Benoit Mandelbrot died this month. He was the guy who came up with fractal theory, which led to all those gorgeous computer graphics like this one:
Last week, my friend and contradance bandmate Tina Fields wrote an essay about Mandelbrot’s ideas on her blog, Indigenize! I found it quite thought-provoking, and it surprised me how much I learned from her post, since I’m the one with the math degree. My next surprise was how Tina’s thoughts on this mathematician inspired me to think about listening to music.
This essay is in response to ideas she raises in her essay, so go read hers first and then come back here!
First I’d like to amplify her comment about coastlines by quoting this passage from Mandelbrot’s obituary in the New York Times, about how coastlines played a role in the genesis of his theory:
Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.
“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”
In the 1950s, Dr. Mandelbrot proposed a simple but radical way to quantify the crookedness of such an object by assigning it a “fractal dimension,” an insight that has proved useful well beyond the field of cartography.
To me, that’s the real genius of his discovery—viewing scale as a dimension. If we measure the coastline or the surface of the broccoli from a mile away we get a much different answer than if we measure it from close up and far different still if we measure under a microscope.
So what is scale, really, but a matter of perspective?
Let’s consider the metaphorical potential: if perspective is a dimension, how does it change the way we view truth about our world? You have some truth, I have some truth, and the differences are not necessarily contradictions but spectral variations along the perspective dimension.
Tina’s big gift to me in her essay isn’t so much her point about Mandelbrot’s focus on verbs rather than nouns, although I enjoy that, too, but her encouraging us to think about new things fractally. The first thing that comes to my mind is Beethoven. (Perhaps I should explain that besides working in statistical software and facilitative leadership, I’m also a professional horn player and hold degrees in music performance and music history.)
Beethoven leads my pantheon, and here’s a bit on why: his compositional technique is extraordinary, and the more you know about musical composition and performance, the more you hear in his work. In addition to doing all the usual classical things—the usual structural designs (four-movement symphonic architecture with movements in sonata, menuet or scherzo, sonata-rondo, etc. forms, linked in a progression of related tonalities, yada yada Haydn, blabbety-blabbety Mozart, blah blah Bach), German-Italianate phrases, symphonic devices of his environment and era—he throws in a few more tricks all his own, chief among them his idea of motivic development.
His every melodic gesture is built up from the smallest motives, e.g. his Fifth Symphony‘s four-note “ba-ba-ba-BOM!” opening. That simple four-note figure is sequenced, layered, mutated, and warped all throughout the first movement, each phrase a new assemblage of basic building blocks, each harmonic gesture arising out of layers and layers of sequences of this tiny musical block and several others.
You can easily find recordings on YouTube if you’d like to remind yourself how it goes, but I’d recommend buying yourself a great recording if you don’t already have one. There are many excellent options; one I’d particularly recommend is Bernstein’s with the New York Philharmonic.
All the composers of Beethoven’s time (and throughout most of history, with differing vocabularies, of course) have adhered to various conventions from the largest possible scale (the arc of their developmental style through their lifetimes) down through the structure of each opus, each movement within, etc., down to the smallest-scale assumptions about harmonic structure, idiomatic styles of individual instruments, and so forth, but Beethoven brings it all to a whole new level, honoring all those formal rules while also constructing everything both melodically and harmonically, both vertically and horizontally in each case, out of these tiniest of musical blocks.
(We later see Wagner up Beethoven’s ante with his Romantic adaptation, the leitmotiv, where each character, event, place, and even philosophical concept is represented by its own fragment of musical DNA, all these leitmotivs swirling in a pan-theatrical operatic swamp of continuous through-composition, rejecting while also embracing formal conventions in a megalomaniacal Gesamtkunstwerk.)
Struggling valiantly now to pull back from this tangent to return to fractal theory, I might suggest that we appreciate Beethoven and indeed all music along fractal dimensions. For many, Beethoven’s Fifth Symphony is, simply, its opening four notes and the loud romp to follow. The scale of observation is large; the perspective is simple. “Fun music!”
Indeed, who wouldn’t appreciate it on such simple terms? When I was hospitalized with pneumonia as a second grader, my parents brought me the best of all possible get-well presents: a portable cassette deck, including a cassette of the first movement of my favorite symphony, which dad had recorded by sitting next to the phonograph (mono, of course) holding the mic near the cabinet speakers while the needle rode its groove. I listened to that tape over and over during my several weeks of long days alone in a hospital room. I’m not sure what I heard, exactly, but I know that by the time I was discharged, I could have sung the whole movement. (I wish he’d recorded the whole symphony for me, because I’l never know the rest of it nearly as well.)
As I’ve developed as a musician, I’ve lost touch with how I used to hear music. I often wonder what normal people hear, and I like to ask people to tell me why they like certain music or what they noticed in a concert.I know that I used to hear the pretty music, and while I can tell you to the minute when it all changed, I can’t for the life of me remember what I used to hear.
It changed the summer after eighth grade. I was at orchestra camp, sitting in a muggy auditorium on a hot summer night, and probably intoxicated by the pheromones of my new friends. We listened to a piano quartet recital. First I noticed that I was hearing a group whose intonation was so tight, they made the freshly, expertly-tuned Steinway sound out of tune. All pianos are out of tune, but it was the first time I heard it for myself. Then I realized I was hearing four virtuosi playing the crap out of their instruments as both individuals and as a collective.
Then my trumpeter friend leaned over and said, “You know, we’re never going to hear music like normal people again,” and for only a moment I wondered what he meant. I spent the rest of the concert hearing, seeing, feeling the compositional structure, the interplay of themes, the exploration of key areas, the work of the individuals and their ensemble, and on and on. The only limits to the depth of scale in my listening were my musical intelligence and attention span.
That night was my awakening as a listener. In the decades that followed, my musical intelligence has evolved tremendously, but I still find that the richness of what I hear is limited only by my abilities and attention span.
So, locating my metaphor in the area of musical perception, I might suggest that our listening has a fractal dimension. Anyone can hear the sounds. But our perspective—the granularity of our musical knowledge and the intensity of our focus—determines in how large a range along the fractal dimension we perceive the music; how much we hear of the infinite possibilities depends on how large or small is the scale of our listening.
How do you hear?
As I said, I’ve long since forgotten how I used to hear music. How do you hear music? What do you hear in Beethoven’s Fifth Symphony? Do you have any musical training? How does this affect your listening (or not)? I’d love your answers, reactions, ideas—please comment!
7 thoughts on “mandelbrot and music: on listening in fractal dimensions”
Comments posted on Facebook:
Thanks. That was a very interesting post. I’m extremely interested in music cognition in general, and am particularly interested in my own music cognition as it compares to other people’s. Unfortunately, I have so much to say on the topic that I usually find it too daunting to even begin.
I do have one relatively concise thing to say: Your experience of not remembering how it used to feel to listen to music reminded me very much of this Radiolab episode in which a man learns the fundamentals of language as an adult, and, when later asked what thinking was like prior to that cognitive change, he can’t really effectively think about or describe it:
I wonder how/whether your discussion might apply fractals to western harmony’s overlapping relationship to the overtone series and our cognition of when harmony becomes dissonance. possibilities?
Wonderful, Erin! I’m honored to have inspired such thoughts. I not only love your insights about learning to listen fractally, but also your point of how our real gift from Mandelbrot’s work is the viewpoint of scale/perspective as yet another dimension. Ah, the things we can accomplish when we apprentice to nature.
Your post is also making me think now about the role of recursivity in all sorts of learning…
Thanks for the link, Zoe—a fascinating tale to start my run today! Many of the ideas they presented about how we use language to think made me wonder about the contrasting points Temple Grandin (Thinking in Pictures, etc.) makes about how animals and autistics think.
I hope you’ll write more on your thoughts about musical cognition. The topic is rich, so please begin.
Sarah, probably yes. Unfortunately my understanding of sound and harmony is almost exclusively trapped in my musical brain. What I have managed to learn about the math behind it stays trapped in my math brain. So far I have not been able to get these two brains introduced to each other, but I’ll keep trying.
How do you think we perceive dissonance? It’s a puzzling one to me.
I believe it’s ironic that you are asking for comments about what people hear, and I, hearing-impaired creature that i am, am responding. I scrolled through the powerfully written comments on Kaddish, and decided to muse a while longer before subjecting you to my thoughts.I thought I’d try lighter fare, such as what I thought was the cooking section of your world. Since my great grandmother used to prepare both wonderfully delicious Mandelbrot as well as comforting tuna noodle casserole, I settled in for a folksy, homey read. I then realized once again I have stepped into another dimension where I am not reading about biscotti, and where a casserole is now a hot diish. At least I am in the right place to contemplate dissonance, albeit not in the sense you’d intended. Well, I guess it is time for another pain pill, and may it ward off non-existent perceived puzzles. If your two brains ever find mine, please put it in a sturdy box, and send it my way. I shall send you a casserole in thanks.