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That the piano, despite its simple, linear appearance, is really a gateway to a non-Euclidean space may come as a surprise. But that's just what music is – and a complicated geometrical space known as an orbifold could provide composers with a logical way of exploiting the mathematics behind chords and melody.
Composer Dmitri Tymoczko of Princeton University compressed the many facets of music into a geometrical space and found the mathematical rules that seem to govern Western music. Tymoczko suggests that famous composers have traditionally valued a kind of harmonic consistency that relates to keeping within a local region of the orbifold. However, he is reluctant to have his study viewed as the be all and end all guide to successful musical composition.
"I'm not in the business of laying down 'rules' or telling people what's good and bad in music," he said. "What I'd like to do is to be able to say: ‘if you want to achieve this musical effect, then you should try this technique'."
In the orbifold, the notes C to B are pitch-wise numbered from zero to 11. Chords, which are clusters of notes, are then multisets of pitch classes, while the musical terms ‘transposition' and ‘inversion' are translate to the mathematical terms ‘translation' and ‘reflection' respectively.
If a chord divides an octave into equal parts it is termed tranpositionally symmetrical (T-symmetrical). Inversionally symmetrical (I-symmetrical) chords, meanwhile, are defined by having an equal number of semi-tones between notes of, for example, a three-note chord. If there are multiple voices playing the same note in a chord, then it is labelled permutationally symmetrical (P-symmetrical).
Perfect symmetry generally produces a dissonant sound. Perfect T-symmetry, for example, results in a tritone (e.g. C F-sharp), which is an eerie chord once denounced by the medieval Church as ‘the Devil's interval', and these days is favoured by heavy metal bands.
However, Tzmoczko found that nearly T-symmetrical chords include those traditionally considered to be most pleasing to the ear, such as the perfect fourth and fifth.
"This has to do with the fact that our sense of pitch is logarithmic with respect to fundamental frequency," he said.
Nearly T-symmetrical chords also lend themselves to efficient 'voice leading', which is a contrapuntal technique favoured by traditional Western music. An example is a round where several people sing the same melody starting at different times. "Row-row-row Your Boat" is a round familiar to most school-children.
Nearly P-symmetrical chords similarly accommodate voice leading. These chords contain notes that are clustered close together (for example, E F G-flat) and are characteristic of modern atonal composition, such as the music of Georgi Ligeti, whose dramatic work featured in the soundtrack to the film 2001: A Space Odyssey.
Nearly I-symmetrical chords are often also nearly T-symmetrical (as in the case of the C-major chord, C E G), or nearly P-symmetrical. 19th century composers such as Schubert, Wagner and Debussy tended towards near-I-symmetry.
"My geometric models show us that there are important strands of commonality running throughout the last thousand years of music," Tymoczko said. "I think these strands have, to some extent, been overlooked, largely because we didn't have a very good vocabulary for talking about the relation between harmony and counterpoint."
The report was published in the July 7 issue of Science. According to a press release by Princeton University, it is the first paper on music theory to appear in the U.S. journal in its entire 127-year history.
