Geometrical approximations to the structure of musical pitch

Suggests that rectilinear scales of pitch can account for the similarity of tones close together in frequency but not for the heightened relations at special intervals that arise when the tones are interpreted musically. Increasingly adequate accounts of musical pitch are provided by increasingly generalized, geometrically regular helical structures: a simple helix, a double helix, and a double helix wound around a torus in 4 dimensions or around a higher-order helical cylinder in 5 dimensions. A 2-dimensional "melodic map" of these double-helical structures provides for optimally compact representations of musical scales and melodies. A 2-dimensional "harmonic map," obtained by an affine transformation of the melodic map, provides for optimally compact representations of chords and harmonic relations. It is isomorphic to the toroidal structure that C. L. Krumhansl and F. J. Kessler use to represent the psychological relations among musical keys. (127 ref)