Labyrinth (4)

The idea for this piece came to me when my daughter and I visited a nearby labyrinth. I had never walked a labyrinth before. There is an endpoint, the center, that you can clearly see. As you follow the path, you find yourself getting closer to that point only to be turned aside.

My daughter started the labyrinth a bit after me, and suddenly we had a very interesting system of three points – two moving, one fixed. The labyrinth was outside but surrounded by curved walls in a courtyard, and there was some great reverb – so of course we started making noises and playing with the echoes. Clearly neither of us understands the proper use of a contemplative labyrinth (in our defense, we were the only ones there). As we walked around, singing and clapping, I had an idea of single tones moving through a labyrinth, like my daughter and me, and the Doppler effect between them would shift the tones into something more interesting.

I did some research into labyrinths, and selected one type as the basis for the piece, the “Cretan” design, which is unicursal (one path) and can be generated quickly by hand. I constrained the labyrinth to a grid and counted the number of nodes along the path including the center (204), to get a sense of how I could start to subdivide the path. I don’t think there’s anything particularly special about trying to stay within this structure – it’s just an interesting constraint. Creating something I was happy with took a lot of experimenting and leaving the strict rules of the initial concept.

A couple of quick definitions regarding note generation: once the labyrinth is created, placed within are an “observer” (moving) and several “emitters” (moving or fixed). Emitters produce a tone periodically. The observer “hears” the tones and its hearing creates the music. Both observer and the moving emitters move through the defined labyrinth at a constant rate (one square per beat).

Once tonic was selected, this pitch was assigned to a fixed emitter at the center point of the labyrinth. Next, odd overtones were assigned to three “voices” that enter the piece a certain number of beats apart (I landed on 29, after experimenting a bit). The first partial being a perfect 5th/dominant, the fifth partial approximately a major 3rd/mediant, and the seventh partial somewhere between the major 6th and minor 7th (I chose the major 6th/submediant to avoid a dominant seventh sound that didn’t feel appropriate here).

The “observer” enters the labyrinth on the first beat, and will reach the center on the 204th beat. The “observer” does not emit a note, but it hears the notes of emitters around it.

The center of the labyrinth is a fixed emitter, producing a note every 6 beats, the fundamental/tonic. This is the first voice. After a fixed duration (29 beats), a moving emitter enters the labyrinth and becomes a second voice. It is emitting the dominant every 4 beats (rather than the true harmonic, I am just using octave equivalence). A second moving emitter (third voice) enters 29 beats after that, emitting one note, the mediant, every 3 beats, and a third emitter (fourth voice) enters after the same duration, emitting one note, the submediant, every 2 beats. Why 29 beats? I was an accident, but it ended up sounding more interesting than other spacings. A good mix of dissonance/consonance.

The emitters are allowed to travel to the center of the labyrinth, with the piece ending when the last emitter reaches the center. Their pitches are shifted to the perception of the observer as noted above. I assigned a unit vector to each node in the path and for each node, evaluate whether the observer and a given emitter are moving toward or away from each other. The observer and moving emitters can only move up, down, left, or right. If moving toward, then the conceptual “Doppler effect” shifts the pitch sharp; if away, then flat. When the observer and each emitter reach the center, they stay. I included a sketch below, showing the labyrinth and the vector at each point (which was also handy for back-checking my math). The starting point and ending point are circled. Relative “velocity” (in nodes per beat) is determined from the instantaneous vector and the dot product is used to determine direction (toward or away). If the observer and emitter are moving in the same direction, the true pitch is heard. If moving at right angles, then the pitch shifts by the a semitone (higher or lower as appropriate). If they are moving head on or in opposite directions, then it shifts by a tone.

For dynamics, I calculated the distance between the observer and each emitter and used that as a general guide to increase/decrease dynamics. It’s not particularly rigorous (I do take the inverse square law into account). If more stuff is closer to the observer, the music gets louder.

Basically, there was a giant spreadsheet.

I tried several different configurations of the space – different starting points for the emitters (came down to personal taste), as many as 7 emitters (it was cacophony), I tried having the “observer” emit its own notes (too droning), I moved voices around between octaves. To create more interesting rhythms, I decided that if an emitter would sound the same note again, I just hold the note for the additional duration.

And that’s pretty much it. It would be an interesting exercise to write variations from the perspectives of each emitter.

You can download the sheet music here: