This dialogue is central to the book, for it contains a set of paraphrases of Gödel's self-referential construction and of his Incompleteness Theorem. One of the paraphrases of the Theorem says, "For each record player there is a record which it cannot play." The Dialogue's title is a cross between the word "acrostic" and the word "contrapunctus", a Latin word which Bach used to denote the many fugues and canons making up his Art of the Fugue. Some explicit references to the Art of the Fugue are made. The Dialogue itself conceals some acrostic tricks.
While you read
- What's the hidden message in this dialogue? What's the second hidden message?
- What kind of record do you think would break Record Player Omega?
Bach's unfinished Contrapunctus can be exactly as tragic as the Tortoise describes it.
This is the last movement in the Art of Fugue (BWV 1080), which like the Musical Offering is a fairly cerebral work, written for no particular instrument.
How to play it is left very much up to the performer. Even the numbering of the pieces is a bit uncertain, which is why you'll see this same piece called "Contrapunctus XIV", "Contrapunctus XV", or even "Contrapunctus XIX".
So let's look at a few performances and how they perform this Contrapunctus, especially what they do at the end.
One very good recording of it is by the Emerson String Quartet. The linked video scrolls the sheet music as they play, indicates where the three themes enter, and points out where the note sequence B-A-C-H appears.
They play the end of the piece as it's notated, with parts trailing off one by one, until the last one ends in the middle of a phrase. Seeing that as a terrible way to end an album, they conclude it with an arrangement of the chorale prelude BWV 668, "Before your throne I now appear", which Bach was also writing at the end of his life, though with a humbler title. (The re-titled piece was apparently suggested by Bach's sons as a conclusion when they published the Art of Fugue.)
Glenn Gould was a world class pianist and also a pretty weird dude. I'm linking his video because it's interesting to see another interpretation of how to end the piece, which is that he plays the last complete measure at full volume, then abruptly and dramatically picks his hands up off the piano.
Gould gives you no nice conclusion about heaven. Bach's dead. It's over. Go home!
Finally, we have the Hungarian team of Gyula Szilágyi on organ, performing a completion of the piece by Zoltán Göncz. The video comes in two parts, which appear to the side of this section.
This video is an even better illustration than the Emerson one, because it shows you a really informative grid of when the four voices are playing the four themes. Yes, all four themes, including one Bach never got to. We know what the fourth theme was supposed to be because all these themes appear elsewhere in the Art of Fugue.
When you see the grid for the part where Göncz puts all four themes together, it's quite clever. All bets are off for what happens after that, especially the stunt he pulls at measure 333. But it is a rather satisfying ending.
However, a commenter points out that Bach's obituary tells us there should be a development in the inversion. So this is a completion, but it is not Bach's, which is still somewhere hidden in the space of possible fugues on these themes.
This term appears only in the index, and points to this dialogue. It presumably is pronounced "turtle-ization" to rhyme with "Gödelization", which names what Tortoise is doing to Crab's record players.
There is a section of the dialogue where the Tortoise asks the question "In the unlikely event that a dialogician should write a contrapuntal acrostics in homage to J.S. Bach, would it be more proper for him to acrostically embed his own name, or that of Bach?" The first letters of this paragraph, the preceding two paragraphs, and the proceeding paragraph answer this question - as they are an acrostic for BACH. In fact, the entire dialogue is an acrostic that reads "Hofstadter's contracrostipunctus acrostically backwards spells 'J.S. Bach'" (including the punctuation, which is added by means of the words 'twould, 'tisn't, and 'pon for apostrophes/single quotes and ellipses for periods). This sentence describes the way you should read it to go one level deeper: it's a backwards acrostic. When you treat J.S. Bach as one term, the initials are HCABSJ, which when reversed is J S Bach.
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- When I first wrote about Göncz's completion of the Contrapunctus, in the 2012 discussion, I thought that his completion ended on a dominant (V) chord, instead of the tonic (i or I) that one would expect from Bach. I complained that the piece "sounded like it was ending on a question mark". It turns out that it really is the I chord, the original tonic from the beginning of the piece, except in D major instead of D minor to give it more finality. I had simply lost track of the "stack" while listening. To borrow a reference from the following dialogue, I got Majotaur'd. --rspeer