"6 Integers": Making Music with Mathematica
On January 25, 2000, concertgoers at the University of
Cincinnati (UC) College Conservatory of Music heard the world premiere of
"6 Integers," a Mathematica-generated musical composition by Carl
a freshman at UC. Computed with Mathematica and rendered using a
synthesizer playing piano and marimbas, "6 Integers" premiered to warm
applause and positive responses. Some listeners called it simply
"awesome," while other composers recognized it as "postminimalism" and "a
fascinating new direction" in algorithmic composition. The former
description is particularly apt given that much of McTague's musical
inspiration is derived from the works of 1960s minimalist composers such
as Philip Glass and Steve Reich.
The piece "6 Integers," as created by McTague, is an expression in
Mathematica (just as x2 + 1 + (p(x))
is an expression). It aspires to
the ideal, espoused by the architect Howard Roark in The Fountainhead by
Ayn Rand: "Nothing can be reasonable or beautiful unless it's made
by one central idea, and the idea sets every detail." According to
McTague, "every detail" (except the timbre of the piano and
marimbas) of "6 Integers" is determined by a simple, "central" formula; neither chance nor
human intervention play any role in the work. The entire piece, including
its double binary structure, arises, essentially, from modulo division and
the integers 1 through 6. Over the course of the composition, a single,
coherent process explores these six integers by means of 5,292 notes and four
independent channels of audio. It is the first full piece generated by the
Hierarchical Functional Inheritance Model (HFIM).
The Hierarchical Functional Inheritance Model is the basis of the "very promising
mathematically-driven approach to composition" that Carl McTague has been
working on for several years now, says Mara Helmuth, director of the
UC Center for Computer Music
studio. After composing by hand for some years, McTague embarked on an
exploration of a new system of musical composition--a system based on the
interactions of simple mathematical patterns. He produced tables and
seemingly endless strings of numbers and permutations by hand.
"Eventually, I realized that the digital computer could be of immense use
in my work," he said. McTague, who has been using Mathematica for three
says he chose to do his musical work in Mathematica because "I
mathematical terms and Mathematica lets me implement my ideas in
mathematical terms." Although his initial progress was slow, over time
McTague developed the HFIM, which is described in a paper on his
web site, to help facilitate his work.
The HFIM provides an abstract computational model for creative work,
enabling McTague to express musical structures in abstract mathematical
terms. The computer then expands these expressions into simple, explicit
instructions for performance. Says Helmuth, "One of the wonderful things
about algorithmic composition is that it can allow people to experience
mathematical and scientific ideas in the realm of the arts and senses,
increasing modes of understanding for the nonscientist." In the case of
"6 Integers," Mathematica evaluated the composition's "expression"
to generate a
tremendous hierarchical structure, which it then symbolically "flattened"
to produce a list of explicit instructions (each command representing a
note). A Perl script then translated these instructions into a binary
format understood by synthesizers.
On his web site, Carl McTague describes himself as an "aspiring composer,
fiddler, hacker, mathematician, minimalist, and scientist." Heading into
only his second year at UC, McTague has already compiled quite an
impressive resume of accomplishments, including numerous local and
national awards. He is a full-scholarship student, triple majoring in
engineering mechanics, mathematics, and medicine, while continuing his
music studies under the direction of Ms. Helmuth. Although he was accepted
to medical school while still in high school, he is undecided about his
future plans. His varied interests include the exploding medical research
field, the world of technical computing, and mathematics--which "is a lot
of fun, too."
When asked where his music fits into the picture, McTague responded, "I
have no idea! I guess it has a lot to do with how good my ideas turn out
being. I was frankly blown away by '6 Integers'--and it was such a simple
application of my ideas." McTague has a lot more ideas that he's working
on now. An alternate recording of "6 Integers" that he's just released
employs bolder orchestration to better illuminate the piece's complexity.
Also, he is currently working with custom audio software to manipulate
sampled and generated sounds directly. In the future he would like to
experiment with traditional musical scores--but this is terribly difficult
as "it's so much easier to communicate with a synthesizer than a human!"
Although it remains to be seen just how far McTague will go with his
algorithmic compositions, his options are obviously wide open. Says
McTague, "The dawn of the information age is an exciting time for
composers. Traditionally we have struggled for live performances of our
pieces: they are expensive and short-lived, and their potential audiences are
quite limited. But today I can create a piece with a digital computer,
alone, and make it permanently available to a worldwide audience. It is
"6 Integers" can be downloaded as a stereo MP3 file.
To read more about Carl McTague and his music, beliefs, writings, and other
accomplishments, visit his web site at http://www.mctague.org/carl.