In
her book Philosophy in a New Key , Susanne Langer characterizes
a historical age not only by the problems and questions which it formulates,
but by the techniques by which they are formulated. "It is the mode
of handling problems," she proposes, " rather than what they are about,
that assigns them to an age" (Langer, 1942). Texts about computer music
projects tend to emphasize those aspects of the project which can be described
definitatively. Many such descriptions of computer music research
are rooted in largely 19th Century notions of musical form and composition
technique, its methodology deeply grounded either in the rationalist tradition
of thesis and argument or, more commonly, in the empiricist tradition of
experiment and conclusion. What seems to be missing in these kinds of descriptions
is a more "holistic" descriptive framework -- one which explicitly addresses
not only the computational models employed for rendering sonic artifacts,
but also the particular stipulative criteria that motivate their development.
What is significant about Hiller's application of information theory to
music composition, for instance, was not that he did it -- what is significant
is that the inventive criteria he sought to realize led him to make that
particular connection. For Hiller -- as for Xenakis and others -- the
incorporation of mathematical theory in music composition was not merely
a technique by which new works could be generated -- it represented an
epistemological model for composition which allowed them to reconceptualize
the process of composition itself.
In
this paper, I attempt to provide an analysis of procedures by which musical
structure might be formulated with the aid of a computer program such that
the software system developed to carry out such a formulation consitutes
a theoretical model for composition itself. Such a model encapsulates the
stipulative criteria motivating a class of composition activity. Its outputs
are useful both as objects themselves -- musical artifacts -- and as verification
of hypotheses which the model formalizes. Viewed as such, a software system
itself becomes the object of compositional investigation, and not merely
a generator of composed artifacts. In order to make this argument, this
paper distinguishes the various roles which computers play in the tradition
of computer-assisted composition and synthesis -- from the more-or-less
"instrumental" use of computers to express musical ideas formulated without
their aid to a "systematic" use of computer systems in which the systems
are themselves regarded as species of composition. What is required for
such a formulation? In order to address this question, we first want to
establish a basic understanding of computational models themselves. Secondly,
we want to investigate the manner in which a computation model might be
mapped to a specific problem domain , such that the mapping is "systematic"
rather than "instrumental."
2. Computation Models
Interest in computation and computing machinery is centuries old. However, what distinguishes machines developed beginning in the 1930's from those which preceded them is, as author Zenon Pylyshyn points out, "that the focus is not primarily on the imitation of movements (as was the case with early clockwork mechanisms) but on the imitation of certain unobservable internal processes" (Pylyshyn, 1989, p.53). This notion was conceived in correlation with a gradual shift in scientists' understanding of mechanisms themselves, becoming more and more concerned with "abstractly defined operations such as storing, retrieving, and altering tokens of symbolic codes" (Pylyshyn, 1989,p. 53). A technology emerged in which the abstract could be literally represented with mechanical parts. The essential model for this technology is the computing device.
A minimal computing device delineates 2 components: an executive component, and an input device. One very basic model of a computing device is a finite automaton (figure 1). A finite automaton consists of a scanner which reads an input tape, and an executive component which interprets the symbols scanned in off the tape as mechanical instructions to do some thing or the other. The 'executive' component represents some set of "states." Two such states are special: the initial state and the accepting state. When
The
Turing Machine or Universal Machine -- or UM -- represents an advancement
over finite state machines. It simulates particular mechanisms by accepting,
as input, a description of the mechanism it is to simulate. Thereafter,
it executes a procedure whose input/output behavior is identical to that
of the simulated mechanism. This is done by partitioning the input of the
UM into two components: one component constitutes a set of input tokens
that are to be interpreted as instructions which specify the function of
the simulated mechanism, and the other constitutes the inputs to that mechanism.
These instructions are read in first, and stored to memory. The UM then
begins executing each such instruction as though it were its own, reading
input, and responding to it according to these instructions. Mechanisms
such as multiplication, division, and later on, more advanced operations
could be simulated by a UM. In fact, it has been proven that virtually
any mechanism could be modeled with a UM. This abstract machine forms the
theoretical basis for Von Neumann based computer systems as we know them
3. Representational vs. Specificational System Formulations
The principle utility of computational devices, then, is their inherent capacity to form abstract representations of specific problem domains. It is solely on the basis of such representations that their output behavior is determined. From the very beginning, the application of computing devices to particular problems represented profoundly different views of the function of computers in scientific research. According to one view, computing devices are seen as useful for the outputs they produce. An entire branch of computational advancements emerged which were directed toward the implementation of high speed mathematical calculation machines and for computing and controlling rockets, balancing accounts, and so on. I call this category of computation representational. It is characteristic of the view of computers taken within mathematics and the physical sciences, a view according to which computer systems (software or hardware) simulate a world which is the object of calculation and control. Even in the case of control of orbital satelites, for example -- where it could be said that the computer is operating upon a real-world object -- it actually does so by virtue of highly specified simulations, embedded within a software system, which form the output behavior controlling those real-world objects. The purpose of such a system is to represent a problem with sufficient accuracy to render a solution attainable. The point of contact between this system and an observer is its output behavior. According to this view, a computational system simulates the behavior of a mechanism and not its organization; its outputs but not its processes.
According to the other view, computing devices are seen as usefull precisely for their capacity to model the processes by which problems can be formulated. Within this area of research, computing devices are viewed as possible models for human cognition, and computational processes as models for cognitive processes. I call this category of computation specificational. It is characteristic of the view of computers taken by many cyberneticians, cognitive scientists and some researchers in AI. I use the word "specificational" in its prescriptive rather than descriptive sense . Thus, to specify a system is to formally state the conditions required for its realization.
With
specificational systems, for example, the design of the control systems
operating upon an orbital satelite would be of interest primarily for the
insight they provide about the processing behavior required to control
a satelite. According to this view, a computational system constitutes
the functional behavior of a mechanism. Its purpose is to allow
for the observation of its processes rather than its outputs.
4. Specificational Systems and Cognitive Systems
In
order to more deeply examine some of these issues, I would first like to
introduce some notions of cognition and cybernetics that are pertinent
to the discussion.
4.1 A Cybernetics of Cognition
In rendering this introduction, I will make reference primarily to ideas put forth by cybernetician and neurophysiologist Humberto Maturana as they are presented in his writings and in those of Winograd and Flores, Kenneth Wilson, and Susan Parenti.
Maturana's ideas stem from observations he first made during his research as a neurophysiologist, working with visual perception. The observations he made called into question traditional notions of perception: i.e. that perception occurs as a direct mapping from real objects "out there" onto structures within our sensory organs. His experiments with frog vision, for example, challenged the traditional assumption that the neurological activity of the optic nerve was a "direct representation of the pattern of light on the retina" (Winogrand, 1986, p.41). He showed, for instance, that fibers within the retina responded not to patterns of light intensity but rather to patterns of local variation on the retina itself. This demonstrated that at least some of the cognitive processes relevant to the survival of the frog occured within the visual system and not at a higher level of neuroanatomy (such as the brain).
In subsequent research in color vision, Maturana noticed that under certain conditions, the retina would produce color messages not actually occuring in the environment. This and many other experiments lead Maturana to question the traditional theories of color vision as a process by which the visual system associates colors with wavelengths on the spectrum and to postulate that the study of color vision is "the understanding of the participation of the retina ... in the generation of the color space of the observer" (Maturana, 1970, p xii). As such, it seemed that perception needed to be studied by viewing "the properties of the nervous system as a generator of phenomena rather than as a filter on the mapping of reality" (Winograd, 1986, p. 42).
This characterization of perception can be applied to cognition in general. Maturana defines a cognitive system as
This leads to a common cybernetic definition of cognition:
This
recursive definition distinguishes cognition as a process -- a process
by which descriptions are generated. A 'description' is, in this context,
understood as the observer's representation of a system or structure in
a linguistic domain such as language, mathematics, or logic.
4.2 Specificational Systems Are Cognitive Systems
Having provided a brief overview of cybernetics as it applies to cognition, we can now deepen our discussion of computation models and their possible consideration as cognitive systems.
Figure
2 is a stripped down illustration of the process by which a computer system
might be developed. It depicts a traditional approach to software development,
Figure
3 depicts a non-traditional approach to software development; one in which
a body of "hypotheses" provide a framework for program design and development.
Hypotheses are changeable and operative within the system itself. Changes
made with respect to hypotheses are motivated within the context of the
designer's engagement with a program's functional and output behavior.
The program acts then as an agent in theverification of hypotheses, which
verification in turn provides a framework for the re-formulation of hypotheses,
and so on.
Specificational
models of computation can be somewhat more rigorously diagrammed then.
As shown in figure 4, the designer specifies both program inputs and program
code itself. In addition, the designer specifies hypotheses. The designer's
specification occurs as a response to his observations. The designer observes
the system from three vantage points: the program's outputs (which are
data oriented), the program's structure (which is procedure oriented),
and abstract hypotheses he himself has formulated (which are theoretically
oriented). This model for system development is highly amenable to experimental
activity.
This model can be somewhat strengthened by reducing our description of the designer's behavior to its procedural functionality. As is shown, observation occurs within the domain of program output, program structure, and hypotheses formulation. Descriptions are formulated on the basis of observation. They form the stipulative environment for the articulation of particular algorithms which in turn determine program structure, program input, and a reformulation of hypotheses.
This
model for system development infers isomorphisms between description
and program, inasmuch as a program comes to be constituted as a
formalized description of a problem and its domain of application. Accordingly,
the separation between description.
and program begin to blur since the articulation of a description becomes the program.A specificational model, then, articulates what I call a "deep" systematization of problem formulation. It activates a topology for experimental gesture in which articulation and representation occur within a unified and mutually interacting sphere of activity.
5. Experimental Research in Synthesis and Composition
We now want to begin to discuss music composition as a "systematic" process. This process differentiates "programming" interfaces from "application" interfaces. Programming allows a composer to interact with a computer at a "deep" level of systematization in as much as it allows him relative freedom in the representation and formulation of a problem domain.
My own research in computer-assisted composition addresses a single question: how can structure be represented symbolically within a program, or more specifically, how are signals to be represented and generated within a computer program?
In the most general sense, we partition the space in terms of control and parameter. "Control" refers to processes which monitor and modify the state of a system or system component. A "parameter" constitutes a point in a vector field. This vector field is defined with respect to a model of some object, and is frequently refered to as a "parameter space." A parameter space can represent a real-world object such as a flute sound, or it can represent a hypothetically stipulated object. A hypothetically stipulated object can itself instantiate a model of a real-world object or process, but its constraints are not subject to those which would render that modeled object or process. In either case, a parameter space encapsulates a description with respect to those structural aspects of the modeled object or process which that description declares as relevant to the program being designed.
It can be said, then, that symbolic representations of sonic and musical processes involve the application of control processes to parameter vectors (Wessel 1979). Control processes activate the generation of information which identifies the object being modeled with respect to a parameter space. The determination of control process and parameter space is a compositional determination if the modeled object occurs as a result of hypothetical experimentation.
In
the following, I will describe certain aspects of the empirical research
which I have been conducting over the last four-and-a-half years in computer-assisted
sound synthesis and composition. This research has involved the development
of software -- and in one case hardware -- used for composition and synthesis,
down to the sample-generating layer. This reflects my desire to experimentally
investigate possible computational representations of musical structure
without the kinds of constraints introduced by more general software systems.
5.1
Non-linear Systems of Granular Synthesis
The first project I will report is one which involved research in non-linear dynamic systems using granular synthesis techniques. Over the last 10 years, granular synthesis has provided a rich model for sound synthesis since it facilitates methods of control over low-level aspects of timbre evolution. The technique itself suggests the possibility of linking control structures determining timbre to those functioning at the level of composition architecture. This investigation was first conducted by Curtis Roads (Roads, 1978, 1985, 1991) and Barry Truax (Truax, 1987, 1988, 1990) and was later taken up by Augostino Di Scipio (Di Scipio, 1990, 1991, 1992) among others. My program Wave summarizes my own preliminary research in composing non-linear musical forms using granular synthesis.
This research began with investigations in understanding the behavior of non-linear functions as computational analogs to the ideas of open form with which I was engaged at the time. The behavior of the logistic difference equation was particularly rich in this respect. The behavior of this function suggested the possibility of generating dynamically evolving timbres whose control functions could be employed on differing temporal dimensions from the smallest grain to an entire class of compositions. This hypotheses became the basis for the development of a composition/synthesis software system in which musical structure constitutes a set of processes which the progrem sets into motion during its execution.
The kernel data structure is a 6-field parameter space:
A particular
child may have multiple parents, each parent-child controlling some unique
aspect of the generative process while a particular parent may have similarly
As a consequence of this complex organization and of the non-deterministic nature of the generative functions used, the behavior of this system -- and thus its outputs -- is indeterminate, though not random. Some aspects of its behavior can be predicted down to a low level, while others are more difficult. As a result, large-scale form evolves only as the program is executed and cannot be absolutely determined beforehand. Because of the fact that non-linear systems are highly sensitive to their initial conditions, minimal alterations in startup data can yield maximally variant formal behaviors . At what point does such variance impact our experience of one output such that we no longer recognize it as related to outputs generated by very similar executions?
This quandary allows us to propose a differentiation between form and pattern -- a differentiation which the biologist C.H. Waddington arrives at with the following reasoning: In a human being, from the time of birth onward, the legs and lower part of the body grow at a greater rate than does the head, so that the proportions between the various parts of the body gradually alter. Is this to be considered a change of form (Waddington, 1962, p. 86)?Waddington offers a qualified definition of form "to mean a structural arrangement which cannot be altered by a mere change in the existing system of growth vectors" (Waddington, 1962, p87). The word "pattern", on the otherhand, is to be used "to focus attention on the spatial interrelations between the various parts into which a form can be analyzed" (Waddington, 1962, p. 87). He further develops this argument by pointing out that "[a] change of form would be produced if a new growth vector made its appearance, for instance, in the development of a third pair of legs as in insects..." (Waddington, 1962, p. 87). This differentiation between form and pattern forms one aspect of the experimental investigation begun with this program.
Three
works for two-channel tape were made: free-Fall, Listing 1 ,
and Listing 2 . Listing 1 and Listing 2 represent
two program executions which are distinguished by small changes in startup
data.
5.2
Dynamically Configurable Feedback/Delay Networks
The program just described models generative processes that are applicable over the range of an entire composition, or family of compositions. The premise which energizes these investigations is that computation allows for a tighter conceptual link between timbre design and composition design, and that both can be designed using hypotheses-based models rather than nature-based models. In more recent sound synthesis research, I am interested in developing models which explicitly address the problem of mapping algorithm to sound such that sound becomes capable of communicating meaningful information with respect to the algorithm with which it is generated. As such, my interest is in transforming sound from the status of material to one of form -- to use Augostino Di Scipio's term (Di Scipio 1994) -- where sound itself becomes a context for computational design. Sound -- as "form" -- systematically encodes information which references the states of the system with which it is formulated. This articulates a sound synthesis model in which individual sounds actually require the contextual framework -- i.e. the larger scale forms -- into which they are projected. It also suggests a compositional procedure by which large-scale form emerges from the computational -- not acoustic -- aspects of individual sounds. Such a procedure begs the question: how can I determine which aspects of a computational model of sound will have significance for the larger form into which it is projected such that the dialectics inherent in the model are recapitulated in its output?
The following is a description of a synthesis program which implements a virtual instrument model in which control parameters for individual sounds both activate and are activated by higher-level processes. By "virtual instrument" I mean a 'physically modelled' synthesis system in which control organizations are modeled around the physical organization of some type of generic musical instrument (Borin, 1992; Woodhouse, 1992; Cadoz, 1993; Smith, 1986, 1987,1993; Garnett, 1988). This type of rendering provides a rich parameter space, as well as methods for its control. In my own work, I am far less interested in imitating the physical characteristics of known musical instruments than I am in formulating hypotheses for the rendering of possible instruments -- instruments which actually extend the controllable parameter space beyond that of traditional instrument models. In the following description, I will first provide an overview of how the basic sound rendering software works. Then I will briefly describe an application of this software and a composition made using this application.
Functionally,
the system delineates two layers: signal propagation and signal control
(see figure 8). At the signal propagation layer, a network of interconnected
In
the example, boxes represent delays, circles represent summers, and triangles
represent multipliers. This particular example diagrams a 1-pole filter.
These module groupings can be interconnected in either feed-forward or
feed-back configurations, allowing for complex re-propagation of the generated
signal throughout the system (figure 10).
The system allows the composer to specify a variety of organizations and, as such, models for synthesis. One example of such a model specifies the organizational structure capable of modeling the behavior of a generic plucked string instrument out into a domain of control not normally associated with such an instrument.
Figure
11 depicts the design of a generic plucked string virtual instrument which
I call "stringX". The propagation layer represents a very basic Karplus-Strong
plucked string model. The excitor provides an initial noise burst which
sets the thing into motion. Besides dispursing signals throughout the propagation
modules in a manner characteristic of plucked string resonances, it causes
signal disruptions within the control layer, activating the generation
of signals there which cause dynamic alteration of those
This piece engaged two experiments. In the first, I was interested in investigating ways in which larger-scale syntax could manifest the organization of signal generation and control algorithms operating at the level of the individual sound. Globally drawn algorithms selected subgroups from the larger signal-propagation network in order to try to make sounds. The composition projects five trajectories of sequences of such subgroupings into time, the five trajectories overlapping and sometimes colliding. The generative structure ends up articulating a topology which composes control parameters within the virtual instrument.
The
second experiment concerned the nature of the sounds as "materials." My
wish was for the presence of absolutely flat sounds and for the
composibility of their envelopes. Within the context of these sounds, it
seemed appropriate to consider only the attack and decay portions of the
envelope, and to reduce these to a duration of between 1 and 1,000 samples.
This allowed me to compose onset and offset clicks with a variety of frequency
and amplitude characteristics
7. Conclusion
In
this paper, I have attempted to frame and demonstrate a description of
computer music which emphasises the articulation of what Otto Laske calls
"possible musics" (Laske, 1994). Generating descriptions of projects that
are not merely narrative recitations of what one thinks about what one
does is a challenge which deserves more attention than it gets. Considerable
progress has been made over the last 30 years in developing rich models
for synthesis technique, spatialization, simulation, and so on. It is my
opinion, however, that insufficient progress has been made in formulating
theoretical bases for these efforts since the 1960s and early 70s,
and what progress is made is woefully under-reported. I consider such theoretical
research -- when occuring alongside empirical investigation -- to be at
least as relevant to composition as the production of musical works, since
it is only by virtue of rigorous inquiry into the manner in which we understand
the problems in front of us, that we can re-formulate them as new and more
interesting ones.
8.
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