To the Edge (1993)                                                                                Gary Lee Nelson (b. 1940)

This piece was composed with a mathematical model from the new field of "chaos."   Chaos theory [ James Gleick.   Chaos: Making a New Science , Viking Press, New York, 1987] was developed to describe continuous phenomena that appear to be random but actually contain a substantial amount of complex order.  

"To the Edge" is based on a simple equation:   x = p*x*(1-x) .   However, the behavior of this equation is decidedly not simple.   In this equation, x ranges from 0.0 to 1.0 and p ranges from 0.0 to 4.0.   After each computation of the equation the resulting x is "fed back" to be used in the next computation.   The number series' that emerge vary in complexity according to the value given to p .   In general, complexity of increases with p but there are "islands" of simplicity.   For me, the most interesting range for p is 3.5 to 4.0 shown in the graph below:

Figure 1

If we take a value for p near the left margin we encounter four lines.   Mapped into music these values of x represent a set of four pitches that will be repeated without variation as long as p remains fixed.   As p increases, we move to the right.   First we encounter a split where the four notes become eight.   Further on, the graph becomes dense and the musical patterns increase in complexity.   The vertical white bands represent areas of relative simplicity (three notes, six notes).   The dense regions appear random but they contain many small identifiable patterns that become motives when they are expressed musically.   The form of this piece is created by moving in a single gesture from left to right, proceeding "to the edge."

In mapping the graph in Figure 1 onto music, each point could become a note.   Obviously, the density of points is too great for a solo marimba piece so the graph was replotted with fewer points as shown in Figure 2.

Figure 2.

Even with this thinning of the points the graph remained unsuitable for transformation into music.   If all of the points are presented as they are computed, a uniform pulse would occur and the result would be without rhythmic character.   Further thinning is necessary.

Another function was created to be supperimposed on the graph of Figure 2.   This function (Figure 3) was made with a small number of points computed from the x = p*x*(1-x) equation.   These points were used to create the boundries of a mask that further thinned the points shown in Figure 2.   The points in the shaded area were discarded and the points that fell within the boundries of the mask remained for musical rendering.

Figure 3.

The chaos function was implemented in MAX, an object-oriented programming system with an interactive graphical interface. [ Puckette, M. Zicarelli, D., MAX: An Interactive Graphic Programming Environment, Opcode Systems,   Menlo Park, CA (1992).]  Notes generated with a MAX patch were recorded in to a MIDI file and transcribed into standard notation using Finale.

James Gleick.   Chaos: Making a New Science , Viking Press, New York, 1987.

Puckette, M. Zicarelli, D., MAX: An Interactive Graphic Programming Environment, Opcode Systems,   Menlo Park, CA (1992).