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Notes: Little Harmonic Labyrinth

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Amanda's posting of her Ant Fugue notes reminded me that I have a bunch of GEB notes lying around. It turns out that my "Ant Fugue" notes are rather worthless, but I did find some other notes that were a little more complete.

The "Little Harmonic Labyrinth" was a pretty interesting essay with a lot of deep structure and detail that I thought was interesting, so I've posted my notes here. As always, notes are rather useless unless you've read the piece yourself.

Setting: 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

Story is isomorphism of Bach�s Little Harmonic Labyrinth
- Isomorphism of musical modulation and recursion in story
- Recursion of stories acts as modulation
- Hofstader moves back and forth between stories-within-stories
- Fools reader in similar way that Bach fools listener by giving resolution in different key; story never returns to primary story

Push/pop
- Pushing-potion and popping-tonic
- Jumping in and out of pictures (Escher�s Convex and Concave)
- Shrinking potion
- Popcorn and pushcorn

Recursive definition
- GOD = GOD Over Djinn
- Infinite, no �top,� merely the sets of all Djinns above a particular Djinn

Infinite recursion
- Granting of meta-wish requires talking to meta-genie, who talks to meta-meta-genie, etc�

Primary story

- Achilles and Tortoise are kidnapped by Goodfortune (Fortune teller had told Achilles that he would have good fortune that day)
- Popcorn at Goodfortune�s place
- Primary storyline never actually resolves; story resolves within secondary story. Isomorphism with Bach�s Little Harmonic Labyrinth, where listener thinks resolution to G modulation is resolution to �tonic�

Secondary story

- Achilles and Tortoise (primary story) read story about their adventures with the Djinn and GOD
- Achilles and Tortoise drinking pushing-potion in order to enter Escher�s Concave and Convex
- Achilles grabs magic lamp from the picture and releases Genie

Isomorphism between wishes and Principia Mathematica:

- At first, Achilles not permitted to wish for a hundred wishes, as such a wish would be a meta-wish.
- Genie cannot answer Achilles comment �I wish you would tell me about meta-wishes,� as such a comment is a meta-meta-wish

Isomorphism with G�del�s Incompleteness Theorem:

- Achilles wishes for his wish to not be granted, which creates a paradox.
- System crashes and sends Achilles and Tortoise to Tumbolia, the land of dead hiccups and extinguished light bulbs.
- System lost context, restores them to Escher�s picture Reptiles
- Idea of context introduced

Structural isomorphism with Reptiles

- Escape from Reptiles by walking �up� out of two-dimension picture, just like gators in Reptiles
- However, only return to Achilles home in secondary story, story never resolves primary story

Tertiary story

- While in Escher�s Reptiles, Achilles and Tortoise (secondary story) again read story about themselves
- Achilles and Tortoise have drunk a shrinking potion and are trapped in a Labyrinth, which is actually the grooves of Bach�s Little Harmonic Labyrinth
- Record groove stops at Bach�s resolution in key of G, instead of at the final resolution in C tonic
- Escape from Majotaur by eating popcorn (instead of pushcorn)
- Majotaur is similar to Goodfortune: evil laugh, Goodfortune alludes to Majotaur story
- Achilles� comment about Goodfortune�s sinister laugh causes Goodfortune to tell Achilles that he has sinister fate in store

Interesting comments

- Ferris wheel: Moves so far yet in reality gets nowhere (Zeno) - Fortune telling: Works even if you don�t believe in it. True statement without proof - Tonic with two meanings: (a) Tonic as home base of modulation, (b) Tonic as potion drink

Comments (1)

Jesse:

I came across this the morning after reading that chapter of GEB, looking for a recording of Little Harmonic Labyrinth. What an interesting recursive journey! Thank you for providing your notes.

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