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."