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Journeys in musical space

Researchers map out the geometric structure of music.

To most of us, a Mozart piano sonata is an elegant succession of notes. To composer and music theorist Dmitri Tymoczko of Princeton University in New Jersey and his colleagues, it is a journey in multidimensional space that can be described in the language of geometry and symmetry.

In a paper appearing in Science1, the trio offer nothing less than a way to map out pitched music (that not constructed from unpitched sounds such as percussion), whether by Monteverdi or Motörhead.

Credit: Getty

Mathematician Rachel Wells Hall of Saint Joseph’s University in Philadelphia, Pennsylvania, says that the work opens up new directions in music theory, and could inspire composers to explore new kinds of music. It might even lead to the invention of new musical instruments, she says.

Musical spaces

Although the work uses some fearsome maths, it is ultimately an exercise in simplification. Tymoczko, with colleagues Clifton Callender and Ian Quinn, has looked for ways to represent geometrically all the equivalences that musicians recognize between different groups or sequences of notes — so that for example C-E-G and D-F#-A are both major triads, or C-E-G played in different octaves is considered basically the same chord.

By recognizing these equivalences, the immense number of ways to arrange notes into melodies and chord sequences can be collapsed from a multidimensional universe of permutations into much more compact spaces. The relationships between ‘musical objects’ made of small groupings of notes can then be understood in geometric terms by mapping them onto the shape of the space. Musical pieces may be seen as paths through this space.

It might sound abstract, but the idea clarifies things that composers and musicologists have been wrestling with for centuries. The researchers say that music interpretation involves throwing away some information so that particular musical structures can be grouped into classes. For example, playing ‘Somewhere Over the Rainbow’ in the key of G rather than, as originally written, the key of E flat, involves a different sequence of notes — but no one would say that it is a different song on that account.

Transforming sounds

The researchers say there are five common kinds of transformation that are used in judging equivalence in music, including octave shifts, reordering of notes (for example, in inversions of chords, such as C-E-G and E-G-C), and duplications (adding a higher E to those chords, say). These equivalences can be applied individually or in combination, giving 32 ways in which, say, two chords can be considered ‘the same’.

Such symmetries ‘fold up’ the vast space of note permutations in particular ways, Tymoczko explains. The geometric spaces that result may still be complex, but they can be analysed mathematically and are often intuitively comprehensible.

“When you are sitting at a piano”, he says, “you are interacting with a very complicated geometry.” In fact, composers in the early nineteenth century were already implicitly exploring such geometries through music that could not have been understood using the mathematics of the time.

In these folded-up spaces, classes of equivalent musical objects — three-note chords, say, or three-note melodies — can each be represented by a point. One point in the space that describes three-note chord types (which is cone-shaped) corresponds to major triads, such as C-E-G, another to augmented chords (in which some notes are sharpened by a semitone), and so on.

A new route to theory

Where does this musical taxonomy get us? The researchers show that all kinds of musical problems can be described with their geometric language. For example, it provides a way to evaluate how similar different sequences of notes or chords are, and thus whether or not they can be regarded as variations of a single musical idea.

“We can identify ways in which chord sequences can be related that music theorists haven’t noticed before”, says Tymoczko. For example, he says the approach reveals how a chord sequence used by Claude Debussy in 'L’Après-Midi d’un Faune' is related to one used slightly earlier by Richard Wagner in the prelude to 'Tristan und Isolde' — something that isn’t obvious from conventional ways of analysing the two sequences.

Debussy is unlikely to have known about this mathematical relationship to Wagner’s work. But Tymoczko says that such connections are bound to emerge as composers explore the musical spaces. Just as a mountaineer will find that only a small number of all the possible routes between two points are actually negotiable, so musicians will have discovered empirically that their options are limited by the underlying shapes and structures of musical possibilities.

For example, he says, composers such as Frederic Chopin in the early nineteenth century began to look for shortcuts to the traditional ways of moving between two nearby keys, say C major to E major. “Music theorists have tended to regard the nineteenth-century experiments in harmony as whimsical and unprincipled”, says Tymoczko. “But we can now see it as taking advantage of certain geometric features of chord space.”

The scheme also supplies a logic for analysing how a technique called voice leading works in chord progressions. This describes the way in which a sequence of chords with the same numbers of notes can be broken apart into parallel melodic lines. For example, the progression C-E-G to C-F-A can be thought of as three melodic lines: the E moves to F, and the G to A, with a constant C root. Finding efficient and effective voice-leading patterns has been challenging for composers and music theorists. But in the geometric scheme, a particular step from one chord to another becomes a movement in musical space between two points separated by a well-defined distance, and one can discover the best routes.

This is just one of the ways in which the new theory could not only illuminate existing musical works but could point to new ways of solving problems posed in musical composition, the researchers claim.

Credit: Getty


  1. Callender, C., Quinn, I. & Tymoczko, D. Science 320, 346-348 (2008).

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Ball, P. Journeys in musical space. Nature (2008).

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