A Princeton researcher has developed an algorithm that turns music into geometry and has uncovered surprising similarities between seemingly disparate compositions.

Three-note music chords are mapped in 3-D. Each point in the prism represents a chord, with its coordinates determined by each note. The chords used more commonly in Western music lie at the center. All images and video by Dmitri Tymoczko

The opening riff of Deep Purple’s “Smoke on the Water” is said to be the most famous in hard rock history. Now listeners can do more than bob their heads to those classic first seven notes—they can also visualize the song’s structure using a unique computer program that creates geometric representations of musical chords.

Dmitri Tymoczko, a music professor at Princeton University who composes his own music, has devised a method of creating multi-dimensional graphs derived from the individual notes that make up a musical chord. The program can be used to reveal the musical structure of a song, and can also show similarities between disparate musical styles, from Chopin to jazz, Tymoczko said. He reported his findings in the July 7th issue of Science

“I got into the business of music theory because I wanted to understand the advanced tonal music that I most enjoy,” Tymoczko said. “And there wasn’t much guidance to be found.”

In Tymoczko’s model, a point on a graph represents each note in a chord. The number of notes in the chord determines the number of dimensions: If a chord has two notes, the graph is 2-D; if it has three notes, the graph is 3-D, and so on. This algorithm can display how a series of chords in a song are structurally related to one another.

It can also be used to compare the structures of songs from various styles of music.

According to Tymoczko, chords employed most often in Western compositions are based on two principles: First, the chords typically sound similar to one another—for example, songs consist of a series of happy-sounding major chords, instead of a major chord followed by a sad-sounding minor chord. Second, songs are put together so that it is easy for a performer to physically move from one chord to the next.

“We can go through the history of Western music and see how composers over the last 1,000 years have basically all been realizing these same two basic principles,” he said. “In a very systematic way, we can explain under what conditions it’s possible to satisfy those two principles at the same time.”

Tymoczko’s program is not the first that makes geometric models from music, but it is unique because it can map any conceivable chord, not just the standard ones found in Western music. This is possible because the program includes not only the standard notes from an instrument like a piano, but also all the notes that fall in-between, like those accessible on an instrument like a violin.

“Since the time of the ancient Greeks, it has been recognized that some sort of mathematical principles underlie musical phenomena, and it has proved very elusive to put your finger on exactly what that relationship is,” said Julian Hook, a professor of music theory at Indiana University. “But I think we’re getting a little closer to that all the time.”

Tymoczko’s model has already yielded some surprises. Using the program, Tymoczko found that the music of late Hungarian composer György Ligeti, who is famous for his work in 2001: A Space Odyssey, is far more traditional than previously thought. This was surprising because Ligeti’s music is considered to be very radical, not conforming to the standard scale of notes used in Western music. 

An alternate representation of the 3-D three-note chords.

Tymoczko says composers can also use his model to explore the space of musical possibilities.   

“To write really good music takes a lot of hard work and some talent, and geometry isn’t going to make the difference between pretty good music and really good music,” Tymoczko said. “However, understanding how it works does help you get from bad music to pretty good music much more quickly.”

Originally published July 10, 2006


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