https:\/\/www.youtube.com\/watch?v=sZffgf4GoMQ<\/a>).<\/p>\nAptly named, this piano etude features factors of 6 in numerous variations throughout the piece \u2013 starting with the most basic; it is in \u00a0time (there are 6 quarter notes in each measure) and is to be played at \u2669=132 = 6 \u2022 22 beats per minute. Impressively, in the recording I used as my exemplar, V\u00edkingur \u00d3lafsson performs this piece at a whopping 216 beats per minute, that is 6 \u2022 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.<\/p>\n
One of the most notable (pun intended) structural features of this etude is Glass\u2019s use of additive sequences, symmetry, and transformations; concepts that clearly parallel mathematical ideas taught in Saskatchewan\u2019s 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 \u201cparing down\u201d 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.<\/p>\n
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 \u00d3lafsson. 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.<\/p>\n
Glass\u2019s 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).<\/p>\n
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\u2019s 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.<\/p>\n
Glass\u2019s 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<\/em> (piano) means soft, mp<\/em> (mezzo piano) means moderately soft, mf<\/em> (mezzo forte) means moderately loud, and f<\/em> (forte) means loud. These dynamic markings shift gradually throughout Etude No. 6<\/em>, 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 \u2013 mp \u2013 mf \u2013 mp<\/em> would correspond to the numerical pattern 1 \u2192 2 \u2192 4 \u2192 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.<\/p>\nHoping 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.<\/p>\n
The connections between mathematics and music are not new; one only needs to look back to Plato\u2019s 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.<\/p>\n","protected":false},"author":11,"template":"","section":[56],"keyword":[],"class_list":["post-19553","article","type-article","status-publish","hentry","section-education-notes"],"toolset-meta":{"author-4-info":{"author-4-surname":{"type":"textfield","raw":""},"author-4-given-names":{"type":"textfield","raw":""},"author-4-honorific":{"type":"textfield","raw":""},"author-4-institution":{"type":"textfield","raw":""},"author-4-email":{"type":"email","raw":""},"author-4-cms-role":{"type":"textfield","raw":""}},"author-3-info":{"author-3-surname":{"type":"textfield","raw":""},"author-3-given-names":{"type":"textfield","raw":""},"author-3-honorific":{"type":"textfield","raw":""},"author-3-institution":{"type":"textfield","raw":""},"author-3-email":{"type":"email","raw":""},"author-3-cms-role":{"type":"textfield","raw":""}},"author-2-info":{"author-2-surname":{"type":"textfield","raw":""},"author-2-given-names":{"type":"textfield","raw":""},"author-2-honorific":{"type":"textfield","raw":""},"author-2-institution":{"type":"textfield","raw":""},"author-2-email":{"type":"email","raw":""},"author-2-cms-role":{"type":"textfield","raw":""}},"author-info":{"author-surname":{"type":"textfield","raw":"Elliott"},"author-given-names":{"type":"textfield","raw":"Sandra"},"author-honorific":{"type":"textfield","raw":""},"author-email":{"type":"email","raw":""},"author-institution":{"type":"textfield","raw":"University of Saskatchewan"},"author-cms-role":{"type":"textfield","raw":""}},"unknown":{"downloadable-pdf":{"type":"file","raw":"https:\/\/notes.math.ca\/wp-content\/uploads\/2025\/05\/7-More-than-fractions_-Where-music-and-mathematics-meet-\u2013-CMS-Notes.pdf","attachment_id":19774},"article-toc-weight":{"type":"numeric","raw":"4"},"author-surname":{"type":"textfield","raw":"Elliott"},"author-given-names":{"type":"textfield","raw":"Sandra"}}},"_links":{"self":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/19553","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/users\/11"}],"version-history":[{"count":12,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/19553\/revisions"}],"predecessor-version":[{"id":19757,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/article\/19553\/revisions\/19757"}],"wp:attachment":[{"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/media?parent=19553"}],"wp:term":[{"taxonomy":"section","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/section?post=19553"},{"taxonomy":"keyword","embeddable":true,"href":"https:\/\/notes.math.ca\/en\/wp-json\/wp\/v2\/keyword?post=19553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}