Tesseract: The artist's introduction to n-dimensional spaces

An artist's function is to take a set of elements (oil paints, cut-up audio, language) and embed them in structures. As an experiment, I have written a piece that stimulates the artist's sense of space, and forces them to consciously be mathematicians.

Tesseract is a piece in four movements for 8 parts. The artist chooses 4 dimensions that can be altered on his or her chosen elements, and throughout Tesseract, the artist will trace out a shape that shares its name with the piece: the 4-dimensional analog of the cube. In each movement, each part alters one dimension, as indicated in the piece, from either 0 to 1 (minimum to maximum) or from 1 to 0 (maximum to minimum.) Group 1 (parts 1-4) will move from the point (0,0,0,0) to (1,1,1,1) and group 2 (parts 5-8) will move from the point (1,0,0,0) to (0,1,1,1), and the paths all the parts take expresses every edge of the tesseract as an embedding in the artist's space.

A quick example of a musical interpretation of Tesseract:

• 8 musicians playing the same (or different) amplified (or unamplified) instrument, each playing one of the 8 parts
• The first dimension they choose is pitch. 0 corresponds to the lowest pitch their instrument produces, and 1 corresponds to the highest pitch their instrument produces (though they could pick the maximum and minimum pitch to be any set range)
• The second dimension they pick is tempo, where 0 corresponds to very infrequently occurring note occurrences, and 1 corresponds to very frequent note occurrences.
• The third dimension they choose is a panning oscillation, where 0 oscillates very slowly from left to right and 1 oscillates very quickly from left to right. 
• The fourth dimension they choose is distortion, where 0 is very clean, and 1 is very distorted

In the first movement,

1. Player 1 plays (+,0,0,0): he steadily increases pitch from minimum to maximum while keeping a very infrequent (slow) tempo, a slow panning oscillation, and a very clean signal.
2. Player 2 plays (0,+,0,0): he steadily speeds up tempo while keeping the other dimensions at their minimum.
3. Player 3 plays (0,0,+,0): increasing panning oscillation while maintaining the other dimensions at 0.
4. Player 4 plays (0,0,0,+): increasing distortion while keeping the other dimensions at 0.
5. Plater 5 plays (-,0,0,0): decreasing pitch from highest to lowest, while maintaining the other dimensions at 0.
6. Player 6 plays (1,0,0,+): keeping pitch at a maximum while increasing distortion and maintaining minimum tempo and panning.
7. Player 7 plays (1,0,+,0): keeping pitch at a maximum while increasing panning oscillation and maintaining minimum tempo and distortion.
8. Player 8 plays (1,+,0,0): keeping pitch at a maximum while increasing tempo and maintaining minimum panning oscillation and distortion.

The other three movements continue in this fashion, and I leave it to the player to think about all the complex relationships between the parts that emerge from this structure.

Though I could attempt to explain the structure of a tesseract, it's much easier and more informative of me to refer you to this video of ProfessorElvisZap, who does an amazing job visually explaining higher dimensional geometry: 

Though this is neither cutting edge mathematics, nor cutting edge music, it's a great way to shake up my writing process.