The annual meeting of the American Association for the Advancement of Science wrapped up last week in Washington, DC. One particularly enjoyable and informative highlight was a session on Mathematics and Music, which presented some work in progress by three prominent researchers in this area.

Noam Elkies of Harvard University presented the first talk, titled “The Entropy of Music: How Many Possible Pieces of Music Are There?” He illustrated his points with virtuosic turns on a keyboard. His basic idea was to apply concepts similar to those used in statistical mechanics and information theory to approach the question posed in his title. Elkies addressed how much a piece of music needs to change before it is a different piece, rather than a variation on the original. He also talked about how much information remains when the redundancy of repeated themes in a piece is accounted for.

Elkies did not address the problem of musical semantics in enough depth to make a compelling case that his statistical approach could generate real insight. But his lecture ended with an impressive performance, from memory, of a piece made from a baroque-style repeating arpeggiation where the root of the chord changed from measure to measure based not on a conventional harmonic progression, but on the digits of π. The result was an intriguingly disorienting congress of order with randomness, evoking something like an inebriated Buxtehude.

Musical geometry

The next presentation was from Dmitri Tymoczko of Princeton University, who talked about what he called the geometry of music. Professor Tymoczko has been engaged in a project to recast traditional music theory in a language that he feels is more rational and consistent—one that better reflects the actual practice of composers. His work is summarized in a book from 2011 called A Geometry of Music (as well as several peer-reviewed papers).

Tymoczko focuses on the harmonic structure and voice leading (the passage of voices from chord to chord, which largely results in counterpoint) of the Western common-practice period. He argues that this is most naturally represented through the language of differential geometry. In his scheme, a note is a point on the real line, turned into a circle with the identification of octaves; chords are higher-dimensional generalizations. Voice leading is represented by vectors in the tangent space of these manifolds.

In differential geometry, a tangent space is a space spanned by vectors that are tangent to a shape at a particular point. If you imagine holding a ring, representing the set of notes, then a piece of paper held against the ring can represent the tangent space for the point where the ring and paper are in contact.

Tymoczko says that his mathematical scheme is a natural and idiomatic description of the structure of music from the common practice period, though his case would be more compelling with some example of how a result on the mathematical side provides new information about structure in music.

Tymoczko is doing an exhaustive, semi-automated gathering of data from a large corpus of music, and he has found that composers routinely violate the purported rules handed down in books of music theory. This is not exactly a secret, but Tymoczko's objective, data-driven approach to musical practice seems to have the power to reveal other significant insights. His love and deep understanding of music was evident in the joy with which he shared his discovery of hidden canons (or rounds, as in “Row, Row, Row Your Boat”) in pieces by Bach and Marenzio.

Musical tuning

The final speaker was David Wright, chairman of the mathematics department at Washington University. He tackled overtones and tuning. The tuning systems are rich with history and inherently mathematical. As the Pythagoreans discovered, harmonious intervals in music are formed from frequencies that are the ratios of small whole numbers.

Performances that preserve these intervals exactly employ just intonation. But keyboard instruments since the time of Bach have been tuned using “equal temperament” (or something similar), which allows music to be played conveniently in any key by dividing the octave into intervals of equal ratios.

Wright mentioned the theorem that an instrument tuned to equal temperament, such as a piano, can not render any interval aside from octaves exactly. He played recorded examples of intervals and chords to allow the audience to hear the difference between just intonation and equal temperament, and other samples of performances by voices and fretless stringed instruments, demonstrating the effects of blue notes and of different intonations.

The highlight here was a recording of a barbershop quartet in which Wright discovered a subtle use of microtonics during a chord change. In shifting from one chord to another, three of the singers changed notes, while the fourth held on to his note across the chord change—or should have, based on the notes on paper. In practice, the sustained note shifted frequency by a small fraction of a semitone to maintain just intervals with the notes of the second chord.

These types of microtonal manipulations are part of what give barbershop quartet performances their intensely harmonious quality. This presentation was a fascinating glimpse of the world of music beyond the constraints of 12 notes, and of an active area of research.

One often hears knowing but vague assurances that music is “mathematical” in nature, with little to back up the truism. Here was a session full of the details. It was entertaining and challenging, and the presentations worked both as tutorials and glimpses into several areas of active research.