Can’t stop humming that tune? Thank math.

While Super Bowl LX is over, the big game still resonates in the minds of many viewers. Not the offensive coordination of the Seattle Seahawks (or the lack of coordination of the New England Patriots), mind you, but all those annoying, catchy commercial jingles. However, it’s not your fault if the melodic hook of a 30-second commercial continues to keep you up at night. The most successful earworms in pop culture are rarely a coincidence: they are often carefully designed to maximize memorability. Today, you can even pursue a bachelor’s degree in commercial songwriting.

The mathematical study of tones dates back at least to the Pythagorean philosophers of ancient Greece in the 5th century BCE, but there is still much to learn about the numbers behind the notes. At the University of Waterloo in Ontario, Canada, computational mechanics researchers recently examined musical melodies from an algebraic perspective and discovered that there is a little-known component to many popular melodies: symmetry. Their results were presented at the 6th AMMCS International Conference on Applied Mathematics, Modeling and Computational Sciences.

“Our goal was to build a clear mathematical bridge between abstract algebra and the music listening experience,” said Olga Ibragimova, co-author of the study. “When we think of melodies as shapes that we can transform, it becomes clear that composers have intuitively used these kinds of symmetries for centuries.”

Ibragimova and her colleagues relied primarily on group theory, a subset of mathematics focused on transformations and mirrored models. They first assigned each of the 12 notes in the chromatic scale a corresponding numerical placeholder from 1 to 12, then divided various melodies into discrete note groups. This allowed them to express melodies in algebraic notation. The team then analyzed how some of the most common techniques affect the overall structure of songs. Among these were melodic concepts such as inversion (flipping a melody), transposition (moving it up or down), retrograde (inversion), and translation (movement in time).

They focused on two main types of symmetry: tonal and positional. With these concepts in mind, they were able to create formulas that illustrate how a melody can evolve while retaining a fundamental structure or deliberately changing it in predictable patterns.

“What surprised us was how clearly the mathematics separated tonal structure from positional structure,” explained Chrystopher Nehaniv, a systems design engineer and co-author of the study. “This duality helps us identify patterns that are not obvious by ear or by looking at a musical score. It also means that we can systematically construct and count all possible symmetrical melodies for a given duration.”

Ibragimova and Nehaniv believe their work can help composers conceptualize new musical possibilities, as well as pave the way for new methods of writing and research. But while this may help some artists, others continue to recognize the ineffable qualities of musical composition.

“Anytime the goal is just to write something that’s really catchy and is going to stick in people’s heads, that’s the least I think about,” songwriter Nick Lutsko said. Popular science. “It’s kind of the first thing that comes to mind and I stick with it. I don’t analyze it too much or think about it too much. It’s not intellectual at all.”

Lutsko has several full-length albums himself, as well as numerous comedy releases that have attracted a devoted international fan base. In 2022, he won a Webby Award for his musical work on an Old Spice commercial. You might even have one of his songs in your head right now. He wrote not one, but two of these absurdly memorable jingles that aired during this year’s Super Bowl.

As is the case in many corporate advertising campaigns, Lutsko was one of several songwriters who received pre-written lyrics with an invitation to set them to music.

“The first time I read these lyrics, the melody came to me as I read them,” he explains. “I know [people] I talk about algebraic equations and the math or science behind songwriting, but to me it’s almost supernatural. I think it’s because it comes from a place of familiarity.

At the same time, Lutsko does not blame anyone who draws inspiration from the mathematics of music. Remember the main route of commercial songwriting? Lutsko took it himself.

“It doesn’t bother me. I went to school for [it]”, he says. “But I’ve always been more interested in creative expression. For example, science was always my worst subject in school. That’s just not how my brain works. I just want to have fun and not think about it too much.

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