Date of this Version
J. Ashley Burgoyne and Lawrence K. Saul, "Visualization of Low Dimensional Structure in Tonal Pitch Space", . September 2005.
In his 2001 monograph Tonal Pitch Space, Fred Lerdahl defined a distance function over tonal and post-tonal harmonies distilled from years of research on music cognition. Although this work references the toroidal structure commonly associated with harmonic space, it stops short of presenting an explicit embedding of this torus. It is possible to use statistical techniques to recreate such an embedding from the distance function, yielding a more complex structure than the standard toroidal model has heretofore assumed. Nonlinear techniques can reduce the dimensionality of this structure and be tuned to emphasize global or local anatomy. The resulting manifolds highlight the relationships inherent in the tonal system and offer a basis for future work in machine-assisted analysis and music theory.
Additional Filestps1.qt (5755 kB)
QuickTime movie: Dimensions 3 and 4 along unrolled polar axis after MDS
tps2.qt (4358 kB)
QuickTime movie: Dimensions 3 and 4 along unrolled polar axis after MVU
Date Posted: 15 November 2005