Indiana University Research & Creative Activity

Humanities, Then and Now

Volume XXIX Number 1
Fall 2006

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The harmonic path of a Beethoven sonata, as visualized on a graph known as a Tonnetz (tone network).
From “Mapping the Music,” Julian Hook, SCIENCE 313:49 (7/7/2006). Reprinted with permission from AAAS.

Seeing the Music

So, a music professor goes to an eye doctor. The professor leans back in the chair for his exam, and the doctor says, "You're a musician, huh? Are you good at math?"

It's a bit like the punch line of an old joke, or maybe an equation: Musical skill equals mathematical prowess. Everyone knows the friend of a friend who's a pianist who double-majored in physics, or a math whiz who plays the trombone. Professor Julian Hook's optometrist was merely dipping into conventional wisdom with his doctor-patient patter.

"It's been recognized since the ancient Greeks that math is the heart of musical structure, but it is elusive to get a handle on just how it works," says Hook, a professor of music theory at the Jacobs School of Music on the Indiana University Bloomington campus.

An award-winning pianist, Hook's career path has taken him from mathematics to architecture to piano performance, finally arriving at music theory. His approach to music theory is to explore tonal space through algebraic and geometric representations.

Simple examples of musical spaces include pitch space (pitches arranged in a straight line from low to high) and pitch-class space (a modified pitch space in which octave-related pitches are considered equivalent, so that the geometry becomes circular). An example of a visual representation of pitch-class space is the "circle of fifths." Different geometries explain different aspects of the structure of a space, Hook explains. A visual representation that progresses chromatically through the notes shows how notes are melodically related, while a representation like the circle of fifths represents a harmonic relationship among the same notes.

"A musical score is a graph whose vertical axis represents pitch and whose horizontal axis represents time," Hook writes in a recent issue of Science magazine. "Given this apparent simplicity, and the recognition since the time of Pythagoras that mathematical principles underlie many musical phenomena, it is perhaps surprising that our understanding of the mathematical structure of the spaces in which musical phenomena operate remains fragmentary."

Before Hook found his calling as a music theorist, he would muse about the math of music and imagined "that surely music theorists had figured out the answers a long time ago." It seems to him still that "the questions I am asking are so basic that I am astonished that no one seems to have worked out the answers before."

To someone not steeped in music theory and sophisticated mathematics, the questions seem far from basic. Hook's research involves an analysis of diatonic and chromatic harmony through the lens of transformation theory. In other words, he looks at the way a set of seven notes interacts with a set of twelve notes. There is seemingly infinite variety in the ways such harmonic relationships can be examined visually and mathematically.

In the July 7, 2006, Science article "Exploring Musical Space," Hook provides context for fellow musicologist Dmitri Tymoczko's article, "The Geometry of Musical Chords." The illustration that accompanies Hook's piece is a representation of the harmonic plan of several measures of Beethoven's Violin Sonata, Opus 24. A grid of lines overlaid with arrows connects notes in a repeating geometric pattern. The grid is called a neo-Riemannian Tonnetz, or tone network. It shows "the transformational relationships among the 12 major and 12 minor triads."

Though anathema to many practicing musicians and even many music theorists, mathematical music theory is gaining popularity, says Hook. He never imagined that he'd be the first person to analyze Beethoven in the pages of Science. "My timing was good," he says. "It turns out that there are quite a few people out there asking the same kinds of questions that I am."

Tymoczko's work, for example, "shows that musical spaces are more complex than we might expect--they have ‘singularities,' points where the space folds back on itself in a non-Euclidean way--and we are now able to understand musical spaces in ways that they have not been understood before," Hook says.

This opens up exciting new possible applications for composition, he notes. "Once you understand the geometry, you could probably devise a multitude of chord progressions that navigate through space in previously unexplored ways."