Listening to Mathematics

I wanted to do something today with math and music, not their formal relationship, but their mingling in our experience.  The seed for the thought came from a 2003 MIT paper I found: Music as Embodied Mathematics. The idea explored in this particular paper was inspired by a composition project that involved software designed for college music students.

Indeed, it was in analyzing musically novice students’ accounts of their work-in-progress, particularly as they experimented with rhythmic possibilities, that we noticed mathematical relationships playing a role in their perception and composition of musical coherence.

It may seem unremarkable that the principal mathematics college students spontaneously put to work involved ratio, proportion, fractions and common multiples.  However, it turns out that these intuitively generated and perceived music/mathematical relationships are some of the important mathematical concepts that are found to be most problematic for middle school children.

And so they considered that music might provide a way for children to manage these troublesome concepts, and also introduce them to a less tool-oriented view of mathematics.

Engaging both domains together might also enhance the children’s appreciation and understanding of aesthetic relations shared by mathematics and music.

The experiment was small and informal.  Researchers met with a group of 6th grade students once or twice a week over a period of three months.  But their observations support the idea that the affinity between math and music is multilayered, involving counting and periodicity (repetition), temporal ruler beats (duration) and their proportional relationships, graphic representations (abstraction) and patterned structure.

More subtle and perhaps more interesting intersections between music and mathematics were discovered as the children composed melodies – particularly as the graphic representations helped them to come to consider patterns such as symmetry, balance, grouping structures, orderly transformations, and structural functions.  Structural functions include, for instance, pitch/time relations that function to “create boundaries,” or entities (e.g. phrase) some of which sound  “incomplete,” thus functioning to resolve or settle on onward motion.

As an aside, I’ve always been intrigued by the sensation of movement in music.

There wasn’t magic in the results of this experiment.  Children didn’t just jump into advanced math ideas by thinking about music.  But they were thinking about the music and the mathematics, and making sense of them.  The authors considered that the idea of a mathematical transformation could be drawn from music, the map of one instance of a given melody, for example, onto another, or one instance of a given drum piece mapped onto another.  And they raised the question: “How far can the mathematics of invariance be drawn out of musical experience?”  They admit that they have not realized this thought experiment but did report confidently that, in composing their tunes, these 6th graders made use of structural relations that included rule-driven transformations such as sequence, fragmentation and extension by repetition.

I think one of their more important observations is this one:

Perhaps the most general aspect of the affinity between mathematics and music might be the perception and articulate study of patterns.  Pursuing this agenda within music might encourage children to become intrigued with looking for patterns in other domains as well.  And it might lend a “sense” to mathematics as a tool for understanding more about what we intuitively have some grasp of and care about.  (italics my own)

Which brings me to a wonderful little piece by mathematician Marcus du Sautoy that appeared in The Guardian this past June. Du Sautoy quickly gets to the heart of the matter.

Just as notes and rhythms are not all there is to music, so arithmetic and counting are not all there is to mathematics. Mathematics is about structure and pattern. As we’ve explored the universe of numbers, we’ve discovered strange connections and stories about numbers that excite and surprise us. Take the discovery by Fermat, the 17th-century French mathematician, that a prime number that has a remainder of 1 after division by 4 (like 41) can always be written as the sum of two square numbers (41=16+25). It was a realisation that linked the seemingly separate worlds of primes and squares.

Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don’t realise about mathematics is that it involves a lot of choice: not about what is true or false (I can’t make the Riemann hypothesis false if it’s true), but from deciding what piece of mathematics is worth “listening to”.

There is, in everyone’s introduction to math and music, much talk about numbers, counting, and arithmetic.  And it is significant that the two disciplines share this.  Yet I think one of the most important things that mathematics accomplishes is its linking of “seemingly separate worlds,” perhaps even more that we have yet imagined.  Every mathematician and musician knows the vast expanse of surprising structure that lies just beyond the bridge of counting. And this is what Marcus du Sautoy speaks to when he says:

…for me, what really binds our two worlds is that composers and mathematicians are often drawn to the same structures for their compositions. Bach’s Goldberg Variations depend on games of symmetry to create the progression from theme to variation. Messiaen is drawn to prime numbers to create a sense of unease and timelessness in his famous Quartet for the End of Time. Schoenberg’s 12-tone system, which influenced so many of the major composers of the 20th century, including Webern, Berg and Stravinsky, is underpinned by mathematical structure. The organic sense of growth found in the Fibonacci sequence of numbers 1,2,3,5,8,13 . . . has been an appealing framework for many composers, from Bartók to Debussy.

Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry.

Both of the pieces I refer to quote Leibniz when he said:

Music is the arithmetic of the soul, which counts without being aware of it.

 

2 comments to Listening to Mathematics

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  • david pinto

    nice 🙂

    we don’t need to dive into the complexities of classical music
    which is like diving into complex math
    we can merely appreciate how our bodies are caught by a simple rhythm
    which is very closely related to the simple process of counting

    i did try to get a bunch of adolescents to move to music
    to translate an audio pattern to a visual one
    and then on to graphs…
    but it was an uphill struggle…

    a rich seam to explore is how kids learn patterns through music
    and i am fairly certain many kids learn to count through noticing patterns in time
    rather than an abstraction from counting things…

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