Hum a note to yourself, even just in your head. Any note will do.
Now double the frequency of that note, and hum that.
“How on Earth,” you’re likely asking, “am I supposed to know the frequency that corresponds to this random note? How am I supposed to know what twice that would sound like?”
Someone with perfect pitch and a good deal of experience in music could probably sing you 440 hertz—A above middle C—because that’s a tuning standard. But would they know what 220 hertz sounds like, or 880?
It turns out they would, even if they didn’t know it, because there’s an extraordinary tie between mathematics and music that not everyone knows about: an octave is a doubling of frequency. Listen for yourself:
Long before you can multiply and divide, well before you can even count to 440, you have an intuitive awareness that 440 hertz and 880 hertz are somehow the same, even though one is higher in pitch. Your intuitive knowledge extends to exponential functions, too—to move up another octave, you’d have to double the frequency of the 880 hertz sound wave, rather than simply tripling the 440.
But that raises an interesting question: What happens when we step outside of integer multiples? How would 1320 hertz—3×440—sound relative to 880? Open up a few instances of this tone generator app and find out for yourself!
Sound familiar? Maybe you recognize that interval—880, 1320, 1720; a frequency multiplication by a factor 1.5, then by 2—from the opening theme to “2000: A Space Odyssey”. Hit play on the 440 hertz tone below, then on the 660 hertz one below that, and you’ll see that it’s the same story.
Harmonies and chord-building, then, are a further manifestation of our extraordinary innate musical-mathematical ability. Of course, part of the reason that two particular notes sound good together to our ears is that we’ve been listening to music featuring these harmonies for generations, but the fact that we hear a doubling of frequency as the “same” note is a clue that something more fundamental is going on—that we can recognize and appreciate the relationships between the physical properties of these sounds.
So what is happening here? Are we really capable of doing math unconsciously?
Of course we are! A classic example, as given by author Douglas Adams—who has elaborated on the “unconscious genius” in each of us more eloquently than I possibly could—follows:
“A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball’s spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes out to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball.
People who call this “instinct” are merely giving the phenomenon a name, not explaining anything.”
Adams goes on to tie this back in to our appreciation for music, though without explicitly mentioning the octave-as-a-doubling-of-frequency phenomenon:
“I think the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts—it has no meaning or purpose other than to be itself.
Every single aspect of a piece of music can be represented by numbers. From the organization of movements in a whole symphony, down through the patterns of pitch and rhythm that make up the melodies and harmonies, the dynamics that shape the performance, all the way down to the timbres of the notes themselves, their harmonics, the way they change over time, in short, all the elements of a noise that distinguish between the sound of one person piping on a piccolo and another one thumping a drum—all of these things can be expressed by patterns and hierarchies of numbers.
And in my experience the more internal relationships there are between the patterns of numbers at different levels of the hierarchy, however complex and subtle those relationships may be, the more satisfying and, well, whole, the music will seem to be.
In fact the more subtle and complex those relationships, and the further they are beyond the grasp of the conscious mind, the more the instinctive part of your mind—by which I mean the part of your mind that can do differential calculus so astoundingly fast that it will put your hand in the right place to catch a flying ball—the more that part of your brain revels in it.
Music of any complexity (and even “Three Blind Mice” is complex in its way by the time someone has actually performed it on an instrument with its own individual timbre and articulation) passes beyond your conscious mind into the arms of your own private mathematical genius who dwells in your unconscious responding to all the inner complexities and relationships and proportions that we think we know nothing about.”
—Douglas Adams, Dirk Gently’s Holistic Detective Agency.
When we listen to a Bach concerto, we’re feeding that subconscious genius, putting it through its paces—but the same goes for when we drive a car, adjusting the angle of the brake pedal to ensure that our velocity will reach zero before the distance to the next car’s bumper does. Even just sitting to watch clouds slowly billow and evolve on a summer day—instinctively, in some part of your brain, you’re estimating the Reynolds number in that particular patch of sky.
You have an innate ability to appreciate—and a tendency to find beauty in—the boundless complexities of the world around us. Seated in the subconscious parts of every human brain, there is a mathematical, musical genius that sees the Fibonacci sequence in the whorls of a flower and the fractal, crack-like geometry in a tree’s barren branches. It speaks in feelings and instincts, in the thrill that runs through us when we feel the “click” of understanding a new concept at a conscious level.
Now if we could only get it to come out and help us with our homework.
—Stephen Skolnick