Thursday, November 24, 2011

Fractals, Feldman, and Flabbergastation

If you were to write “Morton Feldman” right next to "zn+1 = zn2 + c” on a piece of paper, the amount of confused looks you would get would probably be… a lot. According to society, zn+1 = zn2 + c is simply an equation, something incomparable to an artist—but in that equation lies the tangible way of expressing the source, or representation, or one of the most celebrated things in math and nature. If we could reduce Feldman and zn+1 = zn2 + c down to just their meanings in the world of creation, we might have something pretty similar. And in that world, it doesn’t really matter if you’re made up of a full head of slicked back, dark hair or a handful of symbols. 
A couple of weeks ago I got to hear Feldman’s “Clarinet and String Quartet” live at the Church of Beethoven. Not only was that one of the only concerts where it seemed perfectly appropriate to lay down on the floor in a sort of zoned-out, meditative state, but it was the first concert in a while where the explanation of the piece stayed with me as much as the piece itself. The emcee/clarinetist of the piece, James Shields, told the audience about a theory he had on Feldman’s work; if a phrase of music, such as Mozart, was stretched out to reach 40 minutes, maybe we would have a Feldman piece.  

That brought up the idea in me of something seemingly opposite to music—fractals. I wouldn’t doubt that everyone had a phase in middle/high school math where fractals were a small obsession (at least I’m hoping that’s not just me…). If not, a fractal is “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.” zn+1 = zn2 + c  is an equation that describes a large amount of the Mandelbrot set, one of the most famous fractals. Basically, fractals are these things: 

Julia set
Some other set

Mandelbrot set
Fractals aren’t just an idea in the world of graph paper and computer programs, either. Like the Fibonacci sequence, fractals are found in nature, such as in ferns, snowflakes, rivers, and romanesco broccoli.  If something has a self-similar structure, it’s an approximate fractal. That’s the most important thing about fractals that relates them to sound—when magnified, their sequence keeps repeating, expanding, and becoming even more intricate. Fractals themselves are an eerie idea—mathematicians discover them just as celestial bodies thousands of light-years away are found. The Mandelbrot set was first discovered and mapped out in small asterisks like a surfacing underwater creature. 

First picture of the Mandelbrot set

Romanesco broccoli... delicious and mathematically trippy
Morton Feldman and his music, on the ostensibly other hand, are not extraterrestrial-looking shapes on computer screens. But, if looked at with the lens of the theory offered at the concert, Feldman’s pieces could be similar to fractals. While Feldman’s music is part of the indeterminacy movement and fractals are specified to the max, what makes the music of composers like Feldman mind-boggling is their ability to be received in an infinite amount of ways. The theory allows Feldman’s music, and all music in general, to be seen as a medium that is made up of only itself, which, for all we know, is pretty true. This means that, like fractals, music could be self-repeating, and all we hear is its manifestation in certain sizes. Perhaps each symphony is made out of little symphonies, and each note in that symphony is a collection of an infinite amount of compositions. Let’s take, for instance, Feldman’s “Clarinet and String Quartet” to think about.  
In this piece, the string quartet creates a cloud of indecipherable eeriness while the clarinet lurks in the background with a simple chromatic melody: B, C, A, B flat. The melody is stretched and snapped, rarely played the same way twice. While the clarinet continues this, the string quartet changes from echo-filled to sinister and back again. Throughout the entire piece, the clarinet stays as the holder of order. Its repeated notes and different chromatic melodic cells lead the strings like a magnet staying attached to another through a surface.   The music doesn’t seem to follow any set path, but instead wanders through the dark, encountering miniscule areas of action. 
Like the way objects look under microscopes, Feldman’s “Clarinet and String Quartet” is murky and slow moving. If we, as listeners, can let the music take on a life of its own and completely abandon any specified meanings assigned to the piece by Feldman or others, we can put it in the position of a microscope image—if a phrase of Mozart was stretched out to its limits, maybe we would find little pieces of Feldman, Cage, or Wolff inside, like the alternate universes that were found in space by Dave in “2001: A Space Odyssey.” Even though this might seem like a dubious prospect, fractals have technically existed since the dawn of time—it just took sophisticated mathematicians to surface them from the depths of the math world. And even though these mathematicians found the shapes of fractals such as the Mandelbrot set, if they never thought of going further into those shapes, we wouldn’t know about their self-repeating aspects. We can certainly slow down phrases of classical music, but, like a microscope, maybe there is a certain setting that exposes these alternate dimensions of sound that we haven’t found yet. 
Then again, maybe there’s not. Maybe Mozart slowed down sounds like 5 minute-long chords that we’re all familiar with. But the point is this: music can take different forms and identities, from classical to chamber pop to indeterminate, but it’s a medium that has no real physical substance when stripped down to its core meaning. Music isn’t tangible matter that we can stain, put on a slide, and take apart. All we know is that it is made up of itself—kind of like a fractal, no? Maybe, inside of itself, music is made up of separate little compositions that lurk inside of the microscopic nooks and crannies of other compositions. Or, maybe that 80 minute Mahler symphony is a large scale version of the little sounds that make up each vibration that comes out of a cello. Music is mind boggling. And I don’t think asterisks will ever be able to map out something like it. 


  1. Great post Elena. Brings back memories of thinking about fractals in a Schenker class I had years ago...particularly when analyzing Mozart.

  2. Math and Music, the two great tastes that taste great together.

  3. Of course I'm an avid Elena fan. Only you would see the connection between fractals and music. Sure, lots of music is derived from 1 melodic idea, you'll hear a lot about this as a composition major at USC. I'm wrapping up something right now that takes a tritone and spins it out into 4 minutes. And, that's not a simple chromatic melody, that's B-A-C-H in retrograde.

  4. Thanks for reading, everyone. Paul, thanks for the compliment, and that tritone piece sounds insane in the best way possible! I'd like to hear it sometime. How is the composition program at USC?

  5. Elena, sorry for the delay, I haven't checked your blog in a bit. You know you can always call me. Of course I'm biased, but I can't imagine a better composition department than USC.