In this paper, the authors generally summarize an interdisciplinary workshop that integrated interactive activities centered on music and mathematics. They introduce an alternative way of conceptualizing chromatic scales and chords. By representing the chromatic scale—which consists of 12 notes—and chords, which are made up of three notes, the authors construct inscribed triangles that are then audibilized. The workshop is exploratory in nature, inviting participants to make connections between types of triangles (scalene, isosceles, and equilateral) and types of chords (major, minor, augmented, and diminished). The authors argue that this approach opens up playful and innovative possibilities for learning, with the potential to reshape how basic music theory and harmonic composition can be taught alongside, or through, geometry.
My first point of engagement was Figure 2 (see below), where a major chord is visually represented on the chromatic circle as a scalene triangle. I find the connection between mathematics, sound, and music incredibly fascinating. As with the TED Talk we watched—in which particular fractions or ratios were translated into sound—it is rare for mathematics to be experienced aurally.
Building on this idea, we know that sound is transmitted through wavelengths, which presents a promising opportunity to connect mathematical concepts—such as ratios and sinusoidal functions—to sound in more embodied and intuitive ways. For example, one could imagine audibilizing a sinusoidal function by mapping a moving point along the curve to changes in pitch or amplitude, allowing the positive and negative motion of the function to be heard as well as seen. While I have not yet found a concrete example of this, this Youtube video Every Sound Is Sine (https://www.youtube.com/watch?v=UrBZsUBibtk) offers a fun illustration of how complex sounds can emerge from simple sinusoidal waves.
In what ways does audibilizing geometric relationships challenge traditionally visual approaches to mathematics education, and how might this shift support more embodied or intuitive forms of learning?
My second point of engagement was more general and was prompted by the lack of detail in Section 4. It can be difficult to be fully convinced by workshops that are conducted only once, with minimal or no follow-up with participants. In Section 4, the authors write that “the participants choose a song from a list proposed to try to understand what kind of geometry lies beyond by analyzing all the triangles/chords present in that song’s harmonic structure” (p. 650). This description raises several questions: What songs were proposed? Were they modern, romantic, or classical pieces? Who were the composers or artists? Why were certain songs selected over others?
While I recognize that Bridges imposes a short page limit, which may have constrained the level of detail provided, the absence of supporting examples—such as participant quotes, musical excerpts, or visual representations—makes it difficult to fully evaluate or be persuaded by the claims being made about the workshop’s impact.
The general idea and approach taken by the authors is inspiring. Although I was unable to find later papers in which the authors continue or expand upon this workshop, the paper nonetheless presents an enticing way of connecting mathematics and music. In particular, I would be very interested in seeing dynamic visual or virtual representations that show the movement and transformation of triangles over time within a musical piece.
At the same time, because the paper does not describe the participants’ ages or their musical and mathematical backgrounds, questions remain about classroom implementation. How might teachers adapt this geometric–musical approach for students with differing levels of musical familiarity and mathematical understanding? What kinds of scaffolding, representations, or instructional supports would be necessary to make this activity accessible and meaningful in a classroom setting?
Ultimately, do you find the authors’ claims convincing given the limited detail they provided?
Hi Anna,
ReplyDeleteI find the authors' ideas interesting, especially the connection between geometry and music. Personally, I've always felt that using multiple senses can deepen understanding. Back in school, I struggled with visualizing geometric shapes, so I think that if I'd had the chance to listen to sounds related to those shapes, I would have grasped the concepts more naturally. The video Susan shared about math and music sharpened my thoughts on how powerful such an approach could have been for me.
Additionally, bringing sound into learning could help create a more inclusive atmosphere. Many of my students find traditional math intimidating, but they thrive in artistic settings where they can express themselves beyond just visual methods. This multisensory approach could reach a wider range of learners, especially those who connect best through sound and movement.
That said, I’d love to see more research on how effective these methods are over time. Do students who engage with audibilizing geometry retain the concepts better? Do they feel more confident in their math skills? Gathering more long-term feedback would really strengthen the authors' claims and show the real impact of these methods on students' views about math.
If I were to create a workshop based on these ideas, I would focus on making it interactive by allowing students to both listen and create their own sounds or music that represent geometric concepts. This would not only empower them but also bridge the gap between math and artistic creativity.
Ultimately, while the idea of audibilizing geometry is fascinating, I think including more concrete examples and data would help make a stronger case for it. By sharing personal experiences and diverse feedback, we can enhance the conversation and inspire more educators to try out these innovative approaches in their classrooms.
Hi Anna,
ReplyDeleteI found your response of "Make Music Visible, Play Mathematics" very insightful. Your focus on moving beyond the purely visual feels like a direct response to the "stops" we’ve been exploring regarding embodied learning.
I particularly connected with your first point of engagement regarding Figure 2 and the representation of major chords as scalene triangles. In the classroom, we often treat the Cartesian plane or the unit circle as a silent, static map. Your suggestion to map sinusoidal functions to pitch reminds me of how my secondary students struggle to "feel" the period or amplitude of a function. By audibilizing these ratios, we might bypass the rational-analytical struggle and tap into what Fels calls an "imminent phenomenological experience."
I also shared your frustration in your second point regarding the lack of specificity in the workshop data. As educators, we know that the "songs proposed" and the "background of the participants" are not just footnotes -they are the variables that determine if a lesson actually "lands." Without knowing if the students were analyzing Bach or Beyoncé, it’s hard to judge if they were truly engaging with the geometry or just following a pre-set algorithm. It reminds me of how academic papers sometimes flatten the messy, beautiful reality of a classroom into a tidy conclusion.
Excellent points about the centrality of the details in making a lesson work or not! And as a musician, I also have questions about whether it is only the geometry (major, minor, diminished, etc) nature of the chord that is important. What about the relationships among the chords, for example? Lots of good ideas circulating here!
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