The book Gödel, Escher, Bach (“GEB”) is an imaginative exploration of the nature of consciousness, as seen through the metaphorical lenses of mathematician Kurt Gödel's self-referential logic, artist M. C. Escher’s paradoxical images, and composer J. S. Bach’s intricate compositions, among others. I'm not just a fan of the book; I became friends Doug in 1975 as he was writing the book, talked constantly with him as he wrote the book, and contributed many ideas. Among other things we found we had in common, we had both invented ambigrams, and both considered carving a block of wood to cast three different shadows. I had just read Gödel's Proof, and was learning Bach’s Six Part Ricercare from The Musical Offering!
GEB is a feast for the intellectually adventurous, but it can be tough going for many because it calls upon knowledge of so many fields, especially music. So for those of you who want to hear Bach’s music referenced in the book, here is a chapter by chapter collection of links to performances and articles about the pieces. Where possible, I've linked to performances that include visualizations of the musical score, to make the music easier to follow.
The book alternates between fanciful dialogs, and prose chapters, inspired by the structure of Bach's 48 preludes and fugues of The Well-Tempered Clavier. Each dialog introduces ideas explored in the subsequent chapter. In a few dialogs, such as the Crab Canon and the concluding Six-Part Ricercare, the structure of the dialog mimics the structure of the music; usually the connection is not as literal.
Part I: GEB
Introduction: A Musico-Logical Offering 3
GEB opens with the story of The Musical Offering, one of Bach's last works, a suite of canons, fugues all based on a theme given to him as a challenge by King (and musician) Friedrich the Great of Prussia. Like GEB itself, comprehending this monumental work takes effort and many listenings, but is immensely rewarding.
>> The Musical Offering. Listen to a performance by a small orchestra, which makes the individual voices easier to hear than in a performance on keyboard. Read a riveting account of the encounter between Bach and King Friedrich that puts the event in fascinating historical context.
Dialog: Three-Part Invention 29
Bach appropriated the word “Inventions” for short , rigorously constructed contrapuntal pieces. “Contrapuntal” means that the various melodic lines, sometimes called “voices”, are equal musical participants, as opposed to music in which one voice is the melody and other voices are supporting accompaniment. Bach was a master of counterpoint; good jazz trios often strike contrapuntal balance. Bach wrote two collections of inventions for keyboard — the Two-Part Inventions have two voices, and the Three-Part Inventions (originally titled Sinfonias) have three. This dialog, written by Lewis Carroll, is spoken by three characters. Since the Inventions are meant to be played on keyboard — harpsichord was the dominant keyboard instrument of the day — one hand must play two voices at the same time, a feat of manual dexterity that pianists commonly master.
>> Three-Part Inventions. Listen (clavichord and harpsichord). Read.
Chapter I: The MU-puzzle 33
Dialog: Two-Part Invention 43
Chapter II: Meaning and Form in Mathematics 46
Dialog: Sonata for Unaccompanied Achilles 61
A reference to Bach's six Suites for Unaccompanied Cello. These frequently performed compositions create a rich musical universe with a solo instrument.
>> Suites for Unaccompanied Cello. Listen. Read.
Chapter III: Figure and Ground 64
Dialog: Contracrostipunctus 75
The name of this dialog refers to the title Contrapunctus, literally “counterpoint”, which is the idiosyncratic title that Bach used for the 14 fugues that appear in his work The Art of Fugue [recording, article].>> The >> Art of Fugue. Listen to a performance and read an analysis by Jeff Hall, who created a wonderful site containing the entire Art of Fugue, including a completion of the final unfinished fugue.
Chapter IV: Consistency, Completeness, and Geometry 82
Dialog: Little Harmonic Labyrinth 103
A curious work for organ that, true to the title, threads its way tortuously through a series of perilous key changes. As a church organist, Bach needed to know how to get quickly from any key to any other key in order to improvise smooth transitions between hymns written in different keys — this piece is a veritable catalog of harmonic transitions.
>> Little Harmonic Labyrinth. Listen. Read.
Chapter V: Recursive Structures and Processes 127
Dialog: Canon by Intervallic Augmentation 153
To “augment” is to make something bigger. Bach often used temporal augmentation in his compositions — stretching out a theme in time so it takes twice as long to complete. In theory it is possible to augment themes by other ratios, e.g. 3:1, 4:1 or 5:1, though augmentation by ratios other than powers of 2 are very rare. Intervallic augmentation means stretching a theme along the pitch dimension, so note of the scale are spaced twice as far apart — a half-step becomes a whole step, a whole step becomes a major third, and one octave becomes two octaves.
In German musical tradition, B refers to the pitch we call B flat, and H refers to B natural. In this dialog the characters note that BACH played on a keyboard can be intervalically augmented by a ratio of 10:3 to make the name CAGE. John Cage was a 20th century American composer who used chance operations in his music to create soundscapes that are more like listening to nature than listening to classical music.
Intervallic augentation is musical nonsense, so it is no surprise that Bach never used this device. To fill the void, I wrote a canon by intervallic augmentation, in which the slow voice starts with the notes CAGE, and the other starts with the notes BACH. The CAGE voice is also temporally augmented, so it is slower. Both augmentations are by a ratio of 10:3, meaning the fast voice plays ten notes in the same time the slow voice plays three, and the interval of a minor third (3 half steps) becomes a major seventh (10 half steps).
>> BACH/CAGE canon by intervallic and temporal augmentation. (I am working on posting this.)
Chapter VI: The Location of Meaning 158
Dialog: Chromatic Fantasy, And Feud 177
Bach’s Chromatic Fantasy and Fugue is an expansively florid work for keyboard. In music, “diatonic” refers to melodic movement through a conventional major or minor scale, while “chromatic” refers to melodic movement by half-steps, which includes all the intermediate notes omitted in diatonic scales. Doug reinterprets the word to refer literally to color.
>> Chromatic Fantasy and Fugue. Listen.
Chapter VII: The Propositional Calculus 181
Dialog: Crab Canon 199
In a crab canon one voice is the other voice played backwards, making the whole composition a palindrome in time. The dialog follows the same structure: the lines spoken by one character are the same (with small interesting liberties) as the lines spoken by the other character, but in backwards order. This brief canon is particularly easy to comprehend, since the two voices are so clearly different.
>> Crab Canon from The Art of Fugue. Performance with animated score on a Möbius strip.
Chapter VIII: Typographical Number Theory 204
Dialog: A Mu Offering 231
The title is a truncated version of Bach's Musical Offering. "Mu" refers to a mathematical puzzle introduced at the beginning of the book, as well as to the Zen master Mumon.
>> The Musical Offering. Listen.
Chapter IX: Mumon and Gödel 246
Part II EGB
Dialog: Prelude ... 275
The title of this and the following dialog together refer to Bach's many preludes and fugues, which he frequently paired in keyboard works for organ or harpsichord, most notably The Well-Tempered Clavier, two volumes of 48 Preludes and Fugues, one for each major and minor key. Preludes are simply stuctured pieces, often with a thematic textural motif, whereas Fugues have complex dense structures; the two together form a satisfying whole with a dramatic pause between.
>> The Well-Tempered Clavier, book 1. Listen and Watch. Read.
Chapter X: Levels of Description, and Computer Systems 285
Dialog: Ant Fugue 311
Chapter XI: Brains and Thoughts 337
Dialog: English French German Suite 366
Bach was a master at absorbing different national styles, and making them his own. This title refers to Bach's three keyboard volumes, the English Suites, the French Suites, and the German Suites, each containing a number of compositions based on courtly dances.
>> The English Suites. Listen to a piano performance by Murray Perahia.
>> The French Suites. Listen and watch a score of a piano performance by Ingrid Haebler.
>> The German Suites. Listen to a harpsichord performance by Karl Richter.
Chapter XII: Minds and Thoughts 369
Dialog: Aria with Diverse Variations 391
The title of this dialog is the original title of Bach’s famous Goldberg Variations, a cycle of keyboard works all based on a single descending bass line. Like The Well-Tempered Clavier, this work explores a vast number of musical styles within a single collection. In the year 1974, a collection of slight but ingenious canons based on the bass line of the Goldberg variations was found. It is worth a listen too.
>> Goldberg Variations. LIsten. Read.
>> Canons on the theme of the Goldberg Variations. Listen and watch annotated version of the original score, nicely illustrating the structure of each canon.
Dialog: Air on G's String 431
A reference to Bach's Air on the G String, August Wilhelm's 19th-century arrangement for piano and violin of the second movement of Bach's Orchestral Suite No. 3 in D major. By transposing the piece down a step to C major, the arranger was able to play the piece on only the G string of his violin, thus the nickname.
>> Second movement of Bach’s Orchestral Suite No. 3 in D major. Listen and watch the scrolling score.
Chapter XIV: On Formally Undecidable Propositions of TNT and Related Systems 438
Dialog: Birthday Cantatatata ... 461
The title of this work is indeed the name of a lesser-known work by Bach, the Birthday Cantata. The title extends the word "cantata" by adding repeating "tatata", because the letters T and A, along with C and G, are the initials of the four molecules that appear in strands of DNA. The title of the science fiction movie Gattacca is based on the same initials, and for the same reason.
>> Birthday Cantata. Listen to an orchestral performance.
Chapter XV: Jumping out of the System 465
Dialog: Edifying Thoughts of a Tobacco Smoker 480
Another lesser-known work by Bach, which Doug uses to refer to Rene Magritte’s characteristically paradoxical painting This is not a Pipe.
>> Edifying Thoughts of a Tobacco Smoker. Listen to a brief aria.
Chapter XVI: Self-Ref and Self-Rep 495
The Magnificrab, Indeed 549
A pun on Bach's Magnificat in D, a glorious religious work for orchestra and choir. No structural similarity to the musical work, just a clever bit of wordplay.
>> Magnificat in D Major. Listen.
Chapter XVII: Church, Turing, Tarski, and Others 559
Dialog: SHRDLU, Toy of Man's Designing 586
A pun on Bach's chorale Jesu, Joy of Man's Desiring. No structural similarity to the musical work, just more wordplay.
>> Jesu, Joy of Man's Desiring. Listen to an orchestral performance.
Chapter XVIII: Artificial Intelligence: Retrospects 594
Dialog: Contrafactus 633
Another reference to Contrapunctus, literally “counterpoint”, which is the idiosyncratic title that Bach used for the 14 fugues that appear in his work The Art of Fugue [recording, article].
>> Art of Fugue. Listen to a performance and read an analysis by Jeff Hall.
Chapter XIX: Artificial Intelligence: Prospects 641
Dialog: Sloth Canon 681
"Sloth canon" is the clever name Doug gave to the canon by inversion and augmentation (the second voice is an upside down and slower version of the first voice) that appears in Bach's work The Art of Fugue.
>> Canon by Augmentation and Inversion from The Art of Fugue. Listen.
Chapter XX: Strange Loops, Or Tangled Hierarchies 684
Notice that the initials of this chapter spell SLOTH.
Dialog: Six-Part Ricercar 720
GEB ends, as it began, with The Musical Offering, this time a six-person dialog inspired by the structure of the six-voice fugue that concludes Bach’s masterwork. As in a fugue, each new voice opens with the same line, in Hofstadter's case “I can do very well without such a program.” Bach's fugue is eminently playable with two hands on keyboard, despite occupying most of the fingers much of the time.
>> Ricercare a 6. Listen and watch with note by note animation of a synthesized harpsichord. Listen and watch Webern's orchestration, with annotated score.