douglas hofstadter - godel escher bach chapter 04b little harmonic labyrinth

7/18/2019 Douglas Hofstadter - Godel Escher Bach Chapter 04b Little Harmonic Labyrinth

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Little Harmonic Labyrinth

The Tortoise and Achilles are spending a day at Coney Island After buying a couple

of cotton candies, they decide to take a ride on the Ferris wheel.

Tortoise: This is my favorite ride. One seems to move so far, and

reality one gets nowhere.

Achilles: I can see why it would appeal to you. Are you all strapped in?Tortoise: Yes, I think Ive got this !uckle done. "ell, here we go.

Achilles: You certainly are e#u!erant today.

Tortoise: I have good reason to !e. $y aunt, who is a fortune%teller me that a stroke of &ood 'ortune would !efall me today. (o I am tingling with anticipation.

Achilles: )ont tell me you !elieve in fortune%telling*

Tortoise: +o ... !ut they say it works even if you dont !elieve iiAchilles: "ell, thats fortunate indeed.

Tortoise: Ah, what a view of the !each, the crowd, the ocean, the city. . .Achilles: Yes, it certainly is splendid. (ay, look at that helicopter there. It seems to !e

flying our way. In fact its almost directly a!ove us now.Tortoise: (trange%theres a ca!le dangling down from it, which is very close to us. Its

coming so close we could practically gra! it

Achilles: ook* At the end of the line theres a giant hook, with a note

(He reaches out and snatches the note. They pass by and are on their z down.

Tortoise: -an you make out what the note says?

Achilles: Yes%it reads, /owdy, friends. &ra! a hold of the hook time around, for an

0ne#pected (urprise.Tortoise: The notes a little corny !ut who knows where it might lead, 1erhaps its got

something to do with that !it of &ood 'ortune due me. 2y all means, lets try it*

Achilles: ets*

(!n the trip up they unbuckle their buckles, and at the crest of the ride, grab for the

giant hook. All of a sudden they are whooshed up by the ca which "uickly reels

the# skyward into the ho$ering helicopter. A It strong hand helps the# in.

3oice: "elcome a!oard%(uckers.

Achilles: "h%who are you?

ittle /armonic a!yrinth 456



3oce: Allow me to introduce myself. I am /e#achlorophene 7. &oodforttune, 8idnapper

At%arge, and )evourer of Tortoises par 9#cellence, at your service.

Tortoise: &ulp*Achilles whispering to his friend ;: 0h%oh%I think that this &oodfortune is not e#actly

what wed anticipated. To %oodfortune Ah%if I may !e so !old%where are you

spiriting us off to?&oodfortune: /o ho* To my all%electric kitchen%in%the%sky, where I will prepare T/I(

tasty morsel%leering at the Tortoise as he says this;%in a delicious pie%in%the%sky*

And make no mistake%its all <ust for my go!!ling pleasure* /o ho ho*Achilles: All I can say is youve got a pretty fiendish laugh.

&oodfortune laughing fiendishly& /o ho ho* 'or that remark, my friend, you will pay

dearly. /o ho*

Achilles: &ood grief%I wonder what he means !y that*&oodfortune: 3ery simple%Ive got a (inister 'ate in store for !oth of you* 7ust you wait*

/o ho ho* /o ho ho*

Achilles: Yikes*

&oodfortune: "ell, we have arrived. )isem!ark, my friends, into my fa!ulous all%electrickitchen%in%the%sky.

They walk inside.;

et me show you around, !efore I prepare your fates. /ere is my !edroom. /ere ismy study. 1lease wait here for me for a moment. Ive got to go sharpen my knives.

"hile youre waiting, help yourselves to some popcorn. /o ho ho* Tortoise pie*

Tortoise pie* $y favorite kind of pie* 9#it.;

Achilles: Oh, !oy%popcorn* Im going to munch my head off*

Tortoise: Achilles* You <ust stuffed yourself with cotton candy* 2esides, how can you

think a!out food at a time like this?Achilles: &ood gravy%oh, pardon me%I shouldnt use that turn of phrase, should I? I mean

in these dire circumstances ... Tortoise: Im afraid our goose is cooked.

Achilles: (ay%take a gander at all these !ooks old &oodfortune has in his study. =uite acollection of esoterica: 'irdbrains I Ha$e nown) Chess and *#brella+Twirling

ade -asy) Concerto for Tapdancer and !rchestra ... /mmm.

Tortoise: "hats that small volume lying open over there on the desk, ne#t to the

dodecahedron and the open drawing pad?Achilles: This one? "hy, its title is ro$ocati$e Ad$entures of Achilles and the Tortoise

Taking lace in /undry /pots of the %lobe. Tortoise: A moderately provocative

title.Achilles: Indeed%and the adventure its opened to looks provocative. Its called )<inn and

Tonic.

Tortoise: /mm ... I wonder why. (hall we try reading it? I could take the Tortoises part,and you could take that of Achilles.

ittle /armonic a!yrinth 45>



Achilles: Im game. /ere goes nothing . . .

They !egin reading )<inn and Tonic.;

(Achilles has in$ited the Tortoise o$er to see his collection of prints by

his fa$orite artist, . C. -scher.

Tortoise: These are wonderful prints, Achilles.

Achilles: I knew you would en<oy seeing them. )o you have any particular favorite?

Tortoise: One of my favorites is -onve# and -oncave, where two internally

consistent worlds, when <u#taposed, make a completely inconsistentcomposite world. Inconsistent worlds are always fun places to visit,

!ut I wouldnt want to live there.

Achilles: "hat do you mean, fun to visit? Inconsistent worlds dont 9@I(T,so how can you visit one?

Tortoise: I !eg your pardon, !ut werent we <ust agreeing that in

this 9scher picture, an inconsistent world is portrayed?Achilles: Yes, !ut thats <ust a two%dimensional world%a fictitious world%a

picture. You cant visit that world.

Tortoise: I have my ways ...Achilles: /ow could you propel yourself into a flat picture%universe?

Tortoise: 2y drinking a little glass of 10(/I+&%1OTIO+. That does the

trick.Achilles: "hat on earth is pushing%potion?

Tortoise: Its a liuid that comes in small ceramic phials, and which, when

drunk !y someone looking at a picture, pushes him right into the

world of that picture. 1eople who arent aware of the powers of pushing%potion often are pretty surprised !y the situations they wind

up in.

Achilles: Is there no antidote? Once pushed, is one irretrieva!ly lost?Tortoise: In certain cases, thats not so !ad a fate. 2ut there is, in fact, another

potion%well, not a potion, actually, !ut an eli#ir%no, not an eli#ir, !ut a%

aTortoise: /e pro!a!ly means tonic.

Achilles: Tonic?

Tortoise: Thats the word I was looking for* 1O11I+&%TO+I- iu what itscalled, and if you remem!er to carry a !ottle of it in your right hand as

you swallow the pushing%potion, it too will !e pushed into the pictureBthen, whenever you get a hanker ing to pop !ack out into real life,

you need only take a swallow of popping%tonic, and presto* Youre !ack in the rea. world, e#actly where you were !efore you pushed

yourself in.

Achilles: That sounds very interesting. "hat would happen it you took some popping%tonic without having previously pushed yourself into a

picture?

ittle /armonic a!yrinth 45C



Tortoise: I dont precisely know, Achilles, !ut I would !e rather wary of

horsing around with these strange pushing and popping liuids. Once I

had a friend, a "easel, who did precisely what you suggested%and noone has heard from him since.

Achilles: Thats unfortunate. -an you also carry along the !ottle of pushing%

potion with you?Tortoise: Oh, certainly. 7ust hold it in your left hand, and it too will get

pushed right along with you into the picture youre looking at.

Achilles: "hat happens if you then find a picture inside the picture which youhave already entered, and take another swig of pushing%potion?

Tortoise: 7ust what you would e#pect: you wind up inside that picture%in%a%

picture.

Achilles: I suppose that you have to pop twice, then, in order to e#tricateyourself from the nested pictures, and re%emerge !ack in real life.

Tortoise: Thats right. You have to pop once for each push, since a push takes

you down inside a picture, and a pop undoes that.

Achilles: You know, this all sounds pretty fishy to me . . . Are you sure yourenot <ust testing the limits of my gulli!ility?

Tortoise: I swear* ook%here are two phials, right here in my pocket. 0eaches into his lapel pocket, and pulls out two rather large

unlabeled phials, in one of which one can hear a red li"uid sloshing

around, and in the other of which one can hear a blue li"uid sloshing

around .; If youre willing, we can try them. "hat do you say?

Achilles: "ell, I guess, ahm, may!e, ahm ...

Tortoise: &ood* I knew youd want to try it out. (hall we push ourselves into

the world of 9schers Con$e1 and Conca$e?Achilles: "ell, ah, .. .

Tortoise: Then its decided. +ow weve got to remem!er to take along this

flask of tonic, so that we can pop !ack out. )o you want to take thatheavy responsi!ility, Achilles?

Achilles: If its all the same to you, Im a little nervous, and Id prefer letting

you, with your e#perience, manage the operation.Tortoise: 3ery well, then.

(/o saying, the Tortoise pours two s#all portions of pushing+potion. Then

he picks up the flask of tonic and grasps it fir#ly in his right hand, and both he and Achilles lift their glasses to their lips.

Tortoise: 2ottoms up*

(They swallow.

ittle /armonic a!yrinth 45D



'I&0E9 F6. -onve# and -oncave, !y $. -. 9scher lithograph, 4GCC;.

Achilles: Thats an e#ceedingly strange taste.Tortoise: One gets used to it.

Achilles: )oes taking the tonic feel this strange? Tortoise: Oh, thats uite

another sensation. "henever you taste the tonic, you feel a deep senseof satisfaction, as if youd !een waiting to taste it all your life.

Achilles: Oh, Im looking forward to that. Tortoise: "ell, Achilles,

where are we?Achilles taking cognizance of his surroundings;: "ere in a little gondola,

gliding down a canal* I want to get out. $r.&ondolier, please let us

out here.

(The gondolier pays no attention to this re"uest.

Tortoise: /e doesnt speak 9nglish. If we want to get out here, wed !etter <ustclam!er out uickly !efore he

ittle /armonic a!yrinth 45H



9nters the sinister Tunnel of oveJB <ust ahead of us.

(Achilles, his face a little pale scra#bles out in a split second and then pulls his slower friend out.

Achilles: I didnt like the sound of that place, somehow. Im glad we got outhere. (ay, how do you know so much a!out this place, anyway? /ave

you !een here !efore?

Tortoise: $any times, although I always came in from other 9scher pictures.Theyre all connected !ehind the frames, you know. Once youre in

one, you can get to any other one.

Achilles: AmaKing* "ere I not here, seeing these things with my own eyes,

Im not sure Id !elieve you. They wander out through a little arch.;Oh, look at those two cute liKards*

Tortoise: -ute? They arent cute%it makes me shudder <ust to think of them*

They are the vicious guardians of that magic copper lamp hanging

from the ceiling over there. A mere touch of their tongues, and anymortal turns to a pickle.

Achilles: )ill, or sweet?Tortoise: )ill.

Achilles: Oh, what a sour fate* 2ut if the lamp has magical powers, I would

like to try for it.Tortoise: Its a foolhardy venture, my friend. I wouldnt risk it.

Achilles: Im going to try <ust once.

(He stealthily approaches the la#p, #aking sure not to awaken the sleeping lad nearby. 'ut suddenly, he slips on a strange shell+like

indentation in the floor, and lunges out into space. 2urching crazily, he

reaches for anything, and #anages so#ehow to grab onto the la#p withone hand. /winging wildly, with both lizards hissing and thrusting their

tongues $iolently out at hi#, he is left dangling helplessly out in the #iddle

of space.

Achilles: /e%e%e%elp*

(His cry attracts the attention of a wo#an who rushes downstairs and awakens the sleeping boy. He takes stock of the situation, and, with a

kindly s#ile on his face, gestures to Achilles that all will be well. He shouts

so#ething in a strange guttural tongue to a pair of tru#peters high up inwindows, and i##ediately,

ittle /armonic a!yrinth 45L



3eird tones begin ringing out and #aking beats each other. The sleepy

young lad points at the lizards, and Achilles sees that the #usic is ha$ing a

strong soporific effect on the#. /oon, they are co#pletely unconscious.Then the helpful lad shouts to two co#panions cli#bing up ladders. They

both pull their ladders up and then e1tend the# out into space 4ust

underneath the stranded Achilles, for#ing a sort of bridge. Their gestures#ake it clear that Achilles should hurry and cli#b on. 'ut before he does

so, Achilles carefully unlinks the top link of the chain holding the la#p,

and detaches the la#p. Then he cli#bs onto the ladder+bridge and thethree young lads pull hi# in to safety. Achilles throws his ar#s around

the# and hugs the# gratefully.

Achilles: Oh, $r. T, how can I repay them?Tortoise: I happen to know that these valiant lads <ust love coffee, and down

in the town !elow, theres a place where they make an incompara!le

cup of espresso. Invite them for a cup of espresso* Achilles: That

would hit the spot.

(And so, by a rather co#ical series of gestures, s#iles, and words, Achilles#anages to con$ey his in$itation to the young lads, and the party of fi$e

walks out and down a steep staircase descending into the town. They reach

a char#ing s#all cafe, sit down outside, and order fi$e espressos. As they

sip their drinks, Achilles re#e#bers he has the la#p with hi#.

Achilles: I forgot, $r. Tortoise%Ive got this maB lamp with me* 2ut%whats

magic a!out it? Tortoise: Oh, you know, <ust the usual%a genie.Achilles: "hat? You mean a genie comes out when you ru! it, and grants you

wishes?

Tortoise: Eight. "hat did you e#pect? 1ennies fry heaven?Achilles: "ell, this is fantastic* I can have any wish want, eh? Ive always

wished this would happen to me ...

(And so Achilles gently rubs the large letter 526 which is etched on the

la#p6s copper surface ... /uddenly a huge puff of s#oke appears, and in

the for#s of the s#oke the fi$e friends can #ake out a weird, ghostly figure

towering abo$e the#.

ittle /armonic a!yrinth 45G



I

&enie: /ello, my friends M and thanks ever so much for rescuing my amp

from the evil iKard%)uo.

And so saying, the &enie picks up the amp, and stuffs it into a pocket

concealed among the folds of his long ghostly ro!e which swirls out of theamp.;

As a sign of gratitude for your heroic deed, I would like to offer you, on the part of my amp, the opportunity to have any three of your wishes

realiKed.

Achilles: /ow stupefying* )ont you think so, $r. T?

Tortoise: I surely do. &o ahead, Achilles, take the first wish.Achilles: "ow* 2ut what should I wish? Oh, I know* Its what I thought of

the first time I read the Ara!ian +ights that collection of silly and

nested; tales;%I wish that I had a /0+)E9) wishes, instead of <ust

three* 1retty clever, eh, $r. T? I !et YO0 never would have thoughtof that trick. I always wondered why those dopey people in the stories

never tried it themselves.Tortoise: $ay!e now youll find out the answer.

&enie: I am sorry, Achilles, !ut I dont grant metawishes.

Achilles: I wish youd tell me what a meta%wish is*&enie: 2ut T/AT is a meta%meta%wish, Achilles%and I dont grant them, either.

Achilles: "haaat? I dont follow you at all.

Tortoise: "hy dont you rephrase your last reuest, Achilles?

Achilles: "hat do you mean? "hy should I?Tortoise: "ell, you !egan !y saying I wish. (ince youre <ust asking for

information, why dont you <ust ask a uestion?

Achilles: All right, though I dont see why. Tell me, $r. &enie%what is a meta%wish? &enie: It is simply a wish a!out wishes. I am not allowed to

grant meta%wishes. It is only within my purview to grant plain

ordinary wishes, such as wishing for ten !ottles of !eer, to have /elenof Troy on a !lanket, or to have an all%e#penses%paid weekend for two

at the -opaca!ana. You know%simple things like that. 2ut meta%wishes

I cannot grant. &O) wont permit me to.

Achilles: &O)? "ho is &O)? And why wont he let you grant meta%wishes?That seems like such a puny thing compared to the others you

mentioned.




&enie: "ell, its a complicated matter, you see. "hy dont you <ust go ahead

and make your three wishes? Or at least make one of them. I donthave all I time in the world, you know ...

Achilles: Oh, I feel so rotten. I was E9AY /O1I+& wish for a hundred

wishes ...&enie: &ee, I hate to see any!ody so disappointed that. And !esides, meta%

wishes are my favorite k of wish. et me <ust see if there isnt anything

I do a!out this. Thisll <ust take one moment

(The %enie re#o$es fro# the wispy folds of his robe an ob4ect which looks

4ust like the copper 2a#p he had put away, e1cept that this one is #ade of

sil$er) and where the pre$ious one had 626 etched on it, this one has 626 in s#aller letters, so as to co$er the sa#e area.

I

Achilles: And what is that?

&enie: This is my $eta%amp ...

(He rubs the eta+2a#p, and a huge puff of s#oke appears. In the billowsof s#oke, they can all #ake out a ghostly for# towering abo$e the#.

$eta%&enie: I am the $eta%&enie. You summoned me, 5 &enie? "hat is your wish?

&enie: I have a special wish to make of you, 5 )<inn and of &O). I wish for

permission for tempos suspension of all type%restrictions on wishes,

for duration of one Typeless "ish. -ould you ph grant this wish for me?

$eta%&enie: Ill have to send it through -hannels, of course. One half a

moment, please

(And, twice as "uickly as the %enie did, this eta+%enie re#o$es fro# the

wispy folds of her robe an ob4ect which looks 4ust like the sil$er eta+ 2a#p, e1cept that it is #ade of gold) and where the pre$ious one had 626

etched on it, this one has 626 in s#aller letters, so as to co$er the sa#e

area.

Achilles his $oice an octa$e higher than before;: And what is that? $eta%

&enie: This is my $eta%$eta%amp. . .

(/he rubs the eta+eta+2a#p, and a hugs puff of s#oke appears. In the

billows o s#oke, they can all #ake out a ghostly fore towering abo$e

the#.




$eta%$eta%&enie: I am the $eta$eta%&enie. You summoned me,

5 $eta%&enie? "hat is your wish?

$eta%&enie: I have a special wish to make of you, 5 )<inn, and of &O). Iwish for permission for temporary suspension of all type%restrictions

on wishes, for the duration of one Typeless "ish. -ould you please

grant this wish for me?$eta%$eta%&enie: Ill have to send it through -hannels, of course.

One uarter of a moment, please.

And, twice as uickly as the $eta%&enie did, this $eta$eta%

&enie removes from the folds of his ro!e an o!<ect which looks

<ust like the gold $etaamp, e#cept that it is made of ...;

.

.

.

.

.

. .N&O)

.

.

.

. ... swirls !ack into the $eta$eta%$eta%amp, which the $eta%

$eta%&enie then folds !ack into his ro!e, half as uickly as the$eta%$eta%$eta%&enie did.;

Your wish is granted, 5 $eta&enie.

$eta%&enie: Thank you, 5 )<inn, and &O).

(And the eta+eta+%enie, as all the higher ones before hi#, swirlsback into the eta+eta+2a#p, which the eta+%enie then folds back

into her robe, half as "uickly as the eta+eta+%enie did.

Your wish is granted, 5 &enie.

&enie: Thank you, 5 )<inn, and &O).

(And the eta+%enie, as all the higher ones before her,

ittle /armonic a!yrinth 44F



swirls back into the eta+2a#p, which the %enie folds back into his robe,

half as "uickly as the %enie did.

Your wish is granted, Achilles.

(And one precise #o#ent has elapsed since he 7This will 4ust take one#o#ent.7

Achilles: Thank you, 5 )<inn, and &O).&enie: I am pleased to report, Achilles, that you r have e#actly one 4;

Typeless "ish%that is to sa wish, or a meta%wish, or a meta%meta%wish,

as many metas as you wish%even infinitely many if wish;.

Achilles: Oh, thank you so very much, &enie. 2ut curiosity is provoked.2efore I make my wish, would you mind telling me who%or what%

&O) is?

&enie: +ot at all. &O) is an acronym which stands &O) Over )<inn.

The word )<inn is used designate &enies, $eta%&enies, $eta%$eta%&en etc. It is a Typeless word.

Achilles: 2ut%!ut%how can &O) !e a word in own acronym? That doesntmake any sense*

&enie: Oh, arent you acuainted with recursive acronyms? I thought

every!ody knew a!out them. P see, &O) stands for &O) Over )<inn%which can !e e#panded as &O) Over )<inn, O, )<inn%and

that can, in turn, !e e#panded to & Over )<inn, Over )<inn, Over

)<inn%which can its turn, !e further e#panded ... You can go as as you

like.Achilles: 2ut Ill never finish*

&enie: Of course not. You can never totally e#pand &O).

Achilles: /mm ... Thats puKKling. "hat did you me when you said to the$eta%&enie, I have a sped wish to make of you, 5 )<inn, and of

&O)?

&enie: I wanted not only to make a reuest of $eta%&enie, !ut also of all the)<inns over her. I recursive acronym method accomplishes this

naturally. You see, when the $eta%&enie received my reuest, she

then had to pass it upwards to I &O). (o she forwarded a similar

message to I $eta%$eta%&enie, who then did likewise to t $eta%$eta%$eta%&enie ... Ascending the chain this way transmits the message to

&O).




Achilles: I see. You mean &O) sits up at the top of the ladder of d<inns?

&enie: +o, no, no* There is nothing at the top, for there is no top. That is

why &O) is a recursive acronym. &O) is not some ultimate d<innB&O) is the tower of d<inns a!ove any given d<inn.

Tortoise: It seems to me that each and every d<inn would have a different

concept of what &O) is, then, since to any d<inn, &O) is the set of d<inns a!ove him or her, and no two d<inns share that set.

&enie: Youre a!solutely right%and since I am the lowest d<inn of all, my

notion of &O) is the most e#alted one. I pity the higher d<inns, whofancy themselves somehow closer to &O). "hat !lasphemy*

Achilles: 2y gum, it must have taken genies to invent &O).

Tortoise: )o you really !elieve all this stuff a!out &O), Achilles?

Achilles: "hy certainly, I do. Are you atheistic, $r. T? Or are you agnostic?Tortoise: I dont think Im agnostic. $ay!e Im metaagnostic.

Achilles: "haaat? I dont follow you at all.

Tortoise: ets see . . . If I were meta%agnostic, Id !e confused over whether

Im agnostic or not%!ut Im not uite sure if I feel T/AT wayB hence Imust !e meta%meta%agnostic I guess;. Oh, well. Tell me, &enie, does

any d<inn ever make a mistake, and gar!le up a message moving up or down the chain?

&enie: This does happenB it is the most common cause for Typeless "ishes

not !eing granted. You see, the chances are infinitesimal, that agar!ling will occur at any 1AETI-0AE link in the chain%!ut when

you put an infinite num!er of them in a row, it !ecomes virtually

certain that a gar!ling will occur (O$9"/9E9. In fact, strange as it

seems, an infinite num!er of gar!lings usually occur, although theyare very sparsely distri!uted in the chain.

Achilles: Then it seems a miracle that any Typeless "ish ever gets carried

out.&enie: +ot really. $ost gar!lings are inconseuential, and many gar!lings

tend to cancel each other out. 2ut occasionally%in fact, rather seldom%

the nonfulfillment of a Typeless "ish can !e traced !ack to a singleunfortunate d<inns gar!ling. "hen this happens, the guilty d<inn is

forced to run an infinite

ittle /armonic a!yrinth 44>



&auntlet and get paddled on his or her rump, !y &O). Its good fun for the

paddlers, and harmless for the paddlee. You might !e amused !y the

sight.Achilles: I would love to see that* 2ut it only happens when a Typeless "ish

goes ungranted?

&enie: Thats right.Achilles: /mm ... That gives me an idea for my w Tortoise: Oh, really? "hat

is it? Achilles: I wish my wish would not !e granted*

(At that #o#ent, an e$ent+or is 7e$ent7 the word for it8 ++takes place which

cannot be described, and hence no atte#pt will be #ade to describe it.

Achilles: "hat on earth does that cryptic comment mean?Tortoise: It refers to the Typeless "ish Achilles made.

Achilles: 2ut he hadnt yet made it.

Tortoise: Yes, he had. /e said, I wish my wish would not !e

granted, and the &enie took T/AT to !e his wish.

(At that #o#ent, so#e footsteps are heard co#ing down the hallway intheir direction.

Achilles: Oh, my* That sounds ominous.

(The footsteps stop) then they turn around and fade away.

Tortoise: "hew*Achilles: 2ut does the story go on, lets see. or is that the end? Turn the page

and lets see.

(The Tortoise turns the page of 794inn and Tonic7, where they find that the

story goes on ...

Achilles: /ey* "hat happened? "here is my &enie: lamp? $y cup of

espresso? "hat happened to young friends from the -onve# and

-oncave worlds? "hat are all those little liKards doing hi

Tortoise: Im afraid our conte#t got restored incorrectly Achilles.Achilles: "hat on earth does that cryptic comment mean?

Tortoise: I refer to the Typeless "ish you made.

Achilles: 2ut I hadnt yet made it.Tortoise: Yes, you had. You said, I wish my wish would not !e granted,

and the &enie took T/AT to !e your wish.

Achilles: Oh, my* That sounds ominous.TortoiseB It spells 1AEA)O@. 'or that Typeless wish to !e

ittle /armonic a!yrinth 44C



granted, it had to !e denied M yet not to grant it would !e to grant

it.

Achilles: (o what happened? )id the earth come to a standstill? )id theuniverse cave in?

Tortoise: +o. The (ystem crashed. Achilles: "hat does that mean?

Tortoise: It means that you and I, Achilles, were suddenly andinstantaneously transported to Tum!olia. Achilles: To where?

Tortoise: Tum!olia: the land of dead hiccups and e#tinguished light

!ul!s. Its a sort of waiting room, where dormant software waitsfor its host hardware to come !ack up. +o telling how long the

(ystem was down, and we were in Tum!olia. It could have !een

moments, hours, days%even years.

Achilles: I dont know what software is, and I dont know what hardwareis. 2ut I do know that I didnt get to make my wishes* I want my

&enie !ack*

Tortoise: Im sorry, Achilles%you !lew it. You crashed the (ystem, and

you should thank your lucky stars that were !ack at all. Thingscould have come out a lot worse. 2ut I have no idea where we

are.Achilles: I recogniKe it now%were inside another of 9schers pictures.

This time its 0eptiles.

Tortoise: Aha* The (ystem tried to save as much of our conte#t as itcould !efore it crashed, and it got as far as recording that it was

an 9scher picture with liKards !efore it went down. Thats

commenda!le.

Achilles: And look%isnt that our phial of poppingtonic over there on theta!le, ne#t to the cycle of liKards?

Tortoise: It certainly is, Achilles. I must say, we are very lucky indeed.

The (ystem was very kind to us, in giving us !ack our popping%tonic%its precious stuff*

Achilles: Ill say* +ow we can pop !ack out of the 9scher world, into my

house.Tortoise: There are a couple of !ooks on the desk, ne#t to the tonic. I

wonder what they are. He picks up the s#aller one, which is

open to a rando# page.; This looks like a moderately

provocative !ook.Achilles: Oh, really? "hat is its title?

Tortoise: ro$ocati$e Ad$entures of the Tortoise and Achilles Taking

lace in /undry arts of the %lobe. It sounds like an interesting !ook to read out of.

ittle /armonic a!yrinth 44D



'I&0E9 F>. Eeptiles, by . C. -scher (lithograph, :;<=;.

Achilles: "ell, You can read it if you want, !ut as for Im not going totake any chances with t popping%tonic%one of the liKards mightknock it off the ta!le, so Im going to get it right now*

(He dashes o$er to the table and reaches for the popping+tonic, but in

his haste he so#ehow bu#ps the flask of tonic, and it tu#bles off thedesk and begins rolling.

Oh, no* $r. T%look* I accidentally knocked tonic onto the floor, and itsrolling toward towards%the stairwell* =uick%!efore it falls*

(The Tortoise, howe$er, is co#pletely wrapped up in the thin $olu#ewhich he has in his hands. Achilles: "ell, You can read it if you

want, !ut as for Im not going to take any chances with t popping%

tonic%one of the liKards might knock it off the ta!le, so Im goingto get it right

ittle /armonic a!yrinth 44H



Tortoise #uttering ;: 9h? This story looks fascinating.

Achilles: $r. T, $r. T, help* /elp catch the tonic%flask*

Tortoise: "hats all the fuss a!out?Achilles: The tonic%flask%I knocked it down from the desk, and now its

rolling and

(At that instant it reaches the brink of the stairwell, and plu##ets

o$er ...

Oh no* "hat can we do? $r. Tortoise%arent you alarmed? "ere

losing our tonic* Its <ust fallen down the stairwell* Theres only one

thing to do* "ell have to go down one story*

Tortoise: &o down one story? $y pleasure. "ont you <oin me?

(He begins to read aloud, and Achilles, pulled in two directions at

once, finally stays, taking the role of the Tortoise.

Achilles: Its very dark here, $r. T. I cant see a thing. Oof* I !umped

into a wall. "atch out*Tortoise: /ere%I have a couple of walking sticks. "hy dont you take one

of them? You can hold it out in front of you so that you dont

!ang into things.Achilles: &ood idea. He takes the stick .; )o you get the sense that this

path is curving gently to the left as we walk? Tortoise: 3ery

slightly, yes.

Achilles: I wonder where we are. And whether well ever see the light of day again. I wish Id never listened to you, when you suggested I

swallow some of that )EI+8 $9 stuff.

Tortoise: I assure you, its uite harmless. Ive done it scads of times, andnot a once have I ever regretted it. Eela# and en<oy !eing small.

Achilles: 2eing small? "hat is it youve done to me, $r. T?

Tortoise: +ow dont go !laming me. You did it of your own free will.Achilles: /ave you made me shrink? (o that this la!yrinth were

in is actually some teeny thing that someone could (T91 on?

ittle /armonic a!yrinth 44L



FI%*0- >?. Cretan 2abyrinth (Italian engra$ing) /chool of

Finiguerra. @Fro# atthews, azes and 2abyrinths& Their History and 9e$elop#ent (ew Bork& 9o$er ublications, :;D.

Tortoise: a!yrinth? a!yrinth? -ould it Are we in the notorious ittle/armonic a!yrinth of the dreaded $a<otaur?

Achilles: Yiikes* "hat is that?

Tortoise: They say%although I person never !elieved it myself%that an I$a<otaur has created a tiny la!yrinth sits in a pit in the middle of

it, waiting innocent victims to get lost in its fears comple#ity.Then, when they wander and daKed into the center, he laughs and

laughs at them%so hard, that he laughs them to death*Achilles: Oh, no*

Tortoise: 2ut its only a myth. -ourage, Achilles.

(And the dauntless pair trudge on.

Achilles: 'eel these walls. Theyre like o gated tin sheets, or something.2ut the corrugations have different siKes.

ittle /armonic a!yrinth 44G



(To e#phasize 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 cur$ed corridor they are in.

Tortoise alarmed;: "hat was T/AT?Achilles: Oh, <ust me, ru!!ing my walking stick against the wall.

Tortoise: "hew* I thought for a moment it was the !ellowing of the

ferocious $a<otaur* Achilles: I thought you said it was all amyth.

Tortoise: Of course it is. +othing to !e afraid of.

(Achilles puts his walking stick back against the wall, and continueswalking. As he does so, so#e #usical sounds are heard, co#ing fro#

the point where his stick is scraping the wall.

Tortoise: 0h%oh. I have a !ad feeling, Achilles.That a!yrinth may not !e a myth, after all. Achilles: "ait a minute.

"hat makes you change your mind all of a sudden? Tortoise: )oyou hear that music?

(To hear #ore clearly, Achilles lowers the stick, and the strains of

#elody cease.

/ey* 1ut that !ack* I want to hear the end of this piece*

(Confused, Achilles obeys, and the #usic resu#es.

Thank you. +ow as I was a!out to say, I have <ust figured out wherewe are.

Achilles: Eeally? "here are we?

Tortoise: "e are walking down a spiral groove of a record in its <acket.

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: "hat? Arent you over<oyed? /ave you ever had the chance to

!e in such intimate contact with music !efore?

ittle /armonic a!yrinth 4F5



AchKltes: /ow am I ever going to win footraces against full%siKed people

when I am smaller than a flea, $r. Tortoise?Tortoise: Oh, is that all thats !othering you Thats nothing to fret a!opt,

Achilles.

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

Tortoise: I dont know. 2ut one thing for certain is that I dont worry

a!out !eing small. 9specially not when faced with the awfuldanger of the dreaded $a<otaur*

Achilles: /orrors* Are you telling me

Tortoise: Im afraid so, Achilles. The music gave it away.

Achilles: /ow could it do that?Tortoise: 3ery simple. "hen I heard melody B-A-C-H in the top voice, I

immediately realiKed that the grooves were walking through

could only !e 2ittle Har#onic 2abyrinth, one of 2achs er known

organ pieces. It is so named cause of its diKKyingly freuentmodulations.

Achilles: "h%what are they?Tortoise: "ell, you know that most music pieces are written in a key, or

tonality, as - ma<or, which is the key of this oB

Achilles: I had heard the term !efore. )o that mean that - is the noteyou want to on?

Tortoise: Yes, - acts like a home !ase, in a Actually, the usual word is

tonic.

Achilles: )oes one then stray away from tonic with the aim of eventuallyreturning

Tortoise: Thats right. As the piece develops am!iguous chords and

melodies are t which lead away from the tonic. ittle !y little,tension !uilds 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 Ihad waiting my whole life to hear the ton

Tortoise: 9#actly. The composer has uses knowledge of harmonic

progressions to




manipulate your emotions, and to !uild up hopes in you to hear

that tonic.Achilles: 2ut you were going to tell me a!out modulations.

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

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

Achilles: I see ... I think. )o you mean that some seuence of chords

shifts the harmonic tension somehow so that I actually desire toresolve in a new key?

Tortoise: Eight. This makes the situation more comple#, for although in

the short term you want to resolve in the new key, all the while at

the !ack of your mind you retain the longing to hit that originalgoal%in this case, - ma<or. And when the su!sidiary goal is

reached, there is

Achilles suddenly gesturing enthusiastically;: Oh, listen to the gorgeous

upward%swooping chords which mark the end of this 2ittle Har#onic 2abyrinthE

Tortoise: +o, Achilles, this isnt the end. Its merelyAchilles: (ure it is* "ow* "hat a powerful, strong ending* "hat a sense

of relief* Thats some resolution* &ee*

(And sure enough, at that #o#ent the #usic stops, as they e#erge intoan open area with no walls.

You see, it Is over. "hat did I tell you? Tortoise: (omething is verywrong. This record

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

Tortoise: It was e#actly what I was telling you a!out. /ere 2ach hadmodulated from - into &, setting up a secondary goal of hearing

&. This means that you e#perience two tensions at once%waiting

for resolution into &, !ut also keeping in mind that ultimatedesire%to resolve triumphantly into - $a<or.

Achilles: "hy should you have to keep any

ittle /armonic a!yrinth 4FF



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

intellectual e#ercise?

Tortoise: +o, of course not. (ome music is highly intellectual, !ut mostmusic is not. And most of the time your ear or !r the

calculation for you, and lets your emotions know what they

want to hear, dont have to think a!out it consciously in this piece, 2ach 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 asu!sidiary key?

Tortoise: Thats right.

Achilles: It still sounded like an ending to me

Tortoise: 2ach intentionally made it sot way. You <ust fell into his trap. Itwas deli!erately contrived to sound like an ending !ut if you

follow the harmonic progression carefully, you will see that it is

in the wrong key. Apparently not <ust you !ut this misera!le

record company fell for the same trick%and they truncated the piece early.

Achilles: "hat a dirty trick 2ach playedTortoise: That is his whole game%to m lose your way in his a!yrinth* l

$a<otaur is in cahoots with 2ach, And if you dont watch out, he

i laugh you to death%and perhaps n with you*Achilles: Oh, let us hurry up and get here* =uick* ets run !ackwards

grooves, and escape on the outside record !efore the 9vil

$a<otaur finds us.

Tortoise: /eavens, no* $y sensi!ility is delicate to handle the !iKarre thegressions which occur when time versed.

Achilles: Oh, $r. T, how will we ever get out of here, if we cant <ust

retrace our stepsTortoise: Thats a very good uestion.

(A little desperately, Achilles starts runt about ai#lessly in the dark./uddenly t is a slight gasp, and then a 7thud7.




Achilles%are you all right?

Achilles: 7ust a !it shaken up !ut otherwise fine. I fell into some !ig

hole.Tortoise: Youve fallen into the pit of the 9vil $a<otaur* /ere, Ill come

help you out. "eve got to move fast*

Achilles: -areful, $r. T%I dont want You to fall in here, too ...Tortoise: )ont fret, Achilles. 9verything will !e all %%

(/uddenly, there is a slight gasp, and then a 7thud7.

Achilles: $r. T%you fell in, too* Are you all right?

Tortoise: Only my pride is hurt%otherwise Im fine.

Achilles: +ow were in a pretty pickle, arent we?

(/uddenly, a giant, boo#ing laugh is heard, alar#ingly close to

the#.

Tortoise: "atch out, Achilles* This is no laughing matter.

$a<otaur: /ee hee hee* /o ho* /aw haw haw*Achilles: Im starting to feel weak, $r. T ...

Tortoise: Try to pay no attention to his laugh,

Achilles. Thats your only hope.Achilles: Ill do my !est. If only my stomach werent empty*

Tortoise: (ay, am I smelling things, or is there a !owl of hot !uttered

popcorn around here? Achilles: I smell it, too. "here is it coming

from?Tortoise: Over here, I think. Oh* I <ust ran into a !ig !owl of the stuff.

Yes, indeed%it seems to !e a !owl of popcorn*

Achilles: Oh, !oy%popcorn* Im going to munch my head off*

Tortoise: ets <ust hope it isnt pushcorn* 1ushcorn and popcorn are

e#traordinarily difficult to tell apart.

Achilles: "hats this a!out 1ushkin?

Tortoise: I didnt say a thing. You must !e hearing things.

Achilles: &o%golly* I hope not. "ell, lets dig in*

ittle /armonic a!yrinth 4F>



(And the two riends begin #uncnai popcorn (or pushcorn8+and t

once !E I guess it was popcorn) all.

Tortoise: "hat an amusing story. )id you en

Achilles: $ildly. Only I wonder whether the out of that 9vil $a<otaurs pit or r Achilles%he wanted to !e full%siKed again

Tortoise: )ont worry%theyre out, and he is again. Thats what the 1O1

was all a!oAchilles: Oh, I couldnt tell. "ell, now I E9A: find that !ottle of tonic.

'or some reason, !urning. And nothing would taste !ett drink of

popping%tonic.

Tortoise: That stuff is renowned for its thirst powers. "hy, in some places people very craKy over it. At the turn of the century the

(chon!erg food factory stopped maQ and started making cereal

instead. You cai the uproar that caused.

Achilles: I have an inkling. 2ut lets go look fo /ey <ust a moment.Those liKards on the you see anything funny a!out them?

Tortoise: 0mm ... not particularly. "hat do you see of such greatinterest?

Achilles: )ont you see it? Theyre emerging flat picture without

drinking any popQ /ow are they a!le to do that?Tortoise: Oh, didnt I tell you? You can ge picture !y moving

perpendicularly to it you have no popping%tonic. The little li

learned to clim! 01 when they want to ge two%dimensional

sketch!ook world.Achilles: -ould we do the same thing to get 9scher picture were in?

Tortoise: Of course* "e <ust need to go 01 one story. you want to try it?

Achilles: Anything to get !ack to my house* I all these provocativeadventures.

Tortoise: 'ollow me, then, up this way.

(And they go up one story.

Achilles: Its good to !e !ack. 2ut something seems wrong. This isnt my

house* This is YO0E house, $r. TortoiseTortoise: "ell, so it is%and am I glad for that* I wasnt looking

ittle /armonic a!yrinth 4FC



forward one whit to the long walk !ack from your house. I am !ushed,

and dou!t if I could have made it.

Achilles: I dont mind walking home, so I guess its lucky we ended uphere, after all.

Tortoise: Ill say* This certainly is a piece of &ood 'ortune*

douglas hofstadter - godel escher bach chapter 04b little harmonic labyrinth

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