Music + Math: A Common Equation?

One of Noam D. Elkies' earliest memories is climbing onto his mother's piano bench to practice counting the black and white keys and experiment with sounds.

Today, the 22-year-old Elkies' resume is enough to strike anyone but a Nobel laureate dumb with awe. By the time he graduated first in his class from Columbia at age 19, Elkies had triumphed in several national and international mathematics competitions. In 1987, after garnering his Ph.D. in mathematics in two years, he was named a Harvard Junior Fellow and is currently conducting research in number theory.

Meanwhile, Elkies' musicial compositions have been published, performed professionally and awarded several major prizes. His reputation as a gifted modern composer has advanced throughout New England and beyond. A bass-baritone and piano accompanist for Harvard Glee Club, he continues to compose music under the direction of Rosen Professor of Music Leon Kirchner.

Not surprisingly, Elkies is considered a prodigy in both the Mathematics and Music Departments. "Noam is unlike anyone we've ever had," says Jameson N. Marvin, director of the Glee Club. "His gifted musicality, superior musicianship and sight-reading ability are an extraordinary gift. He's like a Bach or a Mozart."

Donna R. D'Fini, administrative assistant in charge of the math graduate studies program, says Elkies has earned a reputation within the department as a Wunderkind--German for "wonder child." She also says that to her knowledge, no recent math graduate student has bested his two-year record for earning a math Ph.D.

At first glance it might seem that there is something incongruous in the pairing of a seemingly dispassionate discipline with one of the arts. The mathematicians who span both activities explain the connections paradoxically. Music has the rigidity of a math problem in many ways, and math requires bursts of inspiration similar to creative composition, they say.

The more one talks to mathemeticians at Harvard, the clearer it becomes: the Math Department is infested with musical talent. Elkies story is exceptional only in degree.

The Math Department hallway abounds with concert flyers and music schedules tacked on doorways and bulletin boards. One office door bears a typewritten paragraph taken from an obscure 1924 article entitled "Mathematics and Music."

The excerpt, which suggests the possibility of a math-music link, quotes the 17th century mathematician Gottfried Leibniz: "Music is a hidden exercise in arithmetic, of a mind unconscious with dealing with numbers."

Yet the Math Department's musical vibrancy is not only suggested on office doors. One math graduate student, when asked if he had musical interests, boasted of his growing prowess at "change-ringing," an experimental technique of ringing church and hand bells according to mathematical permutations.

Another Math Department bystander darted into his office and returned a few seconds later with a large, awkward wooden object which says was a Renaissance-period ancestor of the flute. After demonstrating how to play the instrument, he passed it around among his friends, some of whom acknowledged that they were already familiar with a modern-day woodwind.

"Most of the people in my department play an instrument--many make career choices between math and music," remarks Math Department Chair Arthur M. Jaffe in an interview, as ancient music floated softly from a nearby compact disk player. Jaffe, proficient at piano and clarinet, habitually conducts "business" in his office to the tune of a favorite concerto or madrigal. "Somehow music seems to appeal to mathemeticians more than, say, reading," he notes.

Jaffe's observation was consistently echoed by faculty and students. Like Jaffe, most mathemeticians say they have noticed an unusually high level of musical awareness and talent among their peers.

What is behind this mysterious math-music link?

"I have a feeling it is more or less the same part of my brain which does both [math and music]," reflects Elkies, "they speak to the same place, the same aesthetic."

For Elkies, it is the fundamental simplicity of math and music which lend them their inherent beauty. A stark, basic principle underpins even the most complex symphony or mathematical application, he says.

"You start out with simple concepts," he remarks. To illustrate his point, he pauses to sketch a symmetrical, snowflake-like geometric figure composed of intersecting right triangles. "All of this beautiful edifice of geometry, for example, comes out of this configuration of lines and points."

According to Elkies, there is a similar simplicity in music. "Think of the simple opening--dum dum dum dum!--of Beethoven's Fifth Symphony. In music you start out with a trivial motif and it turns into this beautiful, intricate composition," he says. "Again, it all stems from this small, very simple idea."

Professor of Mathematics Mark W. McConnell, who sings and plays the flute, offered a historical approach to the question. Math and music, he says, share a common history which dates back to ancient Greece. Pythagoras, famous for his theorum on triangles, is also considered to be a pioneer in music theory.

"Pythagoras first developed the idea that the ratio between two musical pitches should be a rational number," McConnell explains. "Some people consider him to be the founder of both math and music."

In the fifth century, Pythagoras' idea of "rational intervals" between musical pitches was resurrected and profoundly influenced Western music through-out the Middle Ages, McConnell continues.

By the late Middle Ages, however, music theorists began to experiment with non-rational intervals to enhance aesthetic appeal. "Musicians began to be disillusioned with the rigidness of the mathematical structure," McConnell says. "Math and music began to diverge--each culture began to build up its own world, with its own great people and its own great ideas...finally the two 'worlds' became separate."

In the modern era, however, a number of composers have once again begun using mathematical techniques to compose music. Norton Lecturer John Cage, who pioneered in this area, is famed for using randomness and probability formulas (often generated by computer) to determine pitch and rhythm.

"It certainly is easy to speculate that because math and music grew out of a common historical framework, it drew them together," says McConnell. "It sounds believable."

Music and math may appeal to individuals who seek aesthetic beauty within a tight structural framework, suggests Mark Spivakovsky, a math junior fellow. "On the one hand, both are rigid and restrictive in structure," he says, "but on the other hand, there's something fantastically creative in both that is hidden."

Jameson Marvin emphasizes that math and music share an underlying sense of order. "In music, there's a dimension of time and ordering time," he says. "Like math, music is ultimately a sense of trying to order. Therefore mathemeticians--just the way their minds work--often have an affinity for music."

In the individual, sensitivity to order often is manifested in a methodical orientation and a painstaking attention to detail, Marvin says. As musical director, he says he has known some math students whose meticulous but uninspired approach to music led them to perform "mechanically."

"If there is a danger [for mathemeticians who pursue music], it is that the notes are technically correct but that the sense of musicality--dynamic nuance, rubatto, phrasing, understanding and projection of the musical gesture, connections between notes--is missing," says Marvin.

Music Professor David Lewin '54 received his undergraduate degree in mathematics, summa cum laude, before choosing music as his profession. "Both fields are very abstract," he says in an interview, "and both give a similar type of pleasure. They both involve the pleasure of puzzle-solving and the pleasure of craftsmanship, and seem to put you in touch with something humanistically profound." Unlike most other fields, he says, social utility is rarely a chief motivation.

Echoing Spivakovsky, Lewin also suggests that success in both fields requires an ability to juggle structural rigidity and creative insight. "A lot of brainstorming goes on and then the precision comes in at the crucial moment," he explains. "There are strong similarities in the way fantasy and precision work together."

Kit Taylor '92 is still deciding between the two fields. In high school, he advanced far beyond the traditional math curriculum while distinguishing himself nationally in piano and composition. An intended math major, Taylor is now considered one of the most promising young concert pianists in the country.

In an interview, Taylor says that specific math-music connections--like the 12-tone scale and the "mathematical" rhythm of music (with beats per measure that usually come in powers of two or three)--are potentially misleading. "You can find these links but they seeem contrived," he says.

Instead, the most convincing parallel is the very self-containment of both systems. "Both math and music take place in a vacuum," says Taylor. "Neither has a tight connection with reality, each has its own internal logic." This isolation may be just what math-musical individuals find so intriguing, Taylor says.

"These self-contained systems are appealing to me because they don't deal with the irrationalities of the human mind," says Taylor. "I guess I just don't find the human psyche and all the various depictions of it all that fascinating. Music fulfills an emotional need in me. But it does this by evoking emotion; not by depicting it or talking about it like literature or theater. I also don't use painting as an outlet because I am not a visual person."

Perhaps it is no accident, Taylor suggests, that math concentrators are stereotyped as sloppy and socially awkward. They tend to be "non-visual people" not especially intrigued by human psychology, he says. They are drawn, as a result, to "self-contained" systems like math and music, and also discover that music fills an emotional need.

Like most of those who were interviewed, Taylor hastens to add a disclaimer after voicing his views. "I know that there certainly seems to be a correlation," he says, "but exactly why? It's a hard thing to answer. My ideas are only speculative. In a way, it baffles me."