Scientists make music into mathematical shapes

The two-note chords form a Mobius strip, whose boundary is a 'trefoil knot'

We know it has pitch and beat. But it turns out that music has a distinct shape too.

In search of the last chord and winter colds In the wake of centuries of effort to seek deep connections between music and mathematics, a team today concludes that music does have geometry.

More than 2000 years ago, Pythagoras discovered that pleasing musical intervals could be described using simple ratios. And the idea of the so-called musica universalis or "music of the spheres" emerged in the Middle Ages which said that the proportions in the movements of the celestial bodies - the sun, moon and planets - could be viewed as a form of music, inaudible but perfectly harmonious.

Now, three music professors - Clifton Callender at Florida State University, Ian Quinn at Yale University, and Dmitri Tymoczko at Princeton University - have devised a new way of analysing and categorising music to reduce musical works to their mathematical essence, suggesting that mathematics is a more fundamental language of nature than music.

Following a pioneernig effort by Prof Tymoczko in 2006, the trio has now outlined a method called "geometrical music theory" in the journal Science that they say can turn music into shapes.

"To me," Prof Tymoczko says "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."

"Our methods are not so great at distinguishing Aerosmith from the Rolling Stones," Tymoczko said. "But they might allow you to visualise some of the differences between John Lennon and Paul McCartney. Paul McCartney's tunes tend to look more traditional, John Lennon's tunes tend to be a little more "rock" - violating more of the traditional rules.

And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.

"The team can take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families."

They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.

This achievement, they expect, will allow researchers to analyse and understand music in much deeper and more satisfying ways. "The music of the spheres isn't really a metaphor - some musical spaces really are spheres," said Tymoczko, who like Callender is also a composer.

"The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."

"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualisation tools - imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."

The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the department of mathematics and computer science at St. Joseph's University in Philadelphia. In Science, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."

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