Assemblying the Geometry of Music

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Approximately 2000 years ago, Pythagoras discovered that pleasing musical intervals can be described using simple mathematical ratios. In the Middle Ages emerged musica universalis or "music of the spheres", a philosophical idea that the proportions of the movement of celestial bodies such as the sun, moon and planets could be viewed as a form of music.

Today, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University - have described something called "geometrical music theory". The theory basically translates the language of musical theory into geometry. They took scales, notes, chords and rhythms and categorized them into separate families. Then they found a way to assign mathematical structure to these families - they are represented by points in complex geometrical spaces, similar to "x" and "y" coordinates in simple algebra.

Different types of categorization produce different geometrical spaces. This in turn reflects the different ways in which musicians have understood music over the past couple of centuries. According to them, this way of analysis will allow musicians to enter the deep and profound hallways of musical mystery.

The method they have used lets them analyze two different types of music - ie. many types of Western and maybe even non-Western music.

"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "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."

What could we possibly do with this?

"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization 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. But to me, 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."

To understand music, authors argue that you have to discard information. For example, if you were a musician and played a "C" on a piano, followed by the note "E" above it, and then by note "G" above that, you would use many different terms to describe this such as "an ascending C major arpeggio" or a "C major chord". The authors on the other hand provide a unified mathematical framework for relating different types of musical aspects.

They also use five different ways of categorizing collections of notes that are similar but not identical. They called these "OPTIC symmetries", with a different letter of this word representing a different way of ignoring musical information - for example, octave the notes are in, in which order they are or how many times each note is repeated. The five symmetries are then combined with each other to produce a cornucopia of different musical concepts. In this way, both the musicians and the researchers are able to reduce music to their mathematical essence.

When notes are translated into numbers and then again into the language of geometry, the result is a broad menagerie of geometrical spaces - each one of these spaces is inhabited by a species of a geometrical object.

This method could possibly explain if there are other different scales out there that have not yet been discovered.

"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."

Moreover, this mathematical sequence could also help in investigating the differences between different musical styles.

Music cannot be separated from complex mathematics. It has been shown that those who are good in mathematics tend to excel in music. Being able to read music from a mathematical perspective is fascinating, even though it would probably require a lot of practice for an ordinary person to read the music in a mathematical form.