The Frequency-domain Grating

It is shown that a complex tone can be •nalyzed into its harmonics by adding to it a series of N -1 similar complex tones with uniformly spaced fundamental frequencies. The harmonics appear in a time-ordered sequence, and this is the basis of a computer music effect employed by composer Jean-Claude Risset. The mathematical derivation involves the summing of a geometric progression to form a slowly varying amplitude factor. The effect is directly analogous to the analysis of white light into its constituent colors by a diffraction grating with Nslits. The temporal ordering of the sequence does not depend upon the relative phases of the harmonics of the complex tone, nor upon the number of tones which is summed• nor is it sensitive toamplitude changes. The order can be changed by a progressive delay of any harmonic in the summed complex tones. Calculations are made to investigate the stability of the effect against randomness in the tones to explain why the effect does not normally appear in vocal or instrumental choruses.