This piece is based on the logistic map, as simple recurrence relation given as:

The intent for this piece was to explore the map, which has some very interesting behavior as the parameter is varied. Initially the value quickly decreases back to 0, then more slowly and slowly, eventually starting to converge on numbers depending on the initial value . Then, as increases, the value of begins oscillating between two (the bifurcation), and then more values. Around , the value of becomes chaotic, returning to oscillation briefly around and then returning to chaos, finally diverging to infinity beyond . On the face, this seemed to hold plenty of potential for interesting musical structure – a slowly ascending pattern breaking into oscillation, bursting into chaos, briefly pulling back from the chaos, and finally diverging to infinity.

As always, much experimentation and many dead-ends, but here’s the final process that generated this piece:

I evaluated the logistic map for four steps ( through ), setting the first value . This created a vertical “column” of numbers (in my scheme) between 0 and 1.

I still needed to set the parameter to some value, so I evaluated across the interval in steps of . I evaluated this in a spreadsheet (of course), always starting with , evaluating for five steps, and varying the parameter across columns, creating a large 5-by-401 grid of numbers, each within the interval .

Next, I evenly binned the numbers into 12 divisions. Each bin was assigned a pitch class starting with D and processing through descending fifths. So became D, became G, and so on.

Each sequence of notes for a given was then assigned to a single measure. I dropped the initial value because having the same note start each measure was not very engaging, and it’s all the stuff that happens after the initial value that I was interested in. Now I had 401 measures of four pitch sequences.

Because the logistic map doesn’t get interesting until or so, I had a really long stretch of just the same note over and over. Which is how the function works. I suppose if I was true to the concept, I’d just trudge ahead, but intent overrode the generating function. I’m happy to get the aesthetic sense of a slowly evolving recurrence relation without having to sit through 100 bars of the same four-note sequence. I temporally “compressed” the repeated measures according to the somewhat arbitrary function , where is the number of identical measures and is the “compressed” number of measures as played. 34 measures became 6, 11 measures became 4, etc.

If all of that was a bit dense, here is a nice color-coded visual of the piece:

Each color represents a pitch class.

Another change is that rather than play the same note, it’s bounced among octaves, and a constant rhythmic scheme is applied across the entire piece. Every four notes you move to a new parameter. The articulation and dynamics were chosen based loosely on the behavior of the map in that region.

That’s pretty much it. Sheet music below. Enjoy!