of a particular rule may be dependent on a sub-rule—leading to more organic, developing forms. Hanspeter Kyburz’s (1960–) Cells for saxophone and ensemble is an example. Martin Supper38 described Kyburz’s use of L-Systems, using results from 13 generations of L-System rewrites to select pre-composed musical motifs. Like Hiller before him, Kyburz uses algorithmic composition techniques to generate and select musical material for the preparation of instrumental scores. However, the listener is probably unaware of the application of software in the composition of such music.

Transitioning L-Systems: Tramontana. As I tend to write music that is concerned with development and transition, my use of L-Systems is somewhat more convoluted. My own Tramontana (2004) for viola and computer14 uses L-Systems in its concluding section. Unlike normal L-Systems, however, I employ Transitioning L-Systems, my own invention, whereby the numbers returned by the L-System are used as lookup indices into a table whose result depends on transitions between related but developing material. The transitions themselves use Fibonacci-based “ folding-in” structures where the new material is interspersed gradually until it becomes dominant; for example, a transition from material 0 to material 1 might look like Figure 6.

In the case of the concluding section of Tramontana, there is slow development from fast, repeated chords toward more and more flageoletst on the C and G strings. Normal pitches and half flageoletsu begin to dominate, with a tendency toward more of the former. At this point, flageolets on the D string are also introduced. All these developments are created with transitioning L-Systems. The score (see Figure 7 for a short extract) was generated with Bill Schottstaedt’s Common Music

 

t Familiar to guitarists, flageolets, and harmonics are special pitches achieved by touching the string lightly with a left-hand finger at a nodal point in order to bring out higher frequencies related to the fundamental of the open string by integer multiples.

u Half flageolets are achieved by pressing the string, as with a full flageolet, but not at a nodal point; the result is a darker, dead-sounding pitch.

CuRTIS RoADS, 1996
It takes a good
composer to design
algorithms that
result in music
that captures
the imagination.

Notation software, taking advantage of its ability to include algorithmically placed nonstandard note heads and other musical signs. Perhaps worth noting is that even before I began work with computers, I was already composing in such a manner. Now, with slippery chicken algorithms, these structures can be programmed to generate the music, test, re-work, and re-gen-erate. A particular advantage of working with the computer here is that it is a simple matter to extend or shorten sections, something that would, with pencil and paper, be so time-consuming as to be prohibitive.

 

Musical Example: Ligeti’s Désordre György Ligeti (1923–2006) is known to the general public mainly through his music in several Stanley Kubrick films: 2001: A Space Odyssey, which included Lux Aeterna and Requiem (without Ligeti’s permission, prompting a protracted but failed lawsuit); The Shining, which included Lontano; and Eyes Wide Shut, which included Musica Ricercata.

After leaving his native Hungary in
the late 1950s, Ligeti worked in the
same studios as Cologne electronic
music pioneers Karlheinz Stockhau-
sen and Gottfried Michael Koenig
though produced little electronic mu-
sic of his own. However, his interest in
science and mathematics led to sev-
eral instrumental pieces influenced
by, for example, fractal geometry and
chaos theory. But these influences did
not lead to a computer-based algo-
rithmic approach.v He was quoted in
Steinitz37 saying, “Somewhere under-
neath, very deeply, there’s a common
place in our spirit where the beauty of
mathematics and the beauty of music
meet. But they don’t meet on the level
of algorithms or making music by cal-
culation. It’s much lower, much deep-
er—or much higher, you could say.”
Nevertheless, as a further example,
we shall consider the structure of Györ-
gy Ligeti’s Désordre from his first book
of Piano Etudes for several reasons:

Structures. The structures of Désordre are deceptively simple in concept v Ligeti’s son, Lukas, confirmed to me that his father was interested conceptually in computers, reading about them over the years, but never worked with them in practice.