A Fifth Is Two Sine Curves
(This post could be seen as a self-narrating zoo-exhibit with bonus spot-the-theory-of-mind. I hope you’ll take it as a follow-up to The Obsessive Joy Of Autism, as equally about the experiences of visual processing impairments and synesthesia as about ~autism, and as a case study in the inadequacy of traditional verbal language. If I’d been taught earlier that experiences which exist independent of words are still meaningful and sufficient, I never would have had to write this.)
(This whole essay represents a dozen years of heartache, a fractured self-image knit back together, and the entire reason why I will never, ever talk about the instinctive simplicity of video editing.)
(Note: as is typical of my writings, pieces are rarely finished in the same day they are started. The voice lesson I refer to as “today” is in reality long past; in fact, due to circumstances regrettably beyond my control, it turned out to be my last. I have been unable to further explore the discovery my voice teacher and I made, and it aches.)
(This entire piece is written in a voice I don’t very much care for. My apologies.)
(I can either keep apologizing in parentheses for a lifetime, or just tell it to you.)
A fifth is two sine curves, and this is the most beautiful thing.
I’ve had a dozen years of classical musical training, and I’ve been playing with calculus for about the same amount of time. I don’t talk about either of these things with much frequency. I don’t like to sing in front of people, and I have a passionate love affair with math but hated and barely passed my classes in high school. I drink up geometry videos on youtube, and once a week I go to my voice teacher’s house and sing for the best hour of the week, and these things are secrets. Very few people know how important music and math are to me, and that’s been a deliberate choice on my part.
I don’t talk about the most sacred things, because it’s the fastest way to destroy them.
I have an intense and embodied relationship with music. Music is a lot like people, and songs are linguistic and therefore social exercises. Music teases and tells and remembers and reflects and transforms and encodes and transmits. I experience music kinesthetically, emotionally, and intimately. It is shattering.
I know a lot about music. I hate talking about it, because the things I know aren’t the same as the things you know.
My left hand doesn’t have enough coordination to permit me to play any instruments. I can, and do, sing, and I’ve spent 12 years learning to do so with surgical precision. I can’t sight-sing–I can peel apart the layers of a piece for theory, no problem, but identifying pitch and tempo and dynamic and a note’s place in the overall piece, and integrating those elements, factoring in timbre and placement, and singing them, in realtime, is simply not possible with my visual processing impairments. I rely on an ear I’ve spent years training–if there’s one other singer, or a pianist accompanying, I can do a passable imitation of first-time sight-singing even pieces I’ve never heard before.
Ever since I first joined an (auditioned, highly-selective, then-prestigious) youth choir at age eight, I’ve known that I do music wrong. At first, that took the form of being shamed for moving when I sang–music shouldn’t be a physical, embodied, tactile experience. In later years, my inability to process visually rapidly enough to sight-sing, and my difficulty holding a harmony, made rehearsal a living hell. Yet I was never kicked out, because I could still sing circles around 2/3 of the other singers, and it wasn’t that I couldn’t harmonize, just that I might sing any one of a range of acceptable harmonies and not necessarily the one I was arbitrarily assigned (or the one the composer had written,) and as long as I didn’t talk about it, no one knew I couldn’t sight-sing. (I did, of course, talk about it; lying didn’t occur to me until later years.) I trained my ear, led the 1st sopranos because the melodic options are necessarily restricted and usually sensical, kept my joy at things I had the wrong words for to myself, and kept singing.
(My life goes on
in endless song
how can I keep from singing?)
It destroyed me. I loved this, I was so good at it, it felt good and right and perfect, and yet any attempts to communicate about this were miserable failures. Singing is the only time I ever feel at home in my body, and yet I felt like a fraud; sure I could learn a new song faster than anyone using only my ear, but I couldn’t sight-sing. Clearly this was some elaborate hoax I’d engineered, clearly I didn’t have any real aptitude, and clearly, my decision not to pursue college or career in music or performance was the only ethical option.
Despite this, despite the guilt and shame and confusion, I’ve kept going to weekly lessons. It’s an hour, just me and my teacher and a piano, and as we pull songs apart and test what my throat can do, the songs resonate between my ribs. It is, essentially, an hour of breathing exercises, so it’s the most effective therapy I’ve ever had. And I’ve kept going, and today, finally, after 12 years, we figured it out.
There were signs. The music teacher at my high school had almost worked it out; he’d play two notes, and I’d hear a third, and not every student could. He talked about harmonics and melodic structure; I listened, and adored him, but had trouble talking to him. With my voice teacher, I complained that the standard system of music notation was nonsensical, bearing no correspondence to the way music actually worked, and relied on a flagrant disregard for basic arithmetic. I struggled with the idea that a fourth is supposed to sound “smaller” than a fifth, and though I could sing any note back perfectly, I couldn’t tell you which notes were higher or lower. None of the language used to communicate about music made any kind of sense to me. The spatial metaphors simply would not map. I had my own rich, dynamic, embodied and visiospatial experience of music, I could feel and see the music move, and sheet music couldn’t come close.
I’ll tell you what my teacher and I learned today, but first I need to give away my other secret: math.
My history with mathematics is, if possible, even more tortured and rapturous and bewildering than my relationship with music. As a small child, I was in love with arithmetic; my mental calculation abilities are noted on my earliest evaluations (and dismissed, of course, as splinter skills.) One of my earliest memories is of not being able to explain to a concerned teacher why 2+3=5. “Because it is,” I said. “And it’s beautiful. It just fits.” Manipulatives were meaningless for me–my eyes couldn’t process them easily, and they didn’t bear any relation to the ladders and scales of numbers in my head. I simply had to close my eyes, and think of an equation, and I would feel myself swing from number to number, landing effortlessly and securely on the answer. Easier than walking.
(There’s a hint to the eventual reveal, there.)
While I had some small troubles learning the language for arithmetic (especially for fractions, which are fun and clever enough as pointless rituals but beautiful when you realize that they actually signify multiplication and division, that you are performing meta math,) mathematics was an easy joy. Note, please, that I am talking about the art of mathematics, not the drivel and formulas we’re drilled on in school. Mathematics really is an art, and it’s inherently playful. It’s a game of “what if” and “how can I.” And when I was eight years old (and still now,) what I wanted most to figure out was how I could model, and therefore predict, events in the world around me. In particular, I was fascinated by the rate at which raindrops crossed and accumulated on my car window as my mom drove me to therapy.
So I invented the fundamental theorem of calculus.
I didn’t know I was doing this at the time, of course. I didn’t know what algebra was yet; if I’d heard of calculus, I thought it was about calculators, which I found woefully inefficient. I’d never heard of Liebnitz, and Newton had only been hit on the head by a terrifying apple, as far as I knew. I didn’t use the greek alphabet or sigma notation; I didn’t use any notation at all. I just ran some dimensional transformations in my head, checked my model for accuracy, and had a nice visual to keep myself comforted when my hands needed to be still.
Eight years later, my AP Calculus/Physics class was asked to prove the relationship between derivatives and integrals, and I got up and drew the model, and not a single one of my classmates understood. After a moment of studying what I did, Mr. Morris said, “spoiler alert, Julia just skipped to the end of the lesson and proved the fundamental theorem of calculus.”
I can’t draw it for you now; I lost the image, along with some of my more unusual human-calculator prowess, when my psychiatrist put me on the anti-psych medication regimen that ended in me dropping out of college. But I could then, and before that day, I hadn’t realized that my classmates couldn’t see that basic, innate relationship, the reason why calculus is beautiful enough to ache. I didn’t have the language for it, had never even heard of the fundamental theorem until my teacher told me that was what I’d done. I’d just been amusing myself with models and transformations, when I should have been paying attention in class.
See, that’s the thing. I struggled in math class from Algebra 1 on. I never actually took the AP calculus exam–Mr. Morris thought I’d do fine, but since I was technically auditing his class, I didn’t want to try and sit for it. And this is where the agony comes in.
Much like music, the way I experience and represent mathematics cognitively is fundamentally different from the conventional language we are taught for it. With arithmetic, this difference only really caused difficulty when new terms were being introduced. But 2+3 is always 5, and I had enough innate understanding and motivation to preserver through even the most ridiculous name choices.
This all fell apart when I got to algebra. It’s not that I lacked understanding–I’d been solving for the missing number since I was nine, and the only hardship I’d had there was, again, in comprehending the linguistic absurdity of mixing letters and numbers and shifting the missing piece to the left side of the equation. As soon as I got over that, I immediately started attempting to teach the game to anyone around me, including my four-year-old sister. Algebra wasn’t the issue. The way we represented it, modeled it, and talked about it? That was a nightmare.
Graphs on a cartesian plane have a lot of meaning for me–for another purpose (clue.) They also bear absolutely no relation to the kinesthetic and visual models I see when I craft algebraic equations. Zero. None. And I couldn’t articulate this in high school, anymore than I could make myself care about pitch or slope whether the equation came in point-slope or some other form. I didn’t understand why we were talking about graphs at all. It wasn’t until algebra 2 that someone explained to me that all of our equations were supposed to be models for those lines.
I was enraged.
I still am, thinking about it. What a stupid waste of my time. Who the fuck cared about lines? I wanted to know how to model the age distribution of my peers around me and how it shifted in relation to me as we moved through the grades, how to fit equilateral triangles with a side length of 2in inside a hexagon with 6in sides as efficiently as possible, how to predict prime numbers. Why on earth would I make an equation for a line I couldn’t even make my hands draw reliably half the time?I wouldn’t! This possibility had never occurred to me! I had better things to do!
So I didn’t care about graphs. Later, in college, my math major friend would teach me the theory behind linear algebra in half an hour, and I would concede that graphs were pretty fucking intense, but in high school, all I knew was that I was being asked to humor, to indulge teachers who were never sure if I was a genius or in the wrong class in their fetish with graphs that my eyes, more often than not, couldn’t even process as a whole.
It did not go well.
There were parts of algebra, of course, that I loved. I enjoyed modeling numeric patterns (“series,” I was told, and “summations,” I just called it numbers) like 2n + 7, and there were moments when the graphs for these, or for my favorite exponents, lined up with the stacks of numbers I saw in my head, and the graph became, fleetingly, meaningful. (It was argued that graphs themselves were by definition simply visual models of numeric patterns, and therefore should have been my favorite thing. I defy anyone to look at something like THIS
and honestly tell me that they instinctively see either 2n + 7 or 9, 11, 13, 15, etc. in that.) Systems of equations were a game that made me laugh and resembled the way I’d play with numbers in my notebook margins when we were supposed to be caring about matrices. Matrices were for our calculators; I was a human calculator, insulted by their existence.
My first three algebra teachers understood none of this. I did well in algebra one, as long as I wasn’t asked to graph anything; as far as I was concerned, I was doing slightly fancier arithmetic and then humoring Mr. Morris by labeling things as he wished. Algebra 2 was all about graphing though, and it made me want to die–and now, in an exciting plot twist, the equations had gotten long enough, and the graphs complex enough, that my visual process started undermining me, scrambling letters and transposing numbers and fragmenting graphs, and with it, my grade. But I passed, by the skin of my teeth, and had a brief reprieve in geometry, and then got a better teacher for precalculus who, in the last two weeks of class, realized I had a language processing problem. We stayed after school one day and hacked enough accommodations that I could FINALLY start to use the conventional vocabulary and start training my eyes to read equations like stories again, the way 2+3=5 is, and then graph the story out accordingly. Just in time for the bonus unit on calculus.
I looked at a derivative for the first time, and my models and raindrops and joy came rushing back, and I was in heaven.
Derivatives and integrals made sense because the transformations they undergo are multi-dimensional, just as the ones I assign my models are. But I didn’t ace calculus. I took an introductory course at Stanford the summer before my senior year of high school and passed on sheer intuition despite missing 1/3 of the classes, but the first few tests I took in my AP class came back with failing grades.
My teacher and I worked it out. It was, as always, a visual processing issue: permitted to solve an equation on a white board, standing up, with more space than I could possibly use, colored markers, writing as large as I needed, and only one problem presented at a time, I could earn 100%. Give me the same test, after I’d already taken it on the whiteboard, but this time package it as a two-page test on an 8×11 inches sheet of paper, and I’d get a 40%.
It seemed easy enough to accommodate: let me take my tests on a whiteboard, with my teacher dictating each problem to me one at a time, and I could be the best in the class. The beauty of reasonable accommodation! But it wasn’t that easy; the teachers were working to contract in protest of their current salary, so my teacher couldn’t stay after school or come in early to test me. If I’d had an IEP I would have been protected, but my parents were told that students with IEPs weren’t permitted to take AP classes. The solution, the administration declared, was that I would audit the class. I could sit in on every class, do every lab and homework, receive no credit, have my motivation and self-esteem destroyed by a year of failing every test but doing any in-class problem on the board perfectly, teach my class the fundamental theorem of calculus but be humiliated every day by my inability to correctly read and copy equations off a sheet of paper, and everybody won.
I don’t remember much of the class. I was an anxious, humiliated, depressive mess, unable to concentrate, still haunted by graphs. It was a combined calculus/physics class: I loved the modeling, but my language processing made the word problems we were given for physics exhausting. There are bright memories, of course–drawing out the fundamental theorem, hacking labs, solving problems at the board, practicing integrals with Tommy for fun, tutoring a classmate over Facebook chat for a test that I failed with a record- breaking amount of transcription errors but she got an A+ on. But I hesitate to call myself mathematically inclined now, and that needlessly hellish year is why.
Math and music are some of the most sacred and meaningful things in the world to me. I don’t talk about them, because that requires a 12-year trauma history and interfacing with a language that treats these two things exactly opposite of how the should be.
(Do you know the secret, yet?)
This is what I learned, at the last music lesson I ever took: I see musical compositions as the graphs I spent years being tortured with perversions of, and I move through numbers like a song.
The spatial metaphors we traditionally use for music (notes are higher or lower; a second is a whole step, a third is two whole steps, etc.) fall apart for me not only because of their flagrant disregard for basic mathematics, but also because my primary concept of music lies in harmonics: in the spaces between tonic and supertonic, in the beats of the frequency of a given pitch, and all the different ways we can stretch and collapse and layer and color them. This is what I see, this is what I hear (and the two are, in this instance, the same,) every time music plays. Sheet music is as pale an imitation of this as a graph of y=3x is of the multiplication table. And yet, the transformations I drag my models through, the process of solving an equation (and I’ve never been much of one for solving, really; just give me a model and I’m happy, and maybe that was the problem with algebra all along,) is actually perfectly encapsulated by the language and metaphors musicians use to interpret sheet music.
And I just wish that once, at some point, just once during those long twelve years, someone said to me “this language isn’t working for you. Let’s find a new one.” Because then I could have told you, at eight years old, before I could even conjugate reliably or deviate from my scripts, that:
this is what C sounds like:
this is what G sounds like:
this is what a fifth sounds like:
and this, Mrs. Lowenthall, is why 2+3=5:
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