Composing while Computing

"We have heard much about the poetry of mathematics, but very little of it has yet been sung," Henry David Thoreau wrote in 1849.

Music professor Dmitri Tymoczko now begs to differ: According to his research, there's an entire world of complex geometric poetry underlying the language of music.

Tymoczko, together with music professors Ian Quinn and Clifton Callender of Yale and Florida State universities, respectively, has found a way to literally illustrate the close connections between music and mathematical topology. The trio has created a "geometrical music theory" that allows them to interpret music as mathematical shapes and patterns and to map sounds onto corresponding spatial representations or diagrams.

The three researchers are, of course, not the first to notice the connection between math and music. More than 2,500 years ago, Pythagoras supposedly noticed that a blacksmith striking his anvil produced different notes depending on the weight of his hammer. Mathematicians and musicians alike have known for centuries that many common musical harmonies are tied to mathematical ratios of sound frequencies, but Tymoczko was convinced the connections ran even deeper. He wanted to find a rigorous, systematic way of describing the patterns that, with slight variations, occur over and over in music.

He started by looking at basic musical building blocks — ordered sequences of pitches — and thinking about ways he could compare them to each other. The key to these comparisons, Tymoczko found, was breaking pieces of music down to their most essential components.

"The world presents us with a blizzard of information, and in order to make use of this information you have to throw away details," he said.

He found that when he discarded minutiae like an object's octave or the order in which its notes were played, a sequence like C, E, G (the C-major chord) can be considered equivalent to the same chord played in a different octave, or to the same notes played in a different order, like E, C, G. The sets of all such equivalent musical groupings form equivalence classes, literally classes of equivalent objects, which are commonly used in many branches of mathematics.

"After we formalized musical discourse into these equivalence classes, we started to think about their geometric representations," Tymoczko said. "Just like an ordered pair of numbers can be represented as a point in the Cartesian plane, a musical object composed of n notes can be mapped into [n-dimensional space]."

The distance between two musical objects in this mapping corresponds to how much a person has to move their fingers on the piano keyboard to shift from playing one to playing the second.

Mapping music in this mathematical fashion has allowed Tymoczko to analyze and understand the historical progression of music as a form of geometric exploration. He found, for instance, that Western music prior to 1600 was confined within a relatively simple spatial lattice because music of that period was largely restricted to the white keys of the piano. When composers began using the black keys, especially in the 1800s, their musical possibilities expanded to a much more complicated lattice, Tymoczko said, but 19th-century listeners, hearing this new music, did not have a way of representing or thinking about it rationally.

"If you look at how people talk about 19th-century music, they tend to say things like ‘in the 19th-century people started taking opium and doing all these crazy musical experiments,’ ” Tymoczko said. "But what really happened in the 19th century was a fairly systematic exploration of this new musical lattice."

Geometrical music theory isn't just important for understanding history, but may also have crucial implications for the future of music. Up-and-coming composers may hope to create entirely new sounds and styles of music, but they should be wary of wandering outside the geometric space of musical possibilities, Tymoczko said. Composers, in developing new musical languages and genres, should be guided by the new geometric maps just as writers are governed by certain linguistic frameworks, he added.

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"If a poet decided he wanted to invent a new language, in order to do that at all competently he would need to know some linguistics," he said. "To a certain extent, I think that some 20th-century composers created languages without a sufficient understanding of musical linguistics. They disregarded some of the most powerful and profound ways of achieving musical coherence."

Atonal composers, like Arnold Schoenberg and Anton Webern, constructed almost incomprehensible musical languages, Tymoczko said, because they ignored the basic musical principles and patterns which his geometric maps have shown to be common to almost all genres of music.

Tymoczko's maps show the ways that many similar components run through different musical styles. These similarities may counter the conception of Western music as just one possible style out of many equivalent musical genres.

"At one point we thought that the earth was one of a number of similarly inhabited planets, but now scientists think that inhabited planets may be extremely rare," Tymoczko said. "Similarly, it could well be that traditional ways of organizing music are enormously special, and there's not a universe of alternatives to them."

This doesn't mean that there's no room for new music in the topological landscape of the future, but it may mean that the most effective method of musical innovation is recombining familiar elements rather than making radical changes. In other words, composers who choose to follow the course set by Tymoczko's maps may find that the future of their musical originality lies in inventing new ways to use existing musical terminology, not in inventing an entirely new language. But this should not limit the depth and variety of tomorrow's music. In fact, Tymoczko said he thinks the opportunities open to composers are still quite similar to those available to other types of creative artists.

"Poets really aren't in the business of fundamentally changing English syntax, but there's still an infinite variety of poetry to write," Tymoczko said.

While young composers may look to his models to glean innovative musical inspiration from modern mathematics, Tymoczko said he believes that there may be just as much that mathematicians can learn from music.

In his analyses of different pieces of music, Tymoczko was particularly struck by the pictorial representations of two musically unusual pieces by Chopin — the E-minor prelude and Chopin's final composition, a mazurka in F minor.

"These are two pieces that people have really struggled to understand musically," Tymoczko said. "It turns out that they explore a very coherent space, a sort of necklace made with four-dimensional hypercube beads that are linked together by a shared vertex."

What's most alarming about this discovery is that Chopin composed during the first half of the 19th century, a time when mathematicians understood very little about conceptualizing four-dimensional space. Still, Tymoczko said, the incredibly close correlation between Chopin's music and four-dimensional geometry could not possibly be a coincidence. In other words, Chopin had some intuitive understanding of a branch of mathematics that would not be formally expressed or understood until decades after his death.

“It was an incredible point in history," Tymoczko said of the early 19th century. "Humanity's knowledge of the four-dimensional structure could only be expressed in the form of beautiful Romantic music."

It's a discovery that gives new meaning to the belief of mathematician and philosopher Gottfried Leibniz that "music is the pleasure the human soul experiences from counting without being aware that it is counting."

With geometrical music theory, however, we can finally keep count.