth between being concave and convex, depending on the direction that you approach them from. There's no way to see them simultaneously as concave AND convex-somehow one's brain doesn't allow that. There are two mutually exclusive "modes" in which one can perceive the bubbles.

Achilles: Just so. Well, I seem to have discovered two somewhat analogous modes in which I can listen to a fugue. The modes are these: either to follow one individual voice at a time, or to listen to the total effect of all of them together, without trying to disentangle one from another. I have tried out both of these modes, and, much to my frustration, each one of them shuts out the other. It's simply not in my power to follow the paths of individual voices and at the same time to hear the whole effect. I find that I flip back and forth between one mode and the other, more or less spontaneously and involuntarily.

Anteater: Just as when you look at the magic bands, eh?

Achilles: Yes. I was just wondering ... does my description of they modes of fugue-listening brand me unmistakably as a naive, inexperienced listener, who couldn't even begin to grasp the deeper mo, perception which exist beyond his ken?

Tortoise: No, not at all, Achilles. I can only speak for myself, but I to myself shifting back and forth from one mode to the other without exerting any conscious control over which mode should he dominant don't know if our other companions here have also experience( thing similar.

Crab: Most definitely. It's quite a tantalizing phenomenon, since you feel that the essence of the fugue is flitting about you, and you can't grasp all of it, because you can't quite make yourself function ways at once.

Anteater: Fugues have that interesting property, that each of their voices is a piece of music in itself; and thus a fugue might be thought o collection of several distinct pieces of music, all based on one theme, and all played simultaneously. And it is up to the listener subconscious) to decide whether it should be perceived as a unit, c collection of independent parts, all of which harmonize.

Achilles: You say that the parts are "independent", yet that can't be literally true. There has to be some coordination between them, otherwise when they were put together one would just have an unsystematic clashing of tones-and that is as far from the truth as could b,

Anteater: A better way to state it might be this: if you listened to each on its own, you would find that it seemed to make sense all by its could stand alone, and that is the sense in which I meant that it is independent. But you are quite right in pointing out that each of individually meaningful lines fuses with the others in a highly nonrandom way, to make a graceful totality. The art of writing a beautiful fugue lies precisely in this ability, to manufacture several diff lines, each one of which gives the illusion of having been written I own beauty, and yet which when taken together form a whole, , does not feel forced in any way. Now, this dichotomy between he a fugue as a whole, and hearing its component voices, is a part: example of a very general dichotomy, which applies to many kit structures built up from lower levels.

Achilles: Oh, really? You mean that my two "modes" may have some general type of applicability, in situations other than fugue-listening?

Anteater: Absolutely.

Achilles: I wonder how that could be. I guess it has to do with alternating between perceiving something as a whole, and perceiving it as a collection of parts. But the only place I have ever run into that dichotomy is in listening to fugues.

Tortoise: Oh, my, look at this! I just turned the page while following the music, and came across this magnificent illustration facing the page of the fugue.

Crab: I have never seen that illustration before. Why don't you pass it 'round?

(The Tortoise passes the book around. Each of the foursome looks at it in a
characteristic way-this one from afar, that one from close up, everyone tipping his
head this way and that in puzzlement. Finally it has made the rounds, and returns
to the Tortoise, who peers at it rather intently.)

Achilles: Well, I guess the prelude is just about over. I wonder if, as I listen to this fugue, I will gain any more insight into the question, "What is the right way to listen to a fugue: as a whole, or as the sum of its parts?"

TTortoise: Listen carefully, and you will!

(The prelude ends. There is a moment of silence; and ...