Journeys in musical space
[This is one of the most stimulating things I’ve read for some time (not my article below, published on Nature’s online news site, but the paper it discusses). The paper itself is tough going, but once Dmitri Tymoczko explained to me where it was headed, the implications it opened up are dizzying – basically, that music is an exploration of complex geometries, giving us an intuitive feel for these spaces that we probably couldn’t get from any other kind of sensory input.]

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 and his colleagues Clifton Callender and Ian Quinn, it is a journey in multidimensional space that can be described in the language of geometry and symmetry.

In a paper in Science, the trio offer nothing less than a way of mapping out all of pitched music (music which is not constructed from unpitched sounds like percussion), whether it is by Monteverdi or Mötörhead.

Commenting on the work, mathematician Rachel Wells Hall of Saint Joseph’s College in Philadelphia says that it 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.

Although the work uses some fearsome maths, it is ultimately an exercise in simplification. Tymoczko and colleagues have looked for ways of representing 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 possible ways of arranging 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 may sound abstract, but the idea brings together things that composers and musicologists have been trying to do in a fragmentary manner for centuries. The researchers say that all 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 is going to say it is a different song on that account.

The Princeton researchers say there are five common kinds of transformation like this 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 different 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’re sitting at a piano”, he says, “you’re 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.

Where does this musical taxonomy get us? The researchers show that all kinds of musical problems can be described using their geometric language. For example, it provides a way of evaluating how related 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 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.

Clearly, Debussy couldn’t have know of 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.

“Music theorists have tended to regard the nineteenth-century experiments in harmony as unmotivated whimsy”, says Tymoczko. But his geometric scheme suggests that they were much more rational than that, governed by rigorous rules that their new approach can now uncover.

For example, the scheme supplies a logic for analysing how so-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.

Reference
1. Callender, C. et al. Science 320, 346-348 (2008).