While Super Bowl LX is over, the big game still echoes in the minds of many viewers. Not the Seattle Seahawks’ offensive coordination (or the New England Patriots’ lack thereof), mind you, but all those annoyingly catchy commercial jingles. It’s not your fault if a 30-second advertisement spot’s melodic hook continues to keep you up at night, however. Pop culture’s most successful earworms are rarely a fluke—they’re often carefully crafted to maximize memorability. Today, you can even pursue a Bachelor’s degree in commercial songwriting.
The mathematical study of tones dates at least as far back as the 5th century BCE Pythagorean philosophers of ancient Greece, but there is still a lot 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 standpoint and found that there is an unsung component to many popular tunes: symmetry. Their findings were presented at the 6th AMMCS-International Conference on Applied Mathematics, Modeling, and Computational Science.
“Our goal was to build a clear mathematical bridge between abstract algebra and the experience of listening to music,” said study co-author Olga Ibragimova. “When we think of melodies as shapes we can transform, it becomes clear that composers have been using these kinds of symmetries intuitively for centuries.”
Ibragimova and her colleagues primarily relied on group theory, a subset of mathematics focused on transformations and mirrored patterns. They first assigned each of the chromatic scale’s 12 notes a corresponding 1-12 numerical placeholder, then broke down various melodies into discrete note groups. This allowed them to express tunes as algebraic notation. The team then analyzed how some of the most common techniques affect overall song structures. Among these were melodic concepts like inversion (flipping a tune), transposition (shifting it up or down scales), retrograding (reversal), and translation (movement over time).
They focused on two primary symmetry types—tonal and positional. With those concepts in mind, they were able to create formulas that illustrate how a melody can evolve while either retaining a foundational structure or purposefully altering it in predictable patterns.
“What surprised us is how cleanly the mathematics separates tonal structure from positional structure,” explained systems design engineer and study co-author Chrystopher Nehaniv. “This duality helps us identify patterns that are not obvious by ear or by looking at a musical score. It also means we can systematically construct and count all possible symmetric melodies for a given length.”
Ibragimova and Nehaniv believe their work may help composers with conceptualizing new music possibilities, as well as pave the way for new methods of songwriting and research. But while this may help some artists, others continue to credit the ineffable qualities of music composition.
“Whenever the goal is just to write something that’s very catchy and is going to get stuck in people’s heads, that’s, like, the least I think about,” songwriter Nick Lutsko tells Popular Science. “It’s kind of the first thing that comes to my head and I just go with it. I don’t overanalyze or overthink it. It’s not intellectual whatsoever.”
Lutsko has multiple full-length albums of his own, as well numerous comedy releases that have garnered an dedicated, international fanbase. In 2022, he won a Webby Award for his musical work on an Old Spice ad. You may even have one of his songs stuck in your head right now. He penned not one, but two of those absurdly memorable jingles that ran during this year’s Super Bowl.
As is the case in many corporate ad campaigns, Lutsko was one of multiple songwriters to receive prewritten lyrics with a prompt to put them to music.
“The first time I read those lyrics, the melody came to me as I was reading them,” he explains. “I know [people] 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 doesn’t fault anyone who turns to the mathematics of music for their inspiration. Remember the commercial songwriting major route? Lutsko took it himself.
“It doesn’t irk me. I went to school for [it],” he says. “But I’ve always been more interested in creative expression. Like, science was always my worst subject in school. It’s just not how my brain works. I just want to have fun and not think about it too much.”
The post Can’t stop humming that tune? Thank math. appeared first on Popular Science.
