Sunday, February 17, 2013

The Möbius Strip and Music

I discovered topology at a fairly young age. I think it was one of the very first things I discovered that truly seemed magical. An object in the real world with only one side? And one edge? That was the Möbius strip. The easiest thing in the world to construct: just take a strip of paper, twist it once and tape the ends together. Voila, something you can perplex your friends with. Here is what it looks like:

It has some interesting properties. Yes, it has just one side which you can demonstrate by taking a pencil and drawing a line down the middle. You can draw the line continuously without lifting the pencil on both sides, because there is only one side. Now, what do you think will happen if you take a pair of scissors and cut the strip in two down the middle? Well, you can't cut it in two! Instead you will get a single strip half as wide and twice as long with two twists.

Now what does this have to do with music? There is a YouTube video that demonstrates the relationship between a Möbius strip and a Bach canon. The canon is the type known as a 'crab' canon, meaning that the melody combines with itself played backwards:

The musical mathematician Vi Hart does even more in this mind-bending video where she demonstrates the folding of space-time with a music box:

If you are interesting in mathematics, music and Bach, this article is a useful introduction.

Let's end with a piece by Bach that has provoked a great deal of commentary. The St. Matthew Passion that Mendelssohn called "the greatest Christian work" is a mammoth depiction of the crucifixion of Christ set for two choruses, two orchestras and six vocal soloists. This article gives some idea of the scope and complexity of the structure. Here is a performance of Part I by Nicholas Harnoncourt:

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