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.