CRAB: I see. But I still don’t see where number theory enters the picture, or what this all has to do with my new records.
ACHILLES: Well, in the mathematics of acoustico-retrieval, there arise many questions which have to do with the number of solutions of certain Diophantine equations. Now Mr. T has been for years trying to find a way of reconstructing the sounds of Bach playing his harpsichord, which took place over two hundred years ago, from calculations involving the motions of all the molecules in the atmosphere at the present time.
ANTEATER: Surely that is impossible! They are irretrievably gone, gone forever!
ACHILLES: Thus think the naïve … But Mr. T has devoted many years to this problem, and came to the realization that the whole thing hinged on the number of solutions to the equation
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 the Bach sounds can be retrieved from the motion of all the molecules in the atmosphere, provided that there exists either 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 no 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 proof of Fermat’s Last Theorem—if one existed—could be described by an elegant formula, which, it so happened, depended on the values of a solution to a certain equation. When I found this second equation, to 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 as, a blueprint for constructing my proof that there were no solutions to the 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 so 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 twenty-four 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 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, the days when I thrilled to each new prelude and fugue, filled with the 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 in that familiarity there is also a kind of depth, which has its own compensations. For instance, I find that there are always new surprises which I 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 depths, out of nowhere. But there are also amazing modulations which it is marvelous to listen to over and over again, and wonder how old Bach dreamt them up.
ACHILLES: I am very glad to hear that there is something to look forward to, after I have been through the first flush of infatuation with the Well-Tempered Clavier—although it also makes me sad that this stage could not last forever and ever.
CRAB: Oh, you needn’t fear that your infatuation will totally die. One of the nice things about that sort of youthful thrill is that it can always be 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 to whom it is a totally new experience—someone such as you, Achilles. Somehow the excitement transmits itself, and I can feel thrilled again.