Capozucca, A., & Fermani, M. (2019). Make music visible, play mathematics. In Proceedings of the Bridges 2019 Conference (pp. 647–650). Bridges Organization.
Introduction
The article Make
Music Visible, Play Mathematics by Andrea Capozucca and Marco Fermani
presents an interdisciplinary, hands-on workshop that connects mathematics and
music through geometry, with the goal of making mathematics audible and music
visible through playful, multisensory learning experiences. The authors argue
that music and mathematics share deep structural relationships that go beyond
counting and ratios, and that these connections become especially clear when
musical harmony is explored visually and spatially. Using the chromatic scale arranged
as a circle, where the twelve notes are evenly spaced and musical intervals
correspond to angles, the workshop provides a concrete geometric framework for
understanding music theory.
The workshop
follows a five-part discovery-based structure that emphasizes active
participation. Participants first explore which musical intervals sound
pleasant through embodied listening activities, then construct geometric
segments and triangles representing these intervals using simple materials.
Through this process, they discover that only four types of triangles can be
perfectly inscribed in the chromatic circle, corresponding to the four
fundamental chord types: major, minor, diminished, and augmented. By rotating
and reflecting these triangles within the circle, participants experience
musical transposition and transformation, learning that geometric rotation
preserves a chord’s identity while symmetry changes it.
In the later
stages, participants apply their geometric understanding to analyze the
harmonic structure of familiar songs and collaboratively compose original music
using geometric “recipes.” The authors conclude that this inquiry-based,
playful approach increases engagement, confidence, and creativity in both
mathematics and music learning. They argue that such interdisciplinary
workshops support open-ended problem solving, authentic collaboration, and
positive attitudes toward learning, and that geometry can function as a shared
language that allows mathematics and music to mutually enrich one another
across educational contexts.
Reflection
This article is
illustrative but not comprehensive; it is a short piece that offers a glimpse
into how mathematics and music can be meaningfully connected through geometry
and hands-on learning. While it does not aim to fully map the theoretical
foundations of either discipline, it succeeds in opening a creative window into
interdisciplinary thinking and shows how mathematical ideas can be experienced
in ways that are intuitive, sensory, and playful.
One point that
made me stop while reading was the authors’ statement that “mathematics is
about structure and pattern.” I really appreciate this way of describing
mathematics because it captures what feels most fundamental about the subject.
I think this is exactly why mathematics can serve as a root for so many
different areas and be investigated across disciplines. In basic terms, much of
the work we do in mathematics, science, and even the arts involves identifying
patterns, finding ways to describe those patterns clearly and systematically,
and then creating or building something new based on them. Seen from this
perspective, mathematics is not just a school subject but a way of organizing
and making sense of the world.
The connection
the article builds between music and mathematics is also especially
interesting, particularly the idea that “music is the sensation of counting
without being aware you were counting.” This insight resonated with me,
although I also think music goes beyond counting alone. Sound itself can be
understood numerically through properties such as frequency measured in hertz,
even if I have not formally studied music from this perspective. Still, the
idea that musical elements like chords can be analyzed mathematically—just as
the authors do through geometric representations—suggests a rich and enjoyable
way to explore music. This approach makes me curious about how much musical
structure, including chord types and harmony, might be better understood by
uncovering the patterns and numbers behind what we hear.
Overall, it has brought
me insight into a way to analyze art. I think it invites a broader view of
mathematics as a deeply human and artistic activity rather than a purely
technical or procedural one. By using geometry to explore musical harmony, the
authors show how mathematics can function as a creative language for interpreting
sound, beauty, and structure. Mathematics here is not presented as a set of
rules to be memorized, but as a way of seeing, shaping, and creating
meaning—much like art and music themselves. I guess that’s why many artworks can
be demonstrated mathematically.
Question
Do you have any
experience investigating a form of art—such as music, visual art, or
dance—through mathematics, or noticing mathematical patterns while creating or
experiencing art?
I really enjoyed reading your post. You explained the article in a way that was super easy to picture, especially the idea of turning music into shapes on a circle. That image of chords as triangles you can rotate and flip is such a cool way to see harmony instead of just hearing it.
ReplyDeleteI also paused at the idea that math is about “structure and pattern.” The way you connected that to art, science, and everyday thinking makes a lot of sense. When you put it like that, math feels less like a list of rules and more like a way our brains look for order in the world.
Your point about music being more than counting also stood out to me. While rhythm and frequency can be measured numerically, the emotional and sensory aspects of music are just as important. What I like about the geometric approach is that it does not reduce music to numbers it offers another doorway into understanding it. It is like math becomes a lens, not a replacement for the artistic experience. Your reflection on math as a creative, human activity really resonated with me, too. When students see math connected to sound, movement, or visual beauty, it can change how they feel about it. It becomes something you explore and create with, not just something you get right or wrong.
To answer your question, even though I haven’t tried many mathematical art activities, I have seen math come alive through a simple symmetry painting I did. To show symmetry, I painted one side of a folded piece of paper and pressed the two sides together. When we opened it, the paint created mirror images on both sides. Without using formal math terms, we were exploring reflection symmetry—both halves matched across an imaginary line through the center.
It felt like a fun art activity, but mathematically, it showed how shapes can be reflected and still retain their form. The final images even resembled butterflies or natural shapes, which helped connect the idea of symmetry to the natural world. This experience showed me that math doesn’t always have to be numbers or equations, it can be something we see, touch, and create.
I really enjoyed reading your reflection. You explain the article clearly and you’re honest about what it does and doesn’t do, which makes your critique feel fair.
ReplyDeleteI especially liked how you picked out the line “mathematics is about structure and pattern” and then expanded it in your own words. That part reads strongly because you connect it to real life, not just school math.
Your response to the “music is counting without noticing” idea is also thoughtful. You agree, but you also push it a bit further by bringing in frequency and the idea that music isn’t only counting. That shows you’re thinking beyond the quote.
But, personally, I connected with the counting part. When I was a kid, I learned piano and an Indian classical dance form, and both were taught through counting. Dance steps were literally built on counts, and in music I had to learn to recognize patterns first, almost like “decoding” them, before I could start creating my own patterns.
Overall, your reflection captures the idea of mathematics as a creative language, not just procedures, and it helped me think about how art can be analyzed through structure, relationships, and form.
Thanks for this wonderful dialogue about the wars that math lets us notice structure and pattern with all our senses, throughout our experiences of the world! Counting can’t quite capture everything of course, but there is so much depth to thinking about our human ways of seeking order.
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