Missouri Classical Music News and Notes

Remarkably, in the 12-tone system of notes, these are precisely the chords that Pythagoras identified almost 2,500 years ago: the chords that sound intrinsically harmonious. Far from arbitrary or haphazard, scales and chords come close to being the unique solutions to the problem of creating two-dimensional musical coherence. Contrary to the hopes of generations of avant-garde composers, it follows that the goal of developing robust alternatives to tonality may be extremely difficult, if not impossible, to achieve.

The shapes of the space of chords we have described also reveal deep connections between a wide range of musical genres. It turns out that superficially different styles--Renaissance music, classical and Romantic music, jazz, rock, and other popular forms--all make remarkably similar use of the geometry of chord space. Traditional techniques for manipulating musical scales turn out to be closely analogous to those used to connect individual chords. And some composers have displayed a profound understanding of the higher-dimensional geometry of musical chords. In fact, one can argue that Romantic composers such as Chopin had an intuitive feel for non-Euclidean higher-dimensional spaces that exceeded the explicit understanding of their mathematical contemporaries. . . .

There are in fact large families of geometrical spaces corresponding to a wide range of musical terms, some of which are considerably more exotic than those described here. (For instance, three-note chord types--such as "major chord" or "minor chord"--live on a cone containing two different flavors of singularity.) Seen in the light of this new geometrical perspective, a wide number of traditional music-theoretical questions become tractable.