The connection between mathematics and music is often touted in awed, mysterious tones, but it is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales. And even a beginning piano student encounters geometry in the "circle of fifths" when learning the fundamentals of music theory.
But according to Dmitri Tymoczko, a composer and music theorist at Princeton University, these well-known connections reveal only a few threads of the hefty rope that binds music and math. To grasp the true structure of music, he says, we need to understand the geometry of hyperdimensional objects. Doing so has given him new ways of understanding pieces of music that have long baffled theorists and even led him to new insights into the history of music.
Tymoczko compares the structure of music to the shape of a rock face that a rock-climber is scrambling up. "If you know the conditions of the rock face, you can predict the motions of the climber," he says. "The structure of the space makes certain choices overwhelmingly natural or convenient. There's something similar that goes on with music. When you think about things abstractly, you can come to understand that the directions that music went aren't completely arbitrary. Composers are exploring the possibilities that musical space presents them with."
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