There’s math hiding in the music we love

MCGILL (CAN) / STANFORD (US) — After analyzing close to 2,000 compositions, researchers have uncovered a mathematical formula governing the rhythmic patterns in music.

“One of the things that we’ve known about music for a couple of decades is that the distribution of pitches and loudness in music follow predictable mathematical patterns,” says Daniel Levitin, a neuroscientist at McGill University and co-author of the study published in the Proceedings of the National Academy of Sciences.

“Rhythm is even more fundamental to our enjoyment of music: it’s rhythm that infants respond to first, it’s rhythm that makes you want to get out of your chair and move, and so it’s not really a surprise to discover that rhythm, too, is governed by a similar mathematical formula.”

Levitin and Vinod Menon, a neuroscientist at Stanford University, led the team that analyzed the scores of close to 2,000 musical compositions written by more than 40 composers over the last 400 years in a large variety of Western musical genres.

They found that all the musical compositions they studied shared the same “fractal” quality, where the part is a more limited repetition of the whole. That is the larger temporal structure of well-formed musical pieces is composed of repeating motifs of their own short-term temporal structure.

At the same time, researchers also discovered that each composer had his or her own highly individual rhythmic signature.

“This was one of the most unanticipated and exciting findings of our research,” asserts Levitin. “Mozart’s notated rhythms were the least predictable, Beethoven’s were the most, and Monteverdi and Joplin had nearly identical, overlapping rhythm distributions. But they each have their own distinctive rhythmic signature that you can capture.

“Our findings also suggest that rhythm may play an even greater role than pitch in conveying a composer’s distinctive style.”

From snowflakes to fern fronds and broccoli florets, fractal patterns are to be found throughout the natural world. The discovery that four centuries of musical compositions obey this same mathematical rule strongly suggests that composers’ own brains may have incorporated certain regularities of the physical world, to recreate self-similarity in works of musical art.

Indeed, the authors suggest, building on work begun in the 1970s that our sensory and motor systems may have a fundamental propensity to both perceive and produce fractal patterns not just across the three dimensions of space, but also across time.

The research was funded by the Natural Sciences and Engineering Research Council of Canada, the Canada Foundation for Innovation, and the National Science Foundation.

More news from McGill University: