So far this year, I have written about two of the four traditional mathematical arts: arithmetic and geometry. I’ve also taken a brief detour into algebra, the mathematical art responsible for one of the largest transitions in the history of mathematical thought because it mended arithmetic and geometry together. Now, I’ll return to music, the third classical mathematical art.
When we discuss music in a mathematical context, we need to understand that historically, what people meant by “music” was not fundamentally what we usually mean. When we think of music, we imagine tuning in to the radio, singing hymns together, putting a record on, lullabying a child to sleep, or bowing away at a violin (hopefully not all at once). All of these are properly called music; but for the Greeks, who laid the foundation of our mathematical thought, these hearable forms of music were low, unideal forms. True music wasn’t something you can hear, but something you can count: specifically, music was the study of ratios.
The Greeks were (rightly) enthralled by the idea that strings and pipes vibrate at specific speeds, and that those vibrations occur together in particular ways relative to each other. Let me explain one example: if I strum the low E string on my guitar, then strum the A string, the two strings make the nice sound we call a fourth (the same sound we hear when we sing “Here Comes the Bride”). That’s great, but the math behind the harmony is where things get really interesting. That E string, because of its thickness, length, and material, vibrates 82.5 times a second. The A string vibrates 110 times a second. If you do a little math, you realize that this means the E string vibrates 3 times for every 4 vibrations in the A string, a ratio of 3 to 4.
Cool, one might say, a little math in the guitar. But if you compare other vibration ratios, like the Greeks did, you find more harmonies. A fifth (the famous beginning to the Star Wars melody) comes from a vibration ratio of 2 to 3; a major sixth (the first two notes in “N-B-C”) has the ratio of 3 to 5; a whole octave (“some-WHERE over the rainbow”) has the ratio of 1 to 2.
And it gets even more fascinating than that. Even if you strum only one string, say, that A that vibrates 110 times a second, it has smaller vibrations that also occur naturally. Some of them go twice as fast as the whole string, some three times as fast, some four, some five, and so on. In other words, those harmonic ratios exist together all within the strumming of one string; every note you play on an instrument contains the whole spectrum of sound! It’s like white light that contains an entire rainbow of color. In fact, that spectrum of sound is what governs the character of an instrument. It’s why violins sound different than flutes; they emphasize different parts of that spectrum.
You can probably guess that I’m only scratching the surface of an incredibly rich field of study, both musically and mathematically. What ratios do you choose to make your harmonies? Do you pick simple ones to sound pure? Do you pick more complex ones to allow for more complicated music? What’s the most perfect set of ratios to use on an instrument? The list of questions goes on, and a lot was written on these topics (just to get a taste of it, you can Google “Fundamentals of Music by Boethius” and scroll to somewhere in the middle; things get quite crazy). This study of ratios and how they interact is what was commonly thought of as that third liberal art, music.
Many students (ours included) get a lot of experience working with ratios, whether that be in schoolwork or in things like Lego-building or cookie-baking. There’s music all over the place, even when we can’t hear it. Thankfully, it’s a huge part of any math class, which means music is (from a Greek perspective) a huge part of it too. The amazing thing is that this kind of math happens to also be something we can hear and enjoy, even just by using our voices, without even thinking about the math involved! Thanks be to God that we have such a beautiful facet of mathematics to enjoy.
In Christ,
Mr. Hahn
