A Bach Canon Visualized as a Mobius Strip

Open Culture

mobius strip

The most impressive of Johann Sebastian Bach’s pieces, musicophiles may have told you, will knock you over with their ingeniousness, or at least their sheer complexity. Indeed, the music of Bach has, over the past two and a half centuries, provided meat and drink to both professional and amateur students of the relationship between ingeniousness and complexity. It’s no mistake, for instance, that the composer has offered such a rich source of intellectual inspiration to Gödel, Escher, Bach author Douglas R. Hofstadter, well beyond having given him a word to fill out the book’s title. Listen to the first canon from Bach’s Musical Offering, and you’ll hear what sounds like a simple beginning develop into what sounds like quite a complex middle. You may hear it and instinctively understand what’s going on; you may hear it and have no idea what’s going on beyond your suspicion that something is happening.

If you process things more visually than you do aurally, pay attention to the video above, a visualization of the piece by mathematical image-maker Jos Leys. You can follow the score, note for note, and then watch as the piece reverses itself, running back across the staff in the other direction. So far, so easy, but another layer appears: Bach wrote the piece to then be played simultaneously backwards as well as forwards. But prepare yourself for the mind-blowing coup de grâce when Leys shows us at a stroke just what the impossible shape of the Möbius strip has to do with the form of this “crab canon,” meaning a canon made of two complementary, reversed musical lines.

  •  
  •  
  •  
  •  
  •  

READ THE ARTICLE >

  •  
  •  
  •  
  •