The opening chord of Richard Wagner's Tristan und Isolde (in the third bar of the score pictured) caused quite a stir when the opera was premièred in 1865. The daring combination of tone intervals — the dissonant augmented fourth; the dynamic, imperfectly consonant major third; the highly consonant perfect fourth — came to be seen as a key change in attitudes to harmony, an overture to the atonality of much subsequent classical music.
The question of how simultaneous notes combine to form aesthetically pleasing (or displeasing) harmonies, and how each harmony best evolves to the next one — a technique known as counterpoint, or voice leading — has exercised music theorists for centuries.
Dmitri Tymoczko (Science 313, 72–74; 2006) is the latest to attempt to systematize such relationships mathematically, with the goal of establishing a universal 'geometry' of musical chords.
A musical octave separates one note from another similar-sounding note of double (or half) the frequency. In the Western tradition, each octave is divided into 12 distinctive tones. Tymoczko therefore uses a simple logarithmic expression to convert the fundamental frequency of each note to a real number in 'pitch space', starting from C (which is 0) and Csharp (1) and going to A (9), Bflat (designated as t) and B (e). After B comes C, which, as the start of the next octave, is 0 again.
The geometrical distance in pitch space between two chords can then be assessed through a 'voice-leading size', a relative measure of how much the constituent notes in one chord must change to make the second. The results can be used to form a geometrical space, called an orbifold, on which the separation between any two chords is proportional to their voice-leading size.
Because of the cyclical nature of the octave, the orbifold loops back on itself: for two-tone chords, for example, it resembles a Möbius strip — with a half-twist along its circumference — whose boundaries, for the purposes of voice leading, act like a mirror. An equivalent picture applies in higher dimensions (corresponding to chords with more notes), creating a unified framework for considering all possible chord combinations.
So what does this teach us? It seems that many of the chord progressions favoured by composers in constructing pleasing counterpoints exploit symmetries of the orbifold's geometry. Chords that divide an octave almost evenly are said to be near transpositional symmetry, and include the familiar consonant chords. These chords cluster into the centre of the orbifold. Nearly permutationally symmetric chords, on the other hand, which involve notes that are close together and therefore sound dissonant, collect towards the boundaries.
A third class of chord, known as nearly inversionally symmetric, is scattered throughout the orbifold. Such chords are prominent in both tonal and atonal music — and particularly in the works of such nineteenth-century pioneers as Richard Wagner.
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Webb, R. Calculated tones. Nature 442, 149 (2006). https://doi.org/10.1038/442149a