Gödel, Escher, Bach: An Eternal Golden Braid (24 page)

Achilles: Oh, really? What is its title?

Tortoise:
Provocative Adventures of the Tortoise and Achilles Taking
Place in Sundry Parts of the Globe
. It sounds like an interesting book to read out of.

FIGURE 24. Reptiles,
by M. C. Escher (lithograph, 1943
).

Achilles: Well, You can read it if you want, but as for I'm not going to take any chances with t popping-tonic-one of the lizards might knock it off the table, so I'm going to get it right now!

(He dashes over to the table and reaches for the popping-tonic, but in
his haste he somehow bumps the flask of tonic, and it tumbles off the
desk and begins rolling.)

Oh, no! Mr. T-look! I accidentally knocked tonic onto the floor, and it's rolling toward towards-the stairwell! Quick-before it falls!

(The Tortoise, however, is completely wrapped up in the thin volume
which he has in his hands.)
Achilles: Well, You can read it if you want, but as for I'm not going to take any chances with t poppingtonic-one of the lizards might knock it off the table, so I'm going to get it right

Tortoise (
muttering
): Eh? This story looks fascinating.

Achilles: Mr. T, Mr. T, help! Help catch the tonic-flask!

Tortoise: What's all the fuss about?

Achilles: The tonic-flask-I knocked it down from the desk, and now it's rolling and

(At that instant it reaches the brink of the stairwell, and plummets
over ... )

Oh no! What can we do? Mr. Tortoise-aren't you alarmed? We're losing our tonic! It's just fallen down the stairwell! There's only one thing to do! We'll have to go down one story!

Tortoise: Go down one story? My pleasure. Won't you join me?

(He begins to read aloud, and Achilles, pulled in two directions at
once, finally stays, taking the role of the Tortoise.)

Achilles: It's very dark here, Mr. T. I can't see a thing. Oof! I bumped into a wall. Watch out!

Tortoise: Here-I have a couple of walking sticks. Why don't you take one of them? You can hold it out in front of you so that you don't bang into things.

Achilles: Good idea. (
He takes the stick
.) Do you get the sense that this path is curving gently to the left as we walk? Tortoise: Very slightly, yes.

Achilles: I wonder where we are. And whether we'll ever see the light of day again. I wish I'd never listened to you, when you suggested I swallow some of that "DRINK ME" stuff.

Tortoise: I assure you, it's quite harmless. I've done it scads of times, and not a once have I ever regretted it. Relax and enjoy being small.

Achilles: Being small? What is it you've done to me, Mr. T?

Tortoise: Now don't go blaming me. You did it of your own free will.

Achilles: Have you made me shrink? So that this labyrinth we're

in is actually some teeny thing that someone could STEP on?

FIGURE 25. Cretan Labyrinth (Italian engraving; School of
Finiguerra). [From N Matthews, Mazes and Labyrinths: Their
History and Development (New York: Dover Publications, 1970).

Tortoise: Labyrinth? Labyrinth? Could it Are we in the notorious Little Harmonic Labyrinth of the dreaded Majotaur?

Achilles: Yiikes! What is that?

Tortoise: They say-although I person never believed it myself-that an I Majotaur has created a tiny labyrinth sits in a pit in the middle of it, waiting innocent victims to get lost in its fears complexity.

Then, when they wander and dazed into the center, he laughs and

laughs at them-so hard, that he laughs them to death!

Achilles: Oh, no!

Tortoise: But it's only a myth. Courage, Achilles.

(And the dauntless pair trudge on.)

Achilles: Feel these walls. They're like o gated tin sheets, or something.

But the corrugations have different sizes.

(To emphasize his point, he sticks out his walking stick against the
wall surface as he walks. As the stick bounces back and forth against
the corrugations, strange noises echo up and down the long curved
corridor they are in.)

Tortoise (alarmed): What was THAT?

Achilles: Oh, just me, rubbing my walking stick against the wall.

Tortoise: Whew! I thought for a moment it was the bellowing of the ferocious Majotaur! Achilles: I thought you said it was all a myth.

Tortoise: Of course it is. Nothing to be afraid of.

(Achilles puts his walking stick back against the wall, and continues
walking. As he does so, some musical sounds are heard, coming from
the point where his stick is scraping the wall.)

Tortoise: Uh-oh. I have a bad feeling, Achilles.

That Labyrinth may not be a myth, after all. Achilles: Wait a minute.

What makes you change your mind all of a sudden? Tortoise: Do

you hear that music?

(To hear more clearly, Achilles lowers the stick, and the strains of
melody cease.)

Hey! Put that back! I want to hear the end of this piece!

(Confused, Achilles obeys, and the music resumes.)

Thank you. Now as I was about to say, I have just figured out where we are.

Achilles: Really? Where are we?

Tortoise: We are walking down a spiral groove of a record in its jacket.

Your stick scraping against the strange shapes in the wall acts like a needle running down the groove, allowing us to hear the music.

Achilles: Oh, no, oh, no ...

Tortoise: What? Aren't you overjoyed? Have you ever had the chance to be in such intimate contact with music before?

Achzltes: How am I ever going to win footraces against full-sized people when I am smaller than a flea, Mr. Tortoise?

Tortoise: Oh, is that all that's bothering you That's nothing to fret abopt, Achilles.

Achilles: The way you talk, I get the impression that you never worry at all.

Tortoise: I don't know. But one thing for certain is that I don't worry about being small. Especially not when faced with the awful danger of the dreaded Majotaur!

Achilles: Horrors! Are you telling me

Tortoise: I'm afraid so, Achilles. The music gave it away.

Achilles: How could it do that?

Tortoise: Very simple. When I heard melody
B-A-C-H
in the top voice, I immediately realized that the grooves we're walking through could only be
Little Harmonic Labyrinth
, one of Bach's er known organ pieces. It is so named cause of its dizzyingly frequent modulations.

Achilles: Wh-what are they?

Tortoise: Well, you know that most music pieces are written in a key, or tonality, as C major, which is the key of this o;

Achilles: I had heard the term before. Do that mean that C is the note you want to on?

Tortoise: Yes, C acts like a home base, in a Actually, the usual word is

"tonic".

Achilles: Does one then stray away from tonic with the aim of eventually returning

Tortoise: That's right. As the piece develops ambiguous chords and melodies are t which lead away from the tonic. Little by little, tension builds up-you feel at creasing desire to return home, to hear the tonic.

Achilles: Is that why, at the end of a pie always feel so satisfied, as if I had waiting my whole life to hear the ton

Tortoise: Exactly. The composer has uses knowledge of harmonic progressions to

manipulate your emotions, and to build up hopes in you to hear that tonic.

Achilles: But you were going to tell me about modulations.

Tortoise: Oh, yes. One very important thing a composer can do is to

"modulate" partway through a piece, which means that he sets up a temporary goal other than resolution into the tonic.

Achilles: I see ... I think. Do you mean that some sequence of chords shifts the harmonic tension somehow so that I actually desire to resolve in a new key?

Tortoise: Right. This makes the situation more complex, for although in the short term you want to resolve in the new key, all the while at the back of your mind you retain the longing to hit that original goal-in this case, C major. And when the subsidiary goal is reached, there is

Achilles (
suddenly gesturing enthusiastically
): Oh, listen to the gorgeous upward-swooping chords which mark the end of this
Little
Harmonic Labyrinth!

Tortoise: No, Achilles, this isn't the end. It's merely

Achilles: Sure it is! Wow! What a powerful, strong ending! What a sense of relief! That's some resolution! Gee!

(And sure enough, at that moment the music stops, as they emerge into
an open area with no walls.)

You see, it Is over. What did I tell you? Tortoise: Something is very wrong. This record

is a disgrace to the world of music. Achilles: What do you mean?

Tortoise: It was exactly what I was telling you about. Here Bach had modulated from C into G, setting up a secondary goal of hearing

G. This means that you experience two tensions at once-waiting for resolution into G, but also keeping in mind that ultimate desire-to resolve triumphantly into C Major.

Achilles: Why should you have to keep any

thing in mind when listening to a piece of music? Is music only an intellectual exercise?

Tortoise: No, of course not. Some music is highly intellectual, but most music is not. And most of the time your ear or br the

"calculation" for you, and lets your emotions know what they want to hear, don't have to think about it consciously in this piece, Bach was playing tricks hoping to lead you astray. And in

your case Achilles, he succeeded.

Achilles: Are you telling me that I responded to a resolution in a subsidiary key?

Tortoise: That's right.

Achilles: It still sounded like an ending to me

Tortoise: Bach intentionally made it sot way. You just fell into his trap.

It was deliberately contrived to sound like an ending but if you follow the harmonic progression carefully, you will see that it is in the wrong key. Apparently not just you but this miserable record company fell for the same trick-and they truncated the piece early.

Achilles: What a dirty trick Bach played

Tortoise: That is his whole game-to m lose your way in his Labyrinth! 'l Majotaur is in cahoots with Bach, And if you don't watch out, he

i laugh you to death-and perhaps n with you!

Achilles: Oh, let us hurry up and get here! Quick! Let's run backwards grooves, and escape on the outside record before the Evil Majotaur finds us.

Tortoise: Heavens, no! My sensibility is delicate to handle the bizarre the gressions which occur when time versed.

Achilles: Oh, Mr. T, how will we ever get out of here, if we can't just retrace our steps

Tortoise: That's a very good question.

(A little desperately, Achilles starts runt about aimlessly in the dark.

Suddenly t is a slight gasp, and then a "thud".)

Achilles-are you all right?

Achilles: Just a bit shaken up but otherwise fine. I fell into some big hole.

Tortoise: You've fallen into the pit of the Evil Majotaur! Here, I'll come help you out. We've got to move fast!

Achilles: Careful, Mr. T-I don't want You to fall in here, too ...

Tortoise: Don't fret, Achilles. Everything will be all --

(Suddenly, there is a slight gasp, and then a "thud".)
Achilles: Mr. T-you fell in, too! Are you all right?

Tortoise: Only my pride is hurt-otherwise I'm fine.

Achilles: Now we're in a pretty pickle, aren't we?