This weekend marks the beginning of the end-of-semester concert season, featuring the Stevens Orchestra’s performance tomorrow, with Concert Band (December 2), Jazz Band (December 8), and Choir (December 9) following Thanksgiving break. There will also be several musical acts at EC’s Winter Wonderland on November 29th.
I’ve enjoyed attending and performing in these events over the years, in part because of the mathematical undertones pervading through all of music. For those of us who have taken or are currently taking music classes or lessons, you’ll know that some of the content inevitably involves at least a bit of music theory, which introduces lots of numbers and their manipulations into music.
For one, we have the concept of “intervals,” which help us determine notes in a chord, or where to jump to for the next note of a melody. Starting with the base note of a scale, we can go up a fifth (to the fifth note), down a third (bringing us to the second of the scale), back to a minor fourth, and so on. This can be a helpful tool for figuring out what a piece of sheet music sounds like before even playing it – if you know the intervals, you have a good sense of where the music is going.
The manipulation of intervals and scales can be captured even more precisely by modular arithmetic. There are 12 notes in a chromatic scale, meaning that every time we go up an octave, we are making (in one sense) 12 steps up to a note that is twice the frequency of the note where we started. Musicians describe these start and end notes as the “same” in a way so that we don’t have as many names for all the notes. So we’re only going up to 12 and then back down to 1, like how we would do addition modulo 12.
One more interesting example is that the definitions of musical terms such as “transpositions” and “inversions” are very close to their mathematical meanings. When we transpose, we shift every note in the scale-up by a certain interval — usually a half or whole step to give us the coveted key change in many famous pop songs. “Inverting” a pair of notes, meanwhile, means inverting them with respect to the octave scale so that, after inversion, the lower note becomes the higher note while maintaining their original interval spacing. It’s quite analogous to reflecting a point about an axis or flipping a shape inside out.
As a counterpoint to all this “look at all the math in music” mantra, I should note that much of this formalism came about long after humans began making music. What fascinates me here is that mathematicians, who generally enjoy finding patterns and noticing their intrinsic beauty, seek out areas like music to study more, and often make exciting discoveries. Even more so, humans, long before having the math of set theory and algebra (which is needed for a lot of these more in-depth mathematical technicalities of music), knew what sounded good, and blended music into many aspects of their cultural experience.
I think that this example is one of many wherein humans are a little more predisposed to mathematics than many people may believe. Sure, mathematics can be quite challenging, and what may come naturally to some can take others a lot longer to gain a working understanding of. But this is ok — it’s the same with music, where some people may have lots of talent and training in the area, while others with less of a background can still appreciate the beautiful noises their fellow humans are making. So as concert season approaches, I hope outlets like this column allow people to generally appreciate mathematics as performances allow the public to engage with and enjoy music. If math can strike the right chord in people’s lives, I firmly believe that it does have wondrous and fulfilling benefits.
