Story Telling in Math and Music
One of the most enrapturing qualities shared by math and music is their relationship to story. This is not always so obvious. Certainly in music, especially lyric based music narrative is apparent. Even when narrative is absent, many of us have had the experience of music “speaking to us,” somehow, in a language we never learned yet seem to understand. But the reason why different pitches, instruments, rhythms, frequencies, and intentions come together to create compelling moments and experiences is less obvious and many times more rich. The story is there in math too. In learning about behaviors of functions, properties of spaces, counterexamples, and techniques, beautiful stories emerge about the structure of abstract thought. Perhaps the cultural evolution of the discipline, need for precise calculation (which is not always present in traditional story telling), or intentionally unemotional language obscure, to the untrained eye, the drama, the questions looking for answers, the affairs and unrequited loves, the buried treasure that lays under the “X” on treasure map.
The stories of that come out of math and music are deeply human, but they are not always easily humanized. It is hard to think of a e^x as a complete human character with a birth and a death, hopes and dreams, fears and fancies. Especially when the only visual reference you have looks like this.
To be fair, you would have the same trouble understanding the melody and form of Benny Golson’s “Along Came Betty” as a person, even though the title includes a human name. Why did Betty show up, What changed when she did, What did she want, What was wanted of her? Unlike the math example, we could all answer this question in our own way, but for me, as a fan of this song and knowing some of the oral history of the jazz tradition which contains the answers to some of the above questions, the music itself doesn’t clarify much. Consider, if I played you the song without knowledge of the title, would you come up with the same answers? Would you even ask the same questions?
So if we aren’t telling stories about people, where do things like plot emerge, if music does something “unexpected” where did the original expectation come from, how can a mathematically mature person tell you how the “story goes” or re-derive complicated theorems from first principles? The answer lies in the notion that good stories in general are not about people, so much as they are about relationships. In good fiction, the plot, tension, resolution, despair, catharsis etc. all come from interesting characters interacting with one another. In human stories can assume certain things like the influence of family, a fear of death and pain, love as transformative and inspiring, revenge as motivation, shame or doubt as sources of dysfunction. We can assume that in the story a protagonist will either to utilize the above toward salvation or redemption or succumb and experience a tragic defeat and many more. These assumptions are common and easy to establish because we all know them intimately from our own lives. Drama unfolds as characters interact with the world of the narrative in ways that are consistent with how the writer developed the characters’ relationship to these basic human experiences.
The structure of relationships is also key to understanding, or at least appreciating, both mathematics and music at deeper level. The difference is that the characters and underlying assumptions are no longer human, but instead live inside of the internal structures of the discipline. We may have an intuitive idea about what a conversation between a dysfunctional couple may sound like, but have no idea what sin(x) and cos(x) can do together, or how groupings of 5 16th notes in 6/8 time might make us feel. But with some knowledge, the potential for these relationships becomes wondrous and deep in the same way that a playwright may find struggling lovers to be. With some study, vast, infinite, multidimensional spaces of functions can be made to look relatively quaint, almost pedestrian, and the ideas used to wrangle infinity from its lofty perch of inconceivability down to simple toy that we can manipulate achieve a real sense of beauty. The epic harmonies of McCoy Tyner become more breathtaking when we understand what ways our system of 12 notes achieved new expressive in the context of his chord voicings, or, in the language of musicians, “what opens up”. If you understand some relationships between sets (or categories), numbers, and maps, or between notes, rhythms, and harmony, the movements of these seemingly abstract objects, their desires and inclinations, fears and limitations, become tangible. And from these characteristics, their stories in all their beauty and drama and humor, shine through clear as day.
I will make the further claim that most of us, culturally in America already have some basic intuition about mathematical and musical relationships. If I told you that 2+2=5, without having to do much thinking, most of you would probably feel some internal dissonance upon reading the faulty equation based on your understanding of how addition is supposed to work and immediately reply that 2+2=4, and also . Similarly, we can tell when a musician, singers especially, hit wrong notes because we have a basic understanding of what certain tonal relationships, (and a nice human voice) should be. With more sophistication you get a wider range of emotional investment and begin to understand more narrative.
So at this point you may accept my premise that math and music tell special stories and still ask “why should I care”? You might fairly claim that these stories are intangible, abstract, removed from the needs of everyday life, and so difficult to decipher that the return vs. time investment is low if you are not a practitioner of either field. If your hang up lives around here, I have three suggestions.
First, I claim that the joy of learning these stories is in the abstraction and distance from everyday life. The process of learning math or music forces a student to build and clarify internal models of thinking; to create abstract “characters” and develop abstract “relationships” between them. Doing this effectively requires imagination to substitute for tangibility. It also requires solitary periods of focus so that you can be with yourself and understand how you put things together. Spending time with a pencil and paper and maybe a friend or 2 just imagining is grounding and humbling in a way that most searches inside the self are. Arriving at a solution or new understanding constitutes a joy indescribable to anyone who hasn’t felt it themselves while failure costs only time.
Second, I claim that these internal models are transferable and using them can open up a new relationship with the world around you. These internal models of logic or harmony or clarity or perfection become reference points. When you see a piece of art, when you connect with another person, when you talk to someone who claims to be an expert. You can check your own abstract stories of what is “harmonious” or what “makes sense” and see if there is alignment. We do this anyway, but because of the rigor and specificity required in math and music, these seem to be great ways to hone this skill set. Now to be clear, I don’t think math and music are the holy grails of self-improvement, nor do I think that musicians always have good taste or that mathematicians are always correct, but spending some time learning about math and music would make a healthy addition to your personal growth regimen.
Third, the stories Math and Music comprise big parts of our shared cultural heritage. Modern mathematics has elements developed both in conversation and separately from Africa, Persia, India, Arabia, Europe, and Asia and there is evidence of mathematical investigation in Central and South America as well. Of course, every culture we know of has music. Mathematical and musical stories, technical stories, abstract stories, stories that, can demand years of work on the parts of the storyteller and sometimes the listener, are deeply human stories, because so much of humanity has invested so much time and effort to tell them; to keep them alive. I would invite you to find out why, and in doing so become part of a world heritage that spans history, gender, geography, tribe and social status. Learn so that you see into the minds of some of our greatest geniuses and discover what they were really trying to say. And then find out what unique and special joy the investigation of music and mathematics has for your life.