Crab Canon from Gödel, Escher, Bach
http://www.youtube.com/watch?v=36ykl2tJwZM
The above is a video of the score of Canon 1 a 2 voci: Canon Cancrizans (Crab Canon). Crabs mark the spot on the score and its corresponding opposite. The title of the piece, which is from Bach’s Musical Offering, a leitmotif of Douglas Hofstadter’s magnum opus, “Gödel, Escher, Bach: an Eternal Golden Braid”, means “crab canon”, and is a reference to the incredible game of form and content played by a true master of the two. Johann Sebastian Bach successfully composed a canon in which one voice plays the inverse of the other voice, making the song palindromic and really, really cool.
Douglas Hofstadter, a more contemporary form-content aficionado as well as a Bach fanatic like myself, weaves together elements of the form of Bach’s Crab Canon, an M.C. Escher print which Hofstadter liberally dubs “Crab Canon” (it features a tessellation of crabs, in which the negative space of some crabs provide the positive space of others. It is particularly impressive because the crabs face opposite directions… not so different from the bass clef of a score forming the treble clef at the other end of the song), and his own literary “crab canon” form. You will see what I mean if you read the dialogue below. The characters from the dialogue are borrowed from Lewis Carroll, who wrote a dialogue between a Tortoise and Achilles for the purposes of demonstrating a mathematical idea. He borrowed the same two characters from the Greek philosopher Zeno, whose paradoxes of motion are fascinating and worth an essay unto themselves. He also introduces a third character, the Crab, whose contribution to the form of the dialogue is an all-important midpoint to serve as a fulcrum for the dialogue.
If the image refuses to post, as my audio clip of the Bach canon from my personal library failed, and my embedding of the youtube video failed as well, here is a link to the Escher print.
http://www.iescarrus.com/edumat/imagenes/baul/curiosidades/Bach_02.jpg
Dialogue, typed word for word by myself:
Crab Canon
Achilles and the Tortoise happen upon each other in teh park one day while strolling.
Tortoise: Good day, Mr. A
Achilles: Why, same to you.
Tortoise: So nice to run into you.
Achilles: That echoes my thoughts.
Tortoise: And it’s a perfect day for a walk. I think I’ll be walking home soon.
Achilles: Oh really? I guess there’s nothing better for you than walking.
Tortoise: Incidentally, you’re looking in very fine fettle these days, I must say.
Achilles: Thank you very much.
Tortoise: Not at all. Here, care for one of my cigars?
Achilles: Oh, you are such a philistine. In this area, the Dutch contributions are of markedly inferior taste, don’t you think?
Tortoise: I disagree, in this case. But speaking of taste, I finally saw that Crab Canon by your favorite artist, M. C. Escher, in a gallery the other day, and I fully appreciate the beauty and ingenuity with which he made one single theme mesh with itself going both backwards and forwards. But I am afraid I will always feel Bach is superior to Escher.
Achilles: I don’t know. But one thing for certain is that I don’t worry about arguments of taste. De gustibus non est disputandum.
Tortoise: Tell me, what’s it like to be your age? Is it true that one has no worries at all?
Achilles: To be precise, one has no frets.
Tortoise: Oh well, it’s all the same to me.
Achilles: Fiddle. It makes a big difference, you know.
Tortoise: Say, don’t you play the guitar?
Achilles: That’s my good friend. He often plays, the fool. But I myself wouldn’t touch a guitar with a ten- foot pole!
(Suddenly, the Crab, appearing from otu of nowhere, wanders up excitedly, pointing to a rather prominent black eye)
Crab: Hallo! Hulloo! What’s up? What’s new? You see this bump, this lump? Given to me by a grump. Ho! And on such a fine day. You see, I was just idly loafing about the park when up lumbers this giant fellow from Warsaw– a colossal bear of a man– playing a lute. He was three meters tall, if I’m a day. I mosey on up to the chap, reach skyward and manage to tap him on the knee, saying , “Pardon me, sir, but you are Pole-luting our park with your mazurkas.” But WOW! he had no sense of humor– not a bit, not a wit– and POW!– he lets loose and belts me one, smack in the eye! Were it in my nature, I would crab up a storm, but in the time-honored tradition of my species, I backed off. After all, when we walk forwards, we move backwards. It’s in our genes, you know, turning round and round. That reminds me– I’ve always wondered, “Which came first– the Crab, or the Gene?” That is to say, “Which came last– the Gene, or the Crab?” I’m always turning thigns round and round, you know. It’s in our genes, after all. When we walk backwards, we move forwards. Ah me, oh my! I must lope along on my merry way– so off I go on such a fine day. Sing “ho!” for the life of a Crab! TATA! ¡Olé!
(and he disappears as suddenly as he arrived.)
Tortoise: That’s my good friend. He often plays the fool. But I myself wouldn’t touch a ten-foot Pole with a guitar!
Achilles: Say, don’t you play the guitar?
Tortoise: Fiddle. It makes a big difference, you know.
Achilles: Oh, well, it’s all teh same to me.
Tortoise: To be precise, one has no frets.
Achilles: Tell me, what’s it like to be your age? Is it true that one has no worries at all?
Tortoise: I don’t know. But one thing for certain is that I don’t worry about arguments of taste. Disputandum non est de gustibus.
Achilles: I disagree, in this case. But speaking of taste, I finally heard that Crab Canon by your favorite composer, J.S. Bach, in a concert hte other day, and I fully appreciate the beauty and ingenuity with which he made one single theme mesh with itself going both backwards and forwards. But I’m afraid I will always feel Escher is superior to Bach.
Tortoise: Oh, you are such a philistine. In this area, the Dutch contributions are of markedly inferior taste, don’t you think?
Achilles: Not at all. Here, care for one of my cigars?
Tortoise: Thank you very much.
Achilles: Incidentally, you’re looking in very fine fettle these days, I must say.
Tortoise: Oh, really? I guess tehres’ nothing better for you than walking.
Achilles: And it’s a perfect day for a walk. I think I’ll be walking home soon.
Tortoise: That echoes my thoughts.
Achilles: So nice to run into you.
Tortoise: Why, same to you.
Achilles: Good day, Mr. T.
As you’ve noticed, the author builds a dialogue COMPLETELY out of form. the content, though often hilarious, and highly informative, is mainly built based on its compatibility with the strict form. Good day is used as both greeting and farewell. Even the word “frets” is used both to describe the wedges on the fingerboard of a guitar, and a synonym for worries, all for the purposes of strict adherence to crab-canon form. The amusing game played with the latin aphorism “On taste, there is no argument” (Hebrew speakers may recognize על טעם וריח אין מה להתוכח) switches it, in the context of the Tortoise’s age, into “Arguments are not about taste”, that is, the Tortoise’s taste buds are old. Not touching a guitar with a ten foot pole, becomes not touching a ten foot Pole (that is, a lute-playing fellow from Warsaw who is three meters tall) with a guitar. The crab’s own “Hallo”! is switched phonetically into ¡Olé!
The dialogue, clever in its own right, closely parallels the structure of Bach’s Canon Cancrizans. Just as the statements of one character become those of the other by the end, Bach’s left-hand and right-hand clefs switch at the midpoint (represented by the crab). He may have named this canon based on the somewhat odd concept that crabs “walk backwards”. A modern day analogy, which Hofstadter makes extensive use of, is the interesting fact that crab DNA is palindromic. That is, the strings of nucleotides when reversed can form double helixes with themselves. That is why Hofstadter cleverly worked in the farewell “tata” and capitalized it. The crab was subtly (in spite of his boorish demeanor) hinting to the Adenine-Thymine bonds in DNA. In the book, illustrations of crab DNA, as well as the score of the canon and the M.C. Escher print, accompany the dialogue. The reason he chose these two in particular corresponds to our characters. Mr. A, and Mr. T, Achilles and the Tortoise, happen to have names initialed by the same letters as that very purine (adenine) and that very pyrimidine (Thymine).
Conclusions: Bach rules. Escher rules. Hofstadter rules. Lewis Caroll rules. Molecular biology rules (I’ll consider that a major work of the Divine Composer, who Rules with a capital “R”). I highly recommend Gödel Escher Bach: An Eternal Golden Braid. I suggest as many people as possible buy it (though it’s kind of pricey. A secondhand might be worth considering. Also kind of dense, and long, and I know of no secondhand that saves money AND time. But it’s very worth the time, and honestly, is anyone in college that busy?). As one reviewer put it, it contains “a complete humanistic education”. My personal review: Greatest Book Ever.
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Cultural Map
"The Original Topography of Manhattan Island from the Battery to 155th Street" From Report on the Social Statistics of Cities, Compiled by George E. Waring, Jr., United States. Census Office, Part I, 1886. (Perry-Castañeda Library Map Collection, University of Texas-Austin)
How come it didn’t say “posted by alex schindler”?
I enjoyed your post.
Here is another great book :
Thinking from A to Z by Nigel Warburton
Routledge Publication 1996
164p
It is a very short and powerful book, it gives you the tools for developing an argument and refuting it. reading and listening critically.
Now that I skimmed the book, I think it’s useful reading for most people. But honestly, most of the terms it defines I’m already aware of. Some I can even express in Aramaic, and most in Latin. I think anyone who has learned a bit of philosophy, law, or debate (or Talmud, which is sort of all three, among other things) is probably aware of most of it.
The other day I nearly responded to an uninitiated friend ” אין הכי נמי “, (yup, that too) which is sort of an answer to an attempted reductio ad absurdum where you accept the so-called absurd conclusion. and I very routinely let the words קל וחומר slip out, which (I’m pretty sure modern Hebrew employs it as well) is a term of extrapolation. It can be pretty embarrassing to explain myself when I lapse into other (archaic) languages in an attempt to clarify.
On the topic of Gödel, Escher, Bach, I think that anyone who enjoys it (which I believe is anyone) would love
Metamagical Themas
The Mind’s Eye
I am a Strange Loop
all by Hofstadter
The Language Instinct
The Stuff of Thought
by Steven Pinker
This book needs no title
What is the name of this book
To Mock a Mockingbird
The Riddle of Scheherezade
by Raymond Smullyan
Brainstorm
by Daniel Dennett (who co-wrote/edited The Mind’s Eye with Hofstadter)
The Selfish Gene
by Richard Dawkins
Anything by Lewis Carroll
by Lewis Carroll
Those are all more or less related to GEB via their focus on consciousness, logic puzzles, the human condition from a scientific perspective, or me liking them a lot.