More than fractions: Where music and mathematics meet

Early last summer, the stars aligned, and I found myself able to attend the CMS Summer Meeting as a presenter in the education arm of the conference. It was thankfully held at the University of Saskatchewan, where I am studying to earn a PhD in Education. I could afford to attend! 

It is hard to describe how nervous I was. I am not a mathematician. I am not a math educator. I am a musician. And for the better part of two decades, I have been a high school music director. The thought of having something to present that would be of interest to those immersed in the world of mathematics seemed ludicrous. However, with a supportive supervisor and an idea that had been percolating in the depths of my mind for years, my presentation began to take shape. 

Music and math are natural allies. Many of the structures present throughout music are also present in math. The parallels seem endless. And, for the record, I am not speaking of the old adage “music teaches fractions”. How cliché. Music is embodied mathematics. The beauty of mathematics can be represented through music. Interestingly, music can represent many of the mathematical concepts taught throughout the Canadian curriculum – from kindergarten to university. 

After much consideration and thoughtful conversations with a friend who is both a mathematician and a musician, we decided the most effective way to convey my thoughts on the confluence between mathematics and music would be a case study. For my presentation, I attempted (in twenty minutes), to identify the mathematical concepts in a single piece of piano music: Philip Glass’s Etude No. 6 (https://www.youtube.com/watch?v=sZffgf4GoMQ).

Aptly named, this piano etude features factors of 6 in numerous variations throughout the piece – starting with the most basic; it is in  time (there are 6 quarter notes in each measure) and is to be played at ♩=132 = 6 • 22 beats per minute. Impressively, in the recording I used as my exemplar, Víkingur Ólafsson performs this piece at a whopping 216 beats per minute, that is 6 • 36. As my presentation focus was on the education angle, I spent much time scouring math curricula and university math syllabi to ground the concepts I presented in the piece with curricular objectives from Saskatchewan.

One of the most notable (pun intended) structural features of this etude is Glass’s use of additive sequences, symmetry, and transformations; concepts that clearly parallel mathematical ideas taught in Saskatchewan’s Pre-Calculus, Calculus, and University-level mathematics curricula. These are especially prominent at the beginning of the piece, where short rhythmic or melodic patterns are gradually expanded by adding small units over time, akin to mathematical sequences or function transformations. In the latter part of the etude, Glass employs a process of minimalism and reduction: rather than introducing new material, he systematically removes or simplifies existing musical layers. This musical “paring down” reflects mathematical ideas of ratio and proportion, where relationships between quantities become more exposed and structurally significant as the texture thins. For instance, repeated patterns may be halved, rhythms slowed in even proportion, or harmonic content distilled to its simplest intervals. While minimalism is often described as a stylistic genre, in this context it serves as a formal mechanism to bring structural closure, mirroring the mathematical elegance found in simplified expressions or concise proofs.

Polyrhythms, when two or more different rhythmic patterns are played at the same time, play a central role throughout the etude. For example, listeners can hear a three-against-two-against-one pattern, where one layer is triplets (three notes per beat), another is duplets (two notes per beat), and a third layer keeps steady quarter notes (one note per beat). This layering becomes even more intricate in section seven of the piece, where a five-against-two-against-one pattern emerges, an exceptionally complex coordination of rhythmic streams performed flawlessly by Ólafsson. These overlapping rhythms create cycles that only align at certain points, depending on their least common multiple (LCM), a concept familiar from middle school math. For instance, a 3-against-2 pattern aligns every 6 beats, while a 5-against-2 pattern aligns every 10. This property offers a direct and engaging way for students to explore number theory concepts in real time. Beyond LCM, polyrhythms also provide an entry point into permutations and combinations, as each rhythmic layer can shift in position, order, or emphasis. Analyzing how these rhythms cycle, intersect, and transform over time gives students a way to experience mathematical structure not only as abstract logic but as something audible, physical, and expressive.

Glass’s etude is grounded in iteration, featuring repetitive motifs (short musical ideas) that gradually evolve through subtle changes in rhythm and harmony. In music, rhythm refers to the timing of sounds and silences, including how long or short notes are and how they are spaced over time. Harmony, on the other hand, involves the combination of notes played simultaneously, creating chords and a sense of tonal color or tension. In this etude, Glass alters both the rhythmic placement and the harmonic context of motifs, creating a dynamic sense of motion within a seemingly repetitive structure. These evolving patterns closely resemble mathematical sequences, where each element is derived from the previous one through a specific rule or transformation. By treating musical motifs as elements in a sequence, or even as functions or sets, students can explore the structure and development of the piece using mathematical concepts such as symmetry (when a motif reflects or rotates), periodicity (when patterns recur at regular intervals), and transformation (how patterns change incrementally over time).

The cyclical nature of these motifs lends itself to analysis through modular arithmetic. Just as modular systems wrap around after reaching a certain value, Glass’s patterns repeat at regular intervals and could be examined using modulo-based approaches (e.g., analyzing a motif that recurs every 8 beats with mod 8). This mathematical lens could help with understanding the compositional structure of the piece and provide a creative entry point for connecting music with mathematical reasoning in the classroom.

Glass’s use of dynamics (the varying levels of loudness in music) is another feature that can be explored through mathematical concepts. In Western musical notation, dynamics are indicated with letters: p (piano) means soft, mp (mezzo piano) means moderately soft, mf (mezzo forte) means moderately loud, and f (forte) means loud. These dynamic markings shift gradually throughout Etude No. 6, creating patterns of exponential growth and decay. For example, if we assign numerical values to these dynamics: p = 1, mp = 2, mf = 4, and f = 8, we can begin to see a geometric progression in the way intensity rises and falls. A dynamic sequence like p – mp – mf – mp would correspond to the numerical pattern 1 → 2 → 4 → 2, illustrating a doubling and halving effect that mirrors exponential behavior. This kind of structured mapping enables students to visualize and analyze dynamic changes as geometric sequences or exponential functions, offering a tangible and expressive way to connect mathematics with musical interpretation.

Hoping to pique the interest of fellow academics, I arrived at the conference nervously prepared to present on a music/math collaboration. Imagine my surprise when the once-empty room suddenly filled with math gurus just before my talk! Shocked (and even more nervous), I powered through my twenty minutes in one piece (again, pun intended), followed by a lively ten-minute Q&A. Then, as quickly as they came, the crowd vanished, leaving me flabbergasted.

The connections between mathematics and music are not new; one only needs to look back to Plato’s Quadrivium to appreciate their shared foundations. Still, the enthusiastic response to my presentation suggested that there remains fertile ground for renewed dialogue, particularly within contemporary pedagogical contexts. Rather than a conclusion, the experience felt like the starting point of a deeper inquiry. I look forward to continuing this work and engaging with others who are interested in exploring the educational possibilities at the intersection of music and mathematics.