From 3D Shapes to Tonal Harmonies and Back


Session Title:

  • Composition (The Next Generation)

Presentation Title:

  • From 3D Shapes to Tonal Harmonies and Back



  • Abstract

    There is a natural way to transform any 3-dimensional shape into a musical harmony, such that nearby points go to pitches which are harmonically related in a simple way, and the transformation can be inverted under certain assumptions. For each point making up the shape and having integer-valued coordinates x, y, z, form a sine wave with frequency 2x3y5z and sum these sine waves. More generally, one can have any real-valued density distribution defined on the Z3 point lattice and use its values as the amplitudes of the corresponding partials. Still more generally, the density can be colored or complex valued. The hue or complex phase angle is taken to be the phase of the sine wave. The 3D transformation can be generalized to any number of dimensions.

    Because 2, 3, and 5 are distinct primes, anything so transformed can uniquely be transformed back, provided one could keep infinite precision. For spectra with imprecise or irrational frequencies, as is always the case in practice, inversion appears to be impossible. There are,
    however, heuristic methods to invert the transformation, like the bounding box method, the tonal centre method, the multiple tonal centre method, and the charge distribution method. I have conjectured that something like this actually happens in the auditory system. In music theory the transformation neatly explains the diatonic scale and the major/minor symmetry. The inverse transformation provides an analysis method for tonaI and microtonal music. The direct transformation may be used as a compositional tool. For example, one can put a cellular automaton to operate on the Zn lattice, and transform the state pattern into music. In prosthetics, the direct transformation gives a potential way to realize virtual vision for the blind, and the inverse transformation a way to visualize music or any sound for the deaf.