J.S. Bach’s Crab Canon and the Möbius Strip

posted by Aji Coronel on Sep 22, 2009, under Music Theory

I caught this fascinating post at BoingBoing.net about a YouTube video showing how J.S. Bach’s Musical Offering (from 1747), specifically the Canon 1 a 2, could be played forwards and reversed, twisted and turned upside-down, split and played simultaneously inside-out by some crazy mathematicians, as a never-ending melodic movement. Well, if that’s a bit confusing, just watch the video below and you’ll get the idea…

J.S. Bach is perhaps the greatest composer of the Baroque era. He was widely regarded for the intellectual depth, technical facility and artistic beauty of his compositions. His output of works was prolific, and he was often commissioned to compose a lot of music for the Church during his time. He also paved the way for the next generation of giants that included Mozart, Beethoven, Robert Schumann, Felix Mendelsshon and even Frederic Chopin.

Bach was also the greatest proponent of the sophisticated art of the counterpoint, and specifically the canon and the fugue, which incidentally I first heard from the Swedish guitar virtuoso Yngwie Malmsteen’s compositions (although I had been listening to classical music too (and Bach specifically) since I was in high school).

The canon is a compositional style whose foundation is in fact the counterpoint but with some very strict and specific requirements. J.S. Bach’s Canon 1 a 2 is an example of an infinite or perpetual canon that is also a retrograde canon (and hence the nickname “Crab Canon”) because it can be played forward or backwards in time indefinitely. I will not go too in-depth about the anatomy of contrapuntal composition and the canon itself as it’s too intricate and technical. Just click on the links below if you need a deeper explanation.

Now, many mathematicians (and some others like Xantox, and John Leys who did the YouTube illustration above) became very intrigued by this melodic snippet, because aside from being a strict canon alright, and an infinite and retrograde canon at that (which means can be played endlessly and backwards too), it is also possible to play it simultaneously forwards and backwards in time. In their own words,

In each of these canons a musical line is played twice (or four times in Canon 10). The second version is always transformed with respect to the first by shifting in time, but it may also be shifted in pitch, turned upside-down, stretched, or played backwards. Each of these transformations occurs in the mathematics of elementary functions; they are examples of how new functions can be made out of old and of how a function can be tailored to fit a new situation.

If you listened intently on the amazing combination and interaction of the contrapuntal melodies, you will realize why it becomes so mind-boggling. The fact that it can be played indefinitely (as in the Möbius Strip transformation) makes it all the more appealing to mathematicians. It’s as if Bach himself composed this seemingly simple piece with the very intention of it being discovered later on (after centuries in fact!) of all these amazing possibilities.

For additional reading: