This semester I am co-moderating a reading group on Douglas Hofstadter’s Gödel, Escher, Bach with a friend. The book’s topics seemed to be a perfect combination of my interest in music and his interest in mathematics. We have only discussed the first few chapters of the book so far, but in our discussion I began to note a distinct parallel between math and music theory.
You may have heard the argument that math and music have a lot in common. In my experience, people most often make this statement when arguing with their local school board about cutting arts education funding, or some similar situation. When pressed, however, most people would find it difficult to explain exactly what they have in common. I myself until earlier this week would have pointed foremost to the ratios between the frequencies of different pitches that determines their intervallic relationship. While this is certainly a valid connection between math and music, it is somewhat arcane. The study and appreciation of music does not depend upon the knowledge that Middle C vibrates at 261.626 Hz, or the frequency of any other note.
The connection that Hofstadter makes between music and math that I didn’t notice for almost 14 years of musical training is this: music and math are both formal systems. In my first music theory class here at the University of Oklahoma, we began doing exercises in counterpoint. In the most basic of these exercises, we were given a simple melody and instructed to compose another melody on top of the first, within the constraint of a set of rules about what was and was not allowed. Within that set of rules we could do whatever we pleased. There were numerous melodies possible within the set of constraints, but we were absolutely required to stay within those boundaries no matter how aesthetically pleasing our rule-breaking melody might be. As Dr. Ken Stephenson so eloquently put it to us freshman year, “borrect is your easiest option–” ‘borrect’ being a portmanteau of the words ‘boring’ and ‘correct.’
Musicians in the 14th century laid down the rules we used for these exercises. As we began to study music written by later musicians, we learned new sets of rules, and sometimes these new formal systems barely resembled the counterpoint exercises with which we began in our earliest theory classes. Nonetheless, skeptics among you might say that educational theory exercises might technically qualify as formal languages, but actual music doesn’t have to follow those rules. I wholeheartedly agree with that critique. We spent three lectures in one of my theory classes on the first three measures of Richard Wagner’s Tristan und Isolde, a roughly 20-second snippet of music that has gained notoriety amongst music theorists for the fact that it nearly defies analysis. So, if the rules laid down by the formal system of music theory don’t perfectly mirror the music written and played by musicians, how can it truly be a formal system?
I had similar questions when I was in my first theory classes. I deeply wanted to break the rules: if real music didn’t follow these rules, then why did I have to? I filed the question away in the back of my mind once I realized my frustration wasn’t going to change the fact that I had to learn music theory, and didn’t think about it very much until the other day, while discussing Gödel, Escher, Bach. There is a passage in the book about how the way numbers behave in number theory doesn’t necessarily mimic the way numbers behave in real life. As Hofstadter poignantly asks, “two raindrops running down a windowpane merge; does one plus one make one?”
Great minds are generally remembered when they discover or invent a new way of examining old subject matter. This is true in the arts as well as in math and science. Just as Euclid, Leibniz, and Gödel are remembered for revolutionizing the way we use numbers, so too are Beethoven, Wagner, and Schönberg remembered for their radical notions about how we use notes. None of their systems was perfect, and each had its own set of rules, and each eventually inspired someone else to reexamine the value of those rules and come up with their own. In Gödel, Escher, Bach, Hofstadter argues that this ability to step outside of the system in which we operate is the root of human consciousness.
In order to be a truly effective mathematician or musician, one must understand the rules and expectations of whatever formal system he is working in. Further, one must also know when to break those rules and leave constraints of the formal system behind–an idea that, while counterintuitive to the idea of a formal system, actually conforms to Gödel’s incompleteness theorem, which Hofstadter periphrastically states: “All consistent axiomatic formulations of number theory include undecidable propositions.” Just as axioms and rules can’t prove all truths in number theory, so to do the rules for melodic composition fail to create all aesthetically pleasing melodies. Our flexibility within (or more accurately, without) these guidelines allow us to discover greater truths and greater beauties than previously imaginable.