It’s a Wednesday afternoon, and classroom B150, tucked away in a corner of the Northwest Building basement, buzzes with energy. Inside the room, students furiously type on their laptops, flip through their worksheets—an interesting blend of colors, chords, and mathematical notation—and consult each other for advice.
“I’m a math major,” George D. Torres ’17 says, “but this is not stuff they teach in the Math Department or the Applied Math Department. This is the stuff that a math major wants to learn, but he can’t.” Torres, a joint concentrator in physics and mathematics, is referring to the coursework of Applied Mathematics 141r: Computational Music Theory. Described as “an inquiry based and hands on exploration” that combines “mathematics, computer programming and aesthetics” in the course catalog, the class seeks to present an intersection between two fields typically seen as operating under different categories: music and math.
Yet Applied Mathematics 141r is only a window into an interdisciplinary area of study within the department that, while small and rife with its own challenges, continues to thrive on campus. The music track of the applied math concentration brings together computation, mathematical modeling, and music theory and composition to produce a uniquely quantitative study of what has traditionally been perceived to be a humanist field. Despite limited course offerings, difficulties in striking the balance between and synthesizing the subject matter of two vastly disparate fields, and the trials of navigating a still-young industry, each year the applied math/music track imparts upon its graduates a perspective, skill set, and breadth of knowledge that serve to redefine Harvard’s liberal arts system for the silicon age.
In the age of big data and the money that often accompanies it, it’s not surprising that the number of undergraduates concentrating in applied mathematics nearly tripled (from 101 to 275) between 2008 and 2014. The meteoric increase is hardly characteristic of a typical concentration at Harvard—but then again, applied mathematics has never been a typical concentration. Rather than remaining bound to the same general topics, concentrators are required to tailor their study to their own interests by choosing one of a variety of fields to which to apply their quantitative knowledge. Many students gravitate toward popular, pragmatic areas of application like economics, computer science, and mechanical engineering; others, breaking from the mold, choose less obvious fields like linguistics, architecture, and government. While the first set of areas of application consistently draws large numbers of concentrators, the second set can in many years see none at all.
Among the less conventional areas of application, however, one subject stands out for the relative reliability with which it attracts students: music. Both Christopher H. Rycroft, director of undergraduate studies for applied mathematics, and his predecessor Michael P. Brenner, professor of applied mathematics and applied physics, confirm that the track has for years seen continued interest from the student body, producing a small number of graduates almost every year.
These students flock to applied mathematics/music for a myriad of reasons. For some, it’s a way to study two disparate fields of interest without accumulating the numerous requirements that adding a joint concentration or secondary field would entail. “I was able to take these music classes that I otherwise probably wouldn’t have taken, because if I did, like, applied math/CS [plus a secondary in music], the course load would have just been so difficult for me,” says Jiho Kang ’16, an applied math/music concentrator.
For others, like Alicia J. Young ’17, it’s a matter of flexibility. “I’m glad I [concentrated in applied mathematics/music] just because I have so much variety. I’m taking classes in the Math Department, the Stat Department, the Computer Science Department, potentially the Ec Department. I get to have classes from all over the departments, and they are all good at making me think about math in general and how it is applied to music,” she says.
And yet for others still, it’s an opportunity to unite the two subjects, to solve musical problems like teaching a computer program to read and digitize handwritten sheet music or to characterize musical phenomena using numerical models. The structure of the applied mathematics concentration provides plenty of room for students to accomplish this task. In addition to their foundational classes, concentrators are required to select five courses that meld, to a significant degree, quantitative study with their chosen area of application. Depending on the specific student, such classes can cover material such as population genetics, decision theory, and natural language processing. Those who choose music as their focus field are able to pursue topics ranging from Schenkerian analysis to tonal counterpoint—but of their options, the course perhaps most archetypal of the applied math/music track is Applied Mathematics 141r.
Computational Music Theory is the culmination of years of work from Elizabeth R. Chen, a lecturer on applied mathematics at the School of Engineering and Applied Sciences—work that she says she started as an undergraduate. With encouragement from Brenner, at the time director of undergraduate studies for the Applied Math Department, she hosted a seminar on the topic for graduate students and postdoctoral fellows over the summer; its positive reception led to the course being approved for undergraduate students for the following school year.
Two years later, Chen is now leading the second iteration of the class, and as she tells it, both her approach and the subject matter are hardly ordinary. “The way I teach the theory is a bit unusual than how most people teach [it],” Chen says. “And I just don’t mean the pedagogy. I also mean the course material is a bit unusual.”
In fact, Applied Math 141r covers a wide breadth of mathematical topics spanning pattern recognition, number theory, vector space modeling, and ratios—to name a few. The 27 students enrolled also work extensively with Mathematica, a mathematical computation software, to encode audio and build their own instruments. These various components come together to help students derive the key mathematical concepts underlying music theory. “It’s just really interesting material that we’re learning,” Ben M. Kelly ’17 says. “Especially… having learned a lot of this stuff from the music theory side of things, to now get to learn it from a mathematical standpoint is really fascinating. I see music now in a different light. I hear it in a different tone.”
Furthermore, the class eschews lectures for a seminar-style format more conducive to collaboration and critical thought. “The way that Beth approaches the class is very unique,” Caroline Teicher ’19 says. “[Chen is] against lectures…. Instead she’ll put something very strange and obscure on the board, and because we have a lot of time in this class, we’ll sit for an hour and a half or three hours to figure this problem out.”
Torres notes the degree of engagement Chen’s pedagogy demands of her students. “[The course is] incredibly self-motivated,” he says. “We try to figure things out because we ourselves are interested in figuring out something. It’s different from a lot of other math classes because it doesn’t derive things from axioms or strict rigor….”
This educational style is what attracted many to Computational Music Theory; in fact, several of those interviewed cited their love of Chen’s teaching as their reason for enrolling in the course. Others chose to take the class for its content. This latter category of students comprises not only applied mathematics concentrators within the music track—though many, such as Young and recent graduate Kara J. Lee ’15, do take the class—but also those within the department who are pursuing other focus fields; biomedical engineers; computer scientists; and even a law student.
Aron I.F. Szanto ’18, an applied mathematics concentrator interested in economics and computer science, highlights the significance of having courses that find applications in two seemingly unrelated fields—particularly when one of those fields is non-STEM. “There is a really large focus on technical sciences being applied to other technical sciences,” he says. “I think there could be a lot more of technical sciences being applied to softer sciences or softer fields. In particular, I’m a big fan of the applied math/music track because [its] students… are really the impressively interdisciplinary ones.”
In fact, students say that what ultimately makes Computational Music Theory an engaging course is its mission to truly highlight and concretize the connections between math and music. “In applied math, a lot of the emphasis is on applied, and it’s kind of ironic because you don’t see many classes like this [class] that are actually focused on the application,” Emily R. Cherkassky ’17 says. “So it’s cool that this class focuses on directly applying math to a topic.”
Chen is leaving Harvard next year, however, and she fears for the fate of the class following her departure. “It’s hard to find people who are applied math and music. I’m actually quite worried about the future because this is my last semester here, so this course probably won’t be offered again, and I’m not sure what they’ll do,” she says. “I think they need to hire more applied math people, period.”
Indeed, according to Rycroft, there currently exist no definite plans to continue Applied Mathematics 141r in Chen’s absence. “That course is very much tied to Beth, really. She developed it. At the moment, the answer is no. People like Beth are hired for a fixed period, and we kind of have a number of people coming through there, and they may offer a number of focus topics in other areas. So we don’t know yet whether there will be anything of that flavor in the near term,” he says.
The loss of what students have described as a crucial element of the applied mathematics/music track hits particularly hard for a focus field that, according to students, faculty, and administrators alike, already suffers from a lack of course offerings. “I would say that SEAS has been a little taken aback by the growth of applied math,” Rycroft says. “Universities only move so quickly in adapting to that…. To expand a program, there’s only kind of a certain rate you can do it in terms of the number of people you have to teach it. But certainly there’s a commitment from SEAS to really grow the applied math program in terms of faculty, in terms of lecturers, and to try and expand the course offerings.”
Brenner also acknowledges the shortage of suitable classes—but in his opinion, music actually fares better than the rest of the humanities. “It’s always difficult…. It’s not like there are classes that were made for this sitting in the course catalogue,” Brenner says. “[But] for music, MIT has classes that students that can take; and also, the music department has classes that work when put in context, and that helps. There are other areas in the humanities where you’d think it would be nice to have such things.”
Indeed, some students do opt to make the trek to MIT, where they can choose from a wider variety of courses on topics ranging from audio engineering to the physics of sound. For example, Michael Scott Cuthbert ’98, a music professor at MIT, teaches classes on music theory, music history, and computational methods of understanding musical notation. “Some of the courses ask questions about, ‘What’s a way of understanding how we extract information from musical scores or from recordings?’” he says.
Others fulfill their credits not with the mathematical modeling classes they had anticipated but rather with music theory, which counts as quantitative under the applied mathematics concentration criteria. But these courses, some of which only loosely touch upon one of the two subjects they are meant to combine, don’t necessarily fit the mold of a focus field touting the strength of its interdisciplinary learning. For instance, applied mathematics/music students have taken Music 167: Introduction to Electroacoustic Music, taught by music professor and Harvard University Studio for Electroacoustic Composition director Hans Tutschku, for concentration credit. Tutschku says, however, that the class—which teaches students to arrange, layer, and edit sounds into full-fledged musical works using technologies like Reaper, a multi-channel digital audio software—treats the scientific component only as a means of music creation rather than something to be understood for its own sake. “There’s no technical background required,” Tutschku says. “What we do in sections is teach them the use of a couple of tools…. But we don’t go into the scientific underpinnings. They don’t learn how to calculate an f(t) or something like that. They really learn the tools in order to produce something they want to write.”
Tutschku’s comments illustrate what students have indicated to be a major shortcoming of the track: Its classes often emphasize, in a binary manner, math or music as their primary focus. One serves merely to facilitate the other, rather than the two being combined in a way that truly enhances students’ understanding of both fields. “I can’t really say that I came out of this experience having a very strong intersectionality between music and math,” Kang says. “I looked at the syllabus of these [music] classes and was like, you know what, this sounds interesting because it has a lot of technological elements—but I never, like, solved a differential equation in these classes…. I like math enough that I wish [the classes] could have incorporated a little more of it. Oftentimes it felt a little separate.”
This perceived lack of synthesis between the musical and mathematical components of applied mathematics/music classes has even influenced students not to declare concentrations in the field. Benjamin S. Wetherfield ’17, for example, is involved extensively in pursuits that combine technology and music. Last semester he embarked on a Visual and Environment Studies multimedia project in which he combined music with live-processed visuals, and this summer he will be working at the Studio for Electroinstrumental Music in the Netherlands, where he says much of his time will be spent working on physical computing and programming. However, Wetherfield has opted to concentrate not in the music track of applied mathematics but rather in math with a secondary in music, in part because he has not found a true connection between the two subjects in his classes—even those that would have counted as applied mathematics/music concentration requirements.
“I don’t really see that much of a bridge in what I do. I guess I use technology a lot or use the computer a lot and things like that [when I make music], but I don’t view anything I do as applied math,” he says. “I think on a micro-level, in creating music, you’re always solving little problems that arise… and sometimes that’ll maybe be a slightly mathematical outlook or maybe a combinatorial way of approaching a music problem. But the problem’s just arisen for creative reasons. And for me, I’m unlikely to have a large-scale problem that I want to tackle using the tools of mathematics. It’d just be an ad hoc solution using any number of tools.”
Administrators within the Applied Mathematics Department recognize the disconnect between the mathematical and musical components of the track but argue that the challenge may, at the same time, actually benefit concentrators. “I’m kind of aware of the issue…. It’s something we’re very aware of and something that we’re trying to continually improve,” Rycroft says. “But I do also think… that one thing about being an applied mathematician is to really be able to communicate across disciplines, so I think that some amount of having to deal with slightly different environments is actually good.”
But it may not be for lack of trying that the Applied Mathematics Department is unable to pilot a fully integrated music track; rather, students attest to a dearth of faculty and advisers who have substantial insight into both fields. While students do praise the overall supportive atmosphere of the Applied Mathematics Department, the strength of its advising, in large part due to this scarcity, has received mixed responses.
Maura D. Church ’14, a recent graduate and applied mathematics/music concentrator, says that she found a welcoming community in her peers in the track and other similar tracks, students at MIT, and faculty members from both the Applied Mathematics and the Music Departments. “The advising was stellar, especially from the Applied Math Department,” she says. “I think they really make an effort… to give you the support you need.”
Lee, on the other hand, agrees that individuals like Brenner, who was director of undergraduate studies while she was an undergraduate, did try their best to guide her—but she says that the Applied Mathematics Department’s lack of offerings and domain knowledge in music hindered her attempts to navigate her focus field. “While the department was extremely supportive and excited... in terms of speaking to professors who really understand the intersection of applied math and music and built classes around that, that was incredibly limiting,” she says.
Her concerns are shared by current undergraduates like Kang, who considered writing a senior thesis but found ultimately that he lacked the expertise to conduct a research project of sufficient scope. “I think something that’s just really challenging is just if the concentration is going to support stuff like applied math/music… the department itself also needs to know a little more about that accompanying concentration to see whether it is a good fit,” he says. Even Church, who did write a thesis detailing an algorithm to correct rhythmic errors in commercial Optical Music Recognition software, found an adviser not in a Harvard professor but in Cuthbert, whom she met while he was at Harvard as a Radcliffe Fellow.
In any case, the absence of a full-fledged, truly cross-disciplinary quantitative music program is hardly a unique shortcoming of the Harvard Applied Math Department. The tech-music sector itself, rising on the shoulders of industry darlings like Spotify and Pandora, has certainly grown in recent years—Spotify went from half a million paid subscribers in July 2010 to 30 million in March 2016—but it has yet to reach critical mass. Job opportunities, while on the rise, remain relatively sparse, and both current students and alumni attest to the difficulty of securing employment. For example, Lee, despite prior internship experience at Spotify, Pandora, and Interscope Records, had not found a full-time position even by the time she graduated the College. And Kang received an offer to intern at Spotify during the summer after his junior year—only to be informed as his start date approached that the company no longer had the funds necessary to support him.
Regardless of the internal challenges that the applied mathematics/music track and its students face, responses to its real-world applicability and utility outside the Harvard bubble have been optimistic. Now graduates, both Church and Lee have found the skills they have derived from the concentration to be extremely useful as they enter the workforce. Church now works as a data scientist at Patreon, a start-up in San Francisco that connects artists and fans through recurring funding, while Lee is at Pandora working on operations, data analysis, and overall production coordination for the company’s genre stations and branded playlists. “I think coming into digital music, being an applied math major has definitely helped,” Lee says. “Just being comfortable with data and seeing music as not only a creative art but also a quantitative data set has given me a leg up so early in my career.”
Church agrees with Lee’s assessment. “I am a great endorser of SEAS’s applied math concentration,” she says. “In terms of being in a cross-disciplinary field in applied math and music, I think it worked very well to my advantage because I was able to say, ‘Hey, I have the hard skills… but also, here’s a field I’m really interested in… that opens up my perspective.’”
But the voices supporting this model of learning do so for reasons that are not only career-oriented but also philosophical. Suzannah E. Clark, professor and director of undergraduate studies for the Music Department, currently teaches Music 223r: Neo-Riemannian Analysis, a graduate course on how mathematical transformations can inform music. Her class often attracts undergraduates interested in blending math and music, and she advocates similar courses for their potential impact in cultivating greater thought in students. “What you get from these kind of courses is the humanistic mind and the scientific mind comes together—either in the same person or in different people coming together at the same thing in two different angles,” she says. “What I think is fruitful with a body of students that come from these different angles is that they come to appreciate different ways of thinking.”
In keeping with Clark’s words, Young says that approaching music in a mathematical way has in fact aided not only her quantitative ability but also her artistic creativity. “I really like [ applied mathematics/music] because I’m a musician… [and] studying music as a math—it almost seems like studying a language class,” she says. “[You’re] putting [music] into a system or a pattern. There’s actually a system that I can go back to. It inspires new ideas for what I want to compose.”
Ultimately, however, Clark feels that the applied mathematics concentration—for all that its name connotes scientific rigor and practicality—is simply a manifestation of Harvard’s liberal arts sensibility. “One of the thing that I think is really important about the liberal arts system is the fact that Harvard has many, many different students and that they are going to have many, many different strengths and interests,” Clark says. “I think what the university should encourage is that if you have different passions, that you can be the person that can bring them together.”
So while the music track of the applied mathematics concentration isn’t without its obstacles—and while it certainly isn’t the first thing that comes to mind when one mentions liberal arts—there exists no better representation of a field of study that encourages its students to do exactly as Clark says: to meld the best elements of disparate subjects to form a more perfect, surprisingly elegant whole.