Gödel, Escher, Bach: An Eternal Golden Braid (132 page)

A Bach Vortex Where All Levels Cross

One cannot help being reminded, when one looks at the diagrams of Strange Loops, of the Endlessly Rising Canon from the Musical Offering. A diagram of it would consist of six steps, as is shown in Figure 147. It is too .

FIGURE 147.
The hexagonal modulation scheme of Bach's Endlessly Rising Canon
forms a true closed loop when Shepard tones are used
.

bad that when it returns to C, it is an octave higher rather than at the exact original pitch.

Astonishingly enough, it is possible to arrange for it to return exactly to the starting pitch, by using what are called Shepard tones, after the psychologist Roger Shepard, who discovered the idea. The principle of a Shepard-tone scale is shown in Figure 148. In words, it is this: you play parallel scales in several different octave ranges. Each note is weighted independently, and as the notes rise, the weights shift. You make the top

FIGURE 148:
Two complete cycles of a Shephard tone scale, notated for piano. The
loudness of each note is proportional to its area, just as the top voice fades out, a new
bottom voice feebly enters. (Printed by Donald Boydś program “SMUT”.)
octave gradually fade out, while at the same time you are gradually bringing in the bottom octave. Just at the moment you would ordinarily be one octave higher, the weights have shifted precisely so as to reproduce the starting pitch ... Thus you can go

"up and up forever", never getting any higher! You can try it at your piano. It works even better if the pitches can be synthesized accurately under computer control. Then the illusion is bewilderingly strong.

This wonderful musical discovery allows the Endlessly Rising Canon to be played in such a way that it joins back onto itself after going "up" an octave. This idea, which Scott Kim and I conceived jointly, has been realized on tape, using a computer music system. The effect is very subtle-but very real. It is quite interesting that Bach himself was apparently aware, in some sense, of such scales, for in his music one can occasionally find passages which roughly exploit the general principle of Shepard tones-for instance, about halfway through the Fantasia from the Fantasia and Fugue in G Minor, for organ.

In his book
J. S. Bach's Musical Offering
, Hans Theodore David writes: Throughout the
Musical Offering
, the reader, performer, or listener is to search for the Royal theme in all its forms. The entire work, therefore, is a ricercar in the original, literal sense of the word.'

I think this is true; one cannot look deeply enough into the
Musical Offering
. There is always more after one thinks one knows everything. For instance. towards the very end of the
Six-Part Ricercar
, the one he declined to improvise, Bach slyly hid his own name, split between two of the upper voices. Things are going on on many levels in the
Musical
Offering.
There are tricks with notes and letters; there are ingenious variations on the King's Theme; there are original kinds of canons; there are extraordinarily complex fugues; there is beauty and extreme depth of emotion; even an exultation in the many-leveledness of the work comes through. The
Musical Offering
is a fugue of fugues, a Tangled Hierarchy like those of Escher and Gödel, an intellectual construction which reminds me, in ways I cannot express, of the beautiful many-voiced fugue of the human mind. And that is why in my book the three strands of Gödel, Escher, and Bach are woven into an Eternal Golden Braid.

Six-Part Ricercar

Achilles has brought his cello to the Crab's residence, to engage in an evening of
chamber music with the Crab and Tortoise. He has been shown into the music
room by his host the Crab, who is momentarily absent, having gone to meet their
mutual friend the Tortoise at the door. The room is filled with all sorts of electronic
equipment-phonographs in various states of array and disarray, television screens
attached to typewriters, and other quite improbable-looking pieces of apparatus.

Nestled amongst all this high-powered gadgetry sits a humble radio. Since the
radio is the only thing in the room which Achilles knows how to use, he walks over
to it, and, a little furtively, flicks the dial and f nds he has tuned into a panel
discussion by six learned scholars on free will and determinism. He listens briefly
and then, a little scornfully, flicks it off.

Achilles: I can get along very well without such a program. After all, it's clear to anyone who's ever thought about it that-I mean, it's not a very difficult matter to resolve, once you understand how-or rather, conceptually, one can clear up the whole thing by thinking of, or at least imagining a situation where ... Hmmm ... I thought it was quite clear in my mind. Maybe I could benefit from listening to that show, after all ...

(Enter the Tortoise, carrying his violin.)

Well, well, if it isn't our fiddler. Have you been practicing faithfully this week, Mr. T?

I myself have been playing the cello part in the Trio Sonata from the
Musical Offering
for at least two hours a day. It's a strict regimen, but it pays off.

Tortoise: I can get along very well without such a program. I find that a moment here, a moment there keeps me fit for fiddling.

Achilles: Oh, lucky you. I wish it came so easily to me. Well, where is our host?

Tortoise: I think he's just gone to fetch his flute. Here he comes.

(Enter the Crab, carrying his flute.)

Achilles: Oh, Mr. Crab, in my ardent practicing of the Trio Sonata this past week, all sorts of images bubbled into my mind: jolly gobbling bumblebees, melancholy buzzing turkeys, and a raft of others. Isn't it wonderful, what power music has?

Crab: I can get along very well without such a program. To my mind.

Achilles, there is no music purer than the Musical Offering.

Tortoise: You can't be serious, Achilles. The Musical Offering isn't programmatic music!

Achilles: Well, I like animals, even if you two stuffy ones disapprove.

Crab: I don't think we are so stuffy, Achilles. Let's just say that you hear music in 'your own special way.

Tortoise: Shall we sit down and play?

Crab: I was hoping that a pianist friend of mine would turn up and play continuo. I've been wanting you to meet him, Achilles, for a long time. Unfortunately, it appears that he may not make it. So let's just go ahead with the three of us. That's plenty for a trio sonata.

Achilles: Before we start, I just was wondering, Mr. Crab-what are all these pieces of equipment, which you have in here?

Crab: Well, mostly they are just odds and ends-bits and pieces of old broken phonographs. Only a few souvenirs (nervously tapping the buttons), a few souvenirs of-of the TC-battles in which I have distinguished myself. Those keyboards attached to television screens, however, are my new toys. I have fifteen of them around here.

They are a new kind of computer, a very small, very flexible type of computer quite an advance over the previous types available. Few others seem to be quite as enthusiastic about them as I am, but I have faith that they will catch on in time.

Achilles: Do they have a special name?

Crab: Yes; they are called "smart-stupids", since they are so flexible, and have the potential to be either smart or stupid, depending on how skillfully they are instructed.

Achilles: Do you mean you think they could actually become smart like, say, a human being?

Crab: I would not balk at saying so-provided, of course, that someone sufficiently versed in the art of instructing smart-stupids would make the effort. Sadly, I am not personally acquainted with anyone who is a true virtuoso. To be sure, there is one expert abroad in the land, an individual of great renown-and nothing would please me more than a visit by him, so that I could appreciate what true skill on the smart-stupid is; but he has never come, and I wonder if I shall ever have that pleasure.

Tortoise: It would be very interesting to play chess against a well-instructed smart-stupid.

Crab: An extremely intriguing idea. That would be a wonderful mark of skill, to program a smart-stupid to play a good game of chess. Even more interesting-but incredibly complicated-would be to instruct a smart-stupid sufficiently that it could hold its own in a conversation. It might give the impression that it was just another person!

Achilles: Curious that this should come up, for I just heard a snatch of a discussion on free will and determinism, and it set me to thinking about such questions once more. I don't mind admitting that, as I pondered the idea, my thoughts got more and more tangled, and in the end I really didn't know what I thought. But this idea of a smartstupid that could converse with you ... it boggles the mind. I mean, what would the smart-stupid itself say, if you asked it for its opinion on the free-will question? I was just wondering if the two of you, who know so much about these things, wouldn't indulge me by explaining the issue, as you see it, to me.

Crab: Achilles, you can't imagine how appropriate your question is. I only wish my pianist friend were here, because I know you'd be intrigued to hear what he could tell you on the subject. In his absence, I'd like to tell you a statement in a Dialogue at the end of a book I came across recently.

Achilles: Not
Copper, Silver, Gold: an Indestructible Metallic Alloy
?

Crab: No, as I recall, it was entitled Gi
raffes, Elephants, Baboons: an Equatorial
Grasslands Bestiary
-or something like that. In any case, towards the end of the aforementioned Dialogue, a certain exceedingly droll character quotes Marvin Minsky on the question of free will. Shortly thereafter, while interacting with two other personages, this droll character quotes Minsky further on musical improvisation, the computer language LISP, and Godel's Theorem-and get this-all without giving one whit of credit to Minsky!

Achilles: Oh, for shame!

Crab: I must admit that earlier in the Dialogue, he hints that he WILL quote Minsky towards the end; so perhaps it's forgivable.

Achilles: It sounds that way to me. Anyway, I'm anxious to hear the Minskian pronouncement on the free will question.

Crab: Ah, yes... Marvin Minsky said, "When intelligent machines are constructed, we should not be surprised to find them as confused and as stubborn as men in their convictions about mind-matter, consciousness, free will, and the like."

Achilles: I like that! Quite a funny thought. An automaton thinking it had free will! That's almost as silly as me thinking I didn't have free will! Tortoise: I suppose it never occurred to you, Achilles, that the three of us-you, myself, and Mr. Crab-might all be characters in a Dialogue, perhaps even one similar to the one Mr. Crab just mentioned. Achilles: Oh, it's occurred to me, of course. I suppose such fancies occur to every normal person at one time or another.

Tortoise: And the Anteater, the Sloth, Zeno, even GOD-we might all be characters in a series of Dialogues in a book.

Achilles: Sure, we might. And the Author might just come in and play the piano, too.

Crab: That's just what I had hoped. But he's always late.

Achilles: Whose leg do you think you're pulling? I know I'm not being controlled in any way by another mentality! I've got my own thoughts, I express myself as I wish-you can't deny that!

Tortoise: Nobody denied any of that, Achilles. But all of what you say is perfectly consistent with your being a character in a Dialogue.

Crab: The---

Achilles: But-but-no! Perhaps Mr. C's article and my rebuttal have both been mechanically determined, but this I refuse to believe. I can accept physical determinism, but I cannot accept the idea that I am but a figment inside of someone else's mentality!

Tortoise: It doesn't really matter whether you have a hardware brain, Achilles. Your will can be equally free, if your brain is just a piece of software inside someone else's hardware brain. And their brain, too, may be software in a yet higher brain .. .

Achilles: What an absurd idea! And yet, I must admit, I do enjoy trying to find the cleverly concealed holes in your sophistry, so go ahead. Try to convince me. I'm game.

Tortoise: Did it ever strike you, Achilles, that you keep somewhat unusual company?