Can You Hear the Music of Mathematics?

Just about anybody who’s anybody can tell you about the math in music. Between time signatures (three-quarter, four-four, cut-time…), fractional notes (whole, half, quarter, eighth, sixteenth, thirty-second, sixty-fourth…), and the rest of the nomenclature (octaves, measures, counting…), music is nothing without math.

Even if you take out the technical aspects, the popular discussion is rife with numbers. Literally. I mean, how many of you have bought a 45 of your Number 1 hit that you just heard on the top 50* countdown? (*These being the top 50 songs as measured by the Billboard 100.)

Again, anybody can talk about the math in music, but can they talk about the music in math?

Now, you might already have known this, but I don’t like to fly. It’s been this way for a long time. Actually, the flying part is OK, it’s the take offs and landings that unnerve me. Way back when, it occurred to me if I didn’t pay attention when the plane took off or landed it seemed I felt much better about it.

Many of you recognize numbers can absorb your thoughts. Indeed, numbers sing to me. Some see them in a cold stark way. To me, they play like a melodious aria. Naturally, then, whenever I boarded a plane, I read books that involved numbers. Space books, science books, and books about the history of space and science. One of the first books I chose to read was Tom Wolf’s The Right Stuff. It’s about the early years of the space program from Chuck Yeager and the test pilots to Gordon Cooper and the Mercury Seven.

A great book for a space buff. Not so great a book for someone who wants to get his mind off of plane crashes. I had to be more deliberate in my choice of reading material if I wanted to alleviate my fear of flying. Heading into a particularly rigorous travel schedule that had me flying nearly every other week, I picked an extremely thick tome that covered the gamut of my interests in a manner I most enjoyed.

Rarely would a month go by when I didn’t read my Scientific American magazine. I most enjoyed Douglas Hofstadter’s Metamagical Themas column. His eclectic style of connecting seeming unconnected ideas, facts, and histories appealed to me. It captured my mind in a manner that so engrossed it, I lost myself amidst my surroundings.

In other words, it was the perfect form of reading for my air trips. Soon, I took my newly purchased copy of Hofstadter’s 1979 book Gödel, Escher, Bach: An Eternal Golden Braid on every plane I boarded.

“Escher” (as in M.C. Escher, the artist best known for creating delightful optical illusions) doesn’t hold much of a place in this column.

But “Gödel” and “Bach” do. Mathematician Kurt Gödel and composer Johann Sebastian Bach appear to come from unrelated fields. In fact, Hofstadter went to great lengths to explain his book was not about the relationship between the areas covered by the men in his title. Ironically, his book was about deriving meaning from meaninglessness. Indeed, Gödel, Escher, Bach wasn’t meant to relate Gödel, Escher and Bach to each other. In the process of writing it, though, this is precisely what Hofstadter did.

All you need to understand to see this is how the three titular characters use recursion and self-reference. For example, Bach’s canon frequently repeated themselves in a mirror-like fashion. Bach was also famous for hiding his name B-A-C-H in the scores of his music. Quite an achievement given there is no musical note “H,” but Bach found a clever way to accomplish this.

Right about now you’re probably wondering “What does this have to do with the music of math?” Granted, all you’ve seen to this point is the math in music. Bach, after all, was often referred to as the mathematical musician. His cryptographical exploits only emphasize this point. And hiding his name in his compositions was only one of many examples.

To tell you the truth, it’s not as easy as it looks. It’s hard (but not impossible) to do it in writing. It’s the devil to do it in music.

Occasionally, you see a bald (yes, “bald,” not “bold”) attempt to map math onto music. You don’t need to (and shouldn’t) use such a blunt hammer. Music naturally alludes to mathematics. It doesn’t need help.

Surely, if you’ve ever read music, you can see how math flows from it.

Zany as it may sound, the same could be said that music flows from math. There is music in math. It only requires a fuller appreciation for what music and math are to see this. Here are some examples.

Around the time you enter Kindergarten, you will have been exposed to songs and nursery rhymes (essentially, songs without a written score). You learn to sing “One Potato, Two Potato” and “The Ants Go Marching” and that reveals the music of counting.

By the time you’re well into elementary school, you expand upon the counting songs. It turns out the music of counting not only reveals addition (and subtraction), but also multiplication (and division). Think of the tunes that went along with counting by twos or counting by fives. Yes, it was math. It was math told through music.

But it is the three basic properties of numbers that sing the loudest. The Associative, Communicative, and Distributive Properties are all about arranging and rearranging. What’s another name for a musical composition? An “arrangement.” Coincidence? I think not. These are themes explored both by Gödel and Bach (OK, and by Escher, too, but this column is about music and math, not art).

Often you will see music repeating itself or scores rearranging themes throughout the piece.

The mathematical properties offer the same movement with numbers. Think of the steady tempo comodo of the multiplication tables, the heightened accelerato of exponentials, and the diminished ritenuto of limits in calculus.

This is the music of math.

Perhaps you can hear it. Maybe you’ll hear it more now that you’re aware of it.

Keep this in mind: If you want your kids to not view math as the cold starkness of their educational years, sing them a song of math. Then tell them the music is in the math, not that the math in the music. This twist will shock them like a “Man Bites Dog” headline.

Whence forth, every time they hear music, they can’t help but think of the math. It’s not that the music is in the math or that the math is in the music. It’s that music is math and math is music.

You may find this is the best way to get your children to like math (or to get them to hate music).