A Strange Loop is a concept created by Douglas R. Hofstadter in his book, Godel, Escher, Bach: An Eternal Braid. The 'Strange Loop' is a continuous cycle of feedback in a system that allows it to 'perceive itself', to talk about itself, to become 'self-aware'." Hofstadter's Inspiration for writing GEB, first published in 1979, was his long-held conviction that his strange loop notion "holds the key to unraveling the mystery that we conscious beings call 'being' or 'consciousness'." Also called an "isomorphism", Hofstadter sees Strange Loops as an analogy for the way human language tracks reality "through the [complex] organic processes taking place inside the brains of carbon based life forms." In Mathematics, the Strange Loop is an "endless mathematical system that perceives itself through 'meaningless' symbols that use patterns to accurately track, or mirror, various phenomena in the world [such as a computer program]." Implicit in Hofstadter's definition of Strange Loops is the concept of infinity - a way of representing an endless process in a finite way.
In GEB, Hofstadter extends his Strange Loop analogy to mathematics, art and music, giving examples of this phenomena discovered by mathematician Kurt Godel (Incompleteness Theorem in Mathematical Logic - a mathematical proof that connects self-referential statements to number theory); intellectually stimulating lithographs and woodcuts by Dutch Graphic artist M.C. Escher; and complex Canons and Fugues by Eighteenth Century musician, J.S. Bach, (specifically, Musical Offering, representing one of Bach's supreme accomplishments in musical counterpoint.) It is Hofstadter's insight on Bach's musical work that I want to discuss here.
Hofstadter explains that on May 7, 1747, Bach first played his Fugue, Musical Offering, extemporaneously for Frederick the Great, King of Prussia, who played the flute and enjoyed evening concerts of chamber music. The King wanted Bach to try out his large collection of pianos, and summoned him to his palace. He asked the King to give him a subject (musical idea) for a Fugue. The King did, and expressed a wish to hear the Fugue based on his idea with six Obligato (accompaniment) parts! To the King's astonishment, Bach was able to perform it. When Bach returned home to Leipzig, he composed it in three and six parts, including ten of the most sophisticated canons he ever wrote, had it engraved under the title of "Musikalisches Opfer" (Musical Offering) and humbly dedicated it to the King.
Hofstadter's discussion of the infinite, escalating nature of Bach's isomorphic, Strange Loop is both fascinating and accessible.
"The idea of a canon is that one single theme is played against itself. This is done by having "copies" of the theme played by the various participating voices. But there are many ways to do this. The most straightforward of all canons is the round, such as "Three Blind Mice", Row, Row, Row Your Boat", or "Frere Jacques". Here the theme enters the first voice and after a fixed time-delay, a "copy" of it enters, in precisely the same key. After the same fixed time-delay in the second voice, the third voice enters carrying the same theme, and so on. Most themes will not harmonize with themselves in this way. In order for a theme to work as a canon theme, each of its notes must be able to serve in a dual (or triple or quadruple) role: it must firstly be part of a melody, and secondly it must be part of a harmonization of the same melody. When there are three canonical voices, for instance, each note of the theme must act in two distinct harmonic ways, as well as melodically. Thus, each note in a canon has more than one musical meaning; the listener's ear and brain automatically figure out the appropriate meaning, by referring to context.
The cannon can then continue to grow in complexity...
The first escalation in complexity comes when the "copies" of the theme are staggered not only in time, but also in pitch; thus, the first voice might sing the theme starting on C, and the second voice, overlapping with the first voice, might sing the identical theme starting five notes higher, on G. A third voice, starting on D, yet five notes higher, might overlap with the first two, and so on. The next escalation in complexity comes when the speeds of the different voices are not equal; thus, the second voice might sing twice as quickly, or twice as slowly, as the first voice. The former is called diminution, the latter augmentation (since the theme seems to shrink or expand.)
Wait, there's more...
The next stage of complexity in a canon construction is to invert the theme, which means to make a melody which jumps down whenever the original theme jumps up, and by exactly the same number of semitones. This is a rather weird melodic transformation, but when one has heard many themes inverted, it begins to seem quite natural. Bach was especially fond of inversions, and they show up often in his work-and the Musical Offering is no exception.
Not done yet...
Finally the most esoteric of "copies" is the retrograde copy- where the theme is played backwards in time. A canon which uses this trick is affectionately known as a crab canon, because of the peculiarities of crab locomotion. Bach included a crab canon in the Musical Offering.
Hofstadter reasons that since every type of 'copy' in Bach's Musical Offering "preserves all the information in the original theme, in the sense that the theme is fully recoverable from any of the copies," it can be called an isomorphism, or a Strange Loop. He suggests pianists try composing a Canon of their own to the tune of Good King Wenceslas. A fan of Bach's music, I was reading GEB around the Christmas holidays a couple of years ago, and couldn't resist exploring this musical puzzle on the piano. It was great fun! This musical process is much like, as Hofstadter describes it - "An Endlessly Rising Canon."
J. S. Bach had an extraordinary gift for producing the most beautifully intelligent, life-giving music - a mirror of his great musical soul. If nothing else, Hofstadter's analogy of Bach's work to his own concept of Strange Loops that mirror what it is to "be", offers inspiring insight into the human drive and ability to innovate. Especially when the choice is -
To be....or NOT be........Your Majesty's most humble and obedient servant!
Don't Wait to Begin Piano Lessons!
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