This video gives a nice musical illustration of the application of the Discrete Fourier Transformation (DFT) to musical chords. If you read “Fourier Transformation” in connection with music you may, perhaps, think of Digital Audio Processing. But Jason’s video is about the usage of DFT as an aid for writing a piece of music in staff notation.

Chords in the 12-Tone System (pitch class sets) can be described as vectors whose 12 coefficients are 0 or 1, depending on whether a note belongs to the chord in question or not. Despite their prominence 0 and 1 are also complex numbers and hence every chord (seen as a 12-dimensional complex vector) has a Fourier Transform also in the form of a 12-dimensional complex vector. In his research articles Jason established an in-depth understanding of the musical meaning of the Fourier amplitudes and phases. References to this research domain are listed below.

In his setting of Shakespeare's sonnet “not marble” for voice and piano Jason features harmonies with significant energy in the 2nd and 3rd Fourier coefficients. This implies that the succession of their phases coveys relevant information about the harmonic process of the piece.

In Jason’s own words:

This is an original composition “Not Marble” in a video with score and animated analysis showing how I composed the piece. The animation shows a phase space on the 2nd and 3rd coefficients of the DFT on pitch-class sets, with the set-types varying from triads and (016) trichords to 4- and 6-notes sets, all relatively compact in the space (i.e. with high values on the 2nd and 3rd coefficients). This is described in my paper “Decontextualizing Contextual Inversions” in the Proceedings of the Sixth International Conference Mathematics and Computation in Music, MCM 2019 (Springer). The composition uses two kinds of progressions, one derived from an upper-right to lower-left diagonal, and the other on a lower-right to upper-left diagonal, which give sequential progressions by fifth or whole step on each of the chord types (always alternating chords related by inversion).