[track art]

Tesseract Fugue

2021/08/26 fugue MSK 013 MSK 007
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Ambient piece based on the symmetry of a tesseract, or four-dimensional hypercube.

Most of the composition was done using constraint logic programming with a fair bit of human supervision. After writing, by hand, 16 different rhythms for a four-bar theme, I used constraint logic programming to write the themes themselves, with some attempt at following counterpoint rules (though not the full classical rule set) such that for each of the 24 faces of the hypercube, the four themes for that face's vertices would sound okay together.

Then (using constraint programming again) I chose a path that would doubly cover the tesseract's faces: start on one face, move to another face sharing an edge with this one, and so on, until we've visited all 24 faces exactly twice each. That generates a sequence of 48 sets of four themes, such that each set in the sequence shares two themes in common with the previous one, and they're all sets that correspond to faces of the hypercube, hitting each face twice. I also had the additional requirement of returning to the start after the end of the sequence. That's a lot of constraints and it's why I needed a computer to choose the sequence, but it meant a very balanced and symmetrical selection of themes.

Each four-bar segment of the piece corresponds to one of the steps in the sequence. I did another round of constraint programming to assign the four themes in each segment to four voices, so that no voice would repeat the same theme in two succcessive segments and every theme would be played exactly three times on each voice in the 48-segment sequence. Finally, I duplicated the first four bars at the end (so that the total length of the piece is actually 49 segments), and applied a sequence of 49 modal transpositions generated by recursing through two levels of a hand-chosen seven-step rule (see my earlier piece on fractal chord progressions).

The overall result is that although there are really many distinct themes involved, the general feel of the piece is repetitive. The extremely high symmetric of the tesseract means that there's no overall progress. Like many of my ambient pieces, it's meant more to present a static texture than to tell a story.

The MIDI sequence was rendered on a modular synthesizer featuring the North Coast Synthesis Middle Path VCO and Leapfrog VCF. Slightly different patch for each of the four voices, and I also added some background pads from a Roland D-05.

PDF sheet music.