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# Iteration

*Program Note*

for alto saxophone, marimba and electronic sound

by Reginald Bain

*Iteration* is now available on the RoseWind Duo's 2016 Equilibrium Records release *Devils Garden/Angels Landing* (EQ 161).
{Spotify}

Commissioned by the RoseWind Duo, ** Iteration** (2012) for
saxophone, marimba and electronic sound is a musical realization of the
logistic map
equation.

Shown in Figure 1, this equation explores the boundary between order and chaos. The title of the
work refers to the feedback-loop nature of the equation, that is, the
result of the current calculation (*x*_{n+1}) becomes the
input (*x*_{n}) for the next calculation. Each repetition
of the calculation is referred to as an *iteration*. This equation has many
interesting properties including a propensity for oscillation,
bifurcation, period doubling, geometric convergence, sudden transition
to chaos, and self-similarity. Musical analogues of these properties are explored in the work. The electronic sound part
presents twenty-one electronic marimba canons that are generated from the equation.
These canons are presented over a rhythmic noise-pulse track that is,
literally, a *sonification* of the equation (i.e., a mapping of its output in the audio domain).
The saxophone and live marimba bring order to whole, echoing and
responding to the algorithmically-generated musical surface, in what
the composer hopes is an exciting encounter with mathematics and
technology.

**Figure 1**. Bifurcation (orbit) map for the iterative logistic equation

Credit: Wikipedia

The world premiere was given by the RoseWind Duo at the 2012 Percussion Arts Society International Conference (PASIC) in Austin, TX.

For a general introduction to chaos theory, and the logistic map equation, see Gleick 1987. Bennett 1996 provides Csound code for sonifying the equation in the audio domain. Cycling '74 2016 provides Max/Javascript code for sonifying the equation in MIDI domain.

Bennett, Gerald. 1996. *Chaos, Self-Similarity, Musical Phrase and Form*. Available online at:

http://www.gdbennett.net/texts/Chaos.pdf

Cycling '74. 2016. *Max 7 JavaScript Tutorial 1: Basic JavaScript*. San Francisco: Cycling '74. Available online at: <https://docs.cycling74.com/max7/tutorials/javascriptchapter01>.

Gleick. James. 1987. *Chaos: Making a New Science*. New York: Penguin Books. {GB}

Updated: March 11, 2017