Despite recent attempts at interdisciplinary study, the sciences and humanities are still marked by a significant emphasis on specialization. This situation often reinforces the traditional, and seemingly unbridgeable, gulf separating notions of objective, empirical science and subjective, metaphysical arts. Such a state of opposition continues to manifest itself in many individual fields, including music, where, for example, positivistic analysis and musicology are still frequently philosophically polarized to non-absolute theories of musical interpretation and criticism.
One of the most persuasive and compelling arguments that has been made to reconcile these oppositions has come from the study of chaos theory—in essence, solving the question of the relationship between order and disorder. In the past, order was viewed as a quantifiable, classifiable state which was subject to rational discussion. Disorder, however, was aligned with chaos and was fundamentally unquantifiable and thus incapable of sustaining rational discussion. Many contemporary scientists have, however, drastically reformulated their early understanding of chaos as the mere existence of disorder, to an understanding of chaos as extremely complex information. Indeed the study of chaos and chaos theory is largely a study of the recognition and understanding of complexity.1 In recent years chaos theory has transgressed the bounds of esoteric scientific thought and found broad popular appeal through studies such as James Gleick's Chaos: Making a New Science, and Ilya Prigogine and Isabelle Stengers' Order out of Chaos: Man's New Dialogue with Nature. Its notoriety has risen to the extent that many scientists have even placed it along side of two other great revolutions in late millennial theory—relativity and quantum mechanics.2 Through realistic computer modeling, scientists are now able to view the evolution of complex dynamic systems. The fact that attempts have been made to observe the phenomenon in a wide variety of fields from biology and physics, to astronomy, economics and geography, stands as some measure of the acceptance which chaos has gained within the scientific community. By far the largest part of work on chaos has been carried out within the physical sciences. However, several notable attempts have also been made to investigate the role of chaos in the humanities, particularly within the fields of art and design and literary criticism.3
The impact of chaos and chaos theory on music has also recently been brought to the fore. Articles by musicologists and theorists such as Richard Steinitz, David Burrows, Paul Attinello, and Judy Lochhead, for example, identify chaotic principles governing compositions by Ligeti, Xenakis, Reich and Jimi Hendrix among others.4 Lochhead's study, in particular, centers on the previously overlooked role played by musicians, particularly composers, during the 1950s and 1960s in "disclosing the new paradigm of "chaotics."5 Less well recognized, however, is the important role of chaos theory in recent critical interpretations of music. My aim in this article is thus to examine some of the manifestations of chaos in music overlooked in recent scholarship and also to examine the relationship of chaos theory to various aspects of critical theory.
Chaos Theory: An Introduction
Order implies a steady and absolutely predictable state. In math and sciences this state has been traditionally described in terms of linear systems, "in which cause and effect are related in a proportionate fashion."6 In a linear system, or in an analogous linear equation, the results are absolute and predictable even given different inputs to the equation. In contrast, randomness describes a state where any outcome is equally probable and, hence, completely unpredictable. Chaos, however, describes a third state which incorporates and, to some degree, subsumes both order and randomness. It is a state in which the outcomes appear to be random at a local or microscopic level of detail, but also simultaneously ordered at a macroscopic level. In math and physics such relationships are analogous to non-linear equations and dynamic systems where "a small change in one variable can have a disproportional, even catastrophic effect on other variables."7 Non-linear relationships exhibit outcomes which are generally predictable but never absolutely quantifiable. The so-called "butterfly effect" is often used to illustrate the concept of a non-linear chaotic system. In this scenario the almost infinitesimal breeze caused by the beating of a butterfly's wings may be the initial seed which eventually generates a hurricane. Weather itself provides an example of a chaotic system which is generally predictable at any given location and time. It falls into large cyclic patterns; in the Northern Hemisphere, for example, it is generally warmer in the summer and cooler in the winter. However, as everyone is painfully well aware, weather is frustratingly unpredictable at a local, microscopic level. It will not, for example, always be colder on a winter day than on a summer day, though it is a more probable occurrence. Thus the study of chaos is concerned with two closely aligned though separate issues: the recognition of hidden order within chaotic systems, and the study of order that appears to spontaneously arise out of chaotic systems. This latter issue concerns the study of self-organizing systems, such as population growth, and theorizes on the constructive role that the complexity of disorder plays in stimulating order.8
Both issues in chaos study involve the concept of attractors. An attractor is "a region representing the equilibrium point of the behavior of a system."9 They are the magnets, so to speak, within systems which create or attract order. Two types of attractors, stable and strange, appear to be the most common. Stable attractors exist in linear, predictable systems and are analogous to the state of rest achieved by a pendulum under the influence of gravity. In some sense the terminal point of a musical work achieves a constantly predictable state and thus functions as a type of ultimate stable attractor.10 Discovered in the mid-sixties by meteorologist Edward Lorenz, strange attractors, on the other hand, exhibit a characteristic of bounded infinity, squeezing an infinitely long line into a finite space. To quote chaos theorist Alexander Argyros, strange attractors are:
. . . both confined within the finite space of a basin of attraction yet display a pattern of behavior which is non periodic. In other words, although a chaotic [strange] attractor never returns to a point it previously occupied in phase space, it is nevertheless still a recognizable attractor.11
The great artistic themes with which the Western artistic community has traditionally been concerned resemble types of strange, or cultural, attractors. Such traditional themes, unrequited love being perhaps the most popular example, are relatively few in number but succeed in maintaining their viability by constantly reappearing in new forms. There perhaps exists the potential for the appearance of new themes to arise, such as those based on feminist ideology for example, but they are often still re-interpretations and re-applications of older themes and ideas (freedom, autonomy, dignity, respect etc., in the case of feminism).12
Closely intertwined with the study of chaos is the geometry-based study of fractals. The word "fractal" was coined by the mathematical physicist Benoit Mandelbrot in the late 1970s, but objects taking fractal form have been known for centuries. A fractal structure is one which exhibits properties of self-similarity and subdivisibility. It is self-similar in that subsections of a whole are comparable to the whole. In other words, "no matter how small a subdivision is taken, the subsection contains no less detail than the whole."13 Fractal objects have been created in art and mathematics for centuries; Albrecht Dürer's Pentagons, for example, were known to him around the year 1500. There are, however, also many instances of naturally-occurring fractal forms, such as exhibited in fern fronds and coastlines. Some fractal objects have perfectly similar subsections but many, especially those occurring in nature, have only "statistically similar subsections, so that subsections have similar forms with some variation."14 It is this state of similar variation that forms much of the relationship between fractal geometry and chaos. Thus in a fractal object, as in a chaotic system, there is often a deep order (or fundamental similarity) to seemingly chaotic, superficial varieties of forms and structures. An apparently random or chaotic outline of a coastline can be observed to exhibit a unified order when smaller and smaller self-similar subsections are taken.
Many of the "organic" methods of musical composition and analytic systems invoke aspects of fractal self-similarity.15 Perhaps the most prevalent example of this is found in the local and global linear descents and fundamental structures which are mapped into the fore, middle, and background level reductions of Schenkerian analysis. Schenker describes the movement between transformation levels in terms that closely resemble fractal self-similarity.
As they move towards the foreground, the transformation levels are actually bearers of the developments and are, at the same time, repetitions of parallelisms in the most elevated sense—if we permit ourselves to use the word "repetition" to describe the movement from transformation level to transformation level. The mysterious concealment of such repetitions is an almost biological means of protection: repetitions thrive better in secret than in the full light of consciousness.16
Indeed, to a large degree Schenker directly presages the study of chaos theory both in tone and in content. The interdisciplinarity and metaphysical nature of his theory is evident in the following aphorism from the introduction to Der Freie Satz:
All that is organic, every relatedness belongs to God and remains His gift, even when man creates the work and perceives that it is organic.
The whole of the foreground, which man calls chaos, God derives from His cosmos, the background. The eternal harmony of His eternal Being is grounded in this relationship.
The astronomer knows that every system is part of a higher system; the highest system of all is God himself, God the creator. . . . Every surface, seen for itself alone, is of necessity confusing and always complex.17
It is somewhat ironic that, given the apparent similarity to chaos theory, Schenkerian analysis has traditionally been regarded as one of the most positivistic fields of musical inquiry.18 The similarity of Schenkerian fore, middle and background levels of analysis to fractal theory is just one instance of the presence of chaotic paradigms in music. Indeed it can be argued that all musical reproductions, whether in the form of mechanical recordings or live performances, can be interpreted as fractal events. In the manner of Jean Baudrillard's concept of simulacra (abstract imitations, or simulations, of reality such as found in the virtual reality of recorded performances or video games), they are all self-similar representations or reproductions of an original conceptual model.19 The concept of a performance, whether live or imagined in the mind's ear, particularly exhibits a high degree of similarity to a dynamic system. This is to say that given a more or less fixed work, the song "Happy Birthday" for example, every performer, through their biological and experiential uniqueness, will conceive of and perform a slightly different version of the piece. Moreover this conception of the piece may change and evolve at different times and places. In this model it is the composition itself that functions more or less as a stable attractor—attracting and organizing many self-similar representations of the original.20
Chaos in Music
In addition to fractal features, chaos is manifest in music in a variety of ways.21 Local level, moment-to-moment relationships between individual tones, harmonies, rhythms, register, and almost all compositional parameters, exhibit essentially random states of being. It is only upon global inspection that these parameters, individual tones in particular, can be recognized as being organized into, more or less, systematic, ordered states corresponding to tonal, atonal, or serial organization.
Another common instance of chaos in music is the recognition of codified formal paradigms, such as sonata form.22 Here the recognition of global paradigms is always mitigated by an infinite, and thus unordered, variety of local individual differentiations. No two works are exactly alike. In short, many works exhibit global similarities of form that are analogous to the seasonal patterns found in weather. Both appear to exhibit an unpredictable or infinite number of local variations. Thus in music, as in the weather or in any other dynamic system, order and entropy exist simultaneously, subsumed in a chaotic organization.
Perhaps one of the most robust arguments for the presence and importance of chaos in music is provided by its temporality. Musical time is variously experienced as metronomically ordered or chaotically free flowing and a-rhythmic. Even when a regularly recurring pulse is present in a composition, our experience of the passage of musical time often ebbs and flows as a seemingly unordered assemblage of fleeting moments. Music carries on a continual negotiation between apparent linearity and non-linearity which distorts the absolute perceptions of real time. The variegated temporal experience of music is thus another manifestation of its complexity, and potential for chaotic behavior. As in the deep order of a superficially complex chaotic system, composers conventionally order sound (tones, harmonies, rhythms etc.) that, though intentionally structured, produce a limitless variety of outcomes and interpretations. Music is of course only one of many experiences that seem capable of creating or incorporating asymmetrical experiences of time. Nonetheless, a large part of how well we value a musical experience is subsumed by some non-linear measure of qualitative timeour degree of being interested. This is to say that time, or music, that is filled with interesting and varied experiences often seems short in passing but often, upon later reflection, lengthy. Conversely, a space of time empty of such experiences seems to take a long time to pass but, upon reflection, paradoxically short. In this sense, as outlined in studies conducted by Henry Orlov in the mid 1970s, there is a double frame of perceived time reference, i.e., "experienced" perceived time and "contemplative" perceived time.23
In addition to the latent existence of chaos in almost all musical works, chaos and aspects of chaos theory have served as the conscious compositional basis for works by many composers.24 Michael McNabb, Charles Wuorinen, Charles Dodge, Tommaso Bolognesi, Jeff Pressing, and Bruno Degazio among many others have experimented with various aspects of computer-generated music based on experiments with non-linear equations and chaos theory.25
Chaos and Critical Theory
It is well known that chaotic paradigms have often been used by twentieth- and early twenty-first-century composers to generate music. Less well recognized, however, is the fact that chaos also pervades many recent critical and interpretive approaches to music. To understand the implications of this it must be understood that ideas are not tangible objects but dynamic systems in their own right, and exhibit a similar state of flux to that of music itself. The fact that there exists considerable potential for two equally capable people to have radically differing views of the same piece of music after hearing the same performance is roughly comparable to a spontaneously chaotic outcome from a seemingly ordered event.26
Related to this observation is my contention that many of the most prevalent critical approaches to music revolve around explanations of apparently chaotic moments which, upon initial inspection appear to defy, or stand apart from, the normatively ordered system employed by the composer. This is to say that these critical approaches seem to intuitively recognize and focus on compositionally irregular or extreme moments in otherwise, more or less, normatively ordered texts.
It is precisely such recognition of momentary chaos that is embodied in the metaphors of "sexual excess" and gendered narratives, central to the criticism of Susan McClary. Many of the textual anomalies, to which she persuasively assigns a subtext of masculine-framed, feminine madness, for example, are also instances of relative disorder within a work. In describing the obsessive quality of the nymph's soliloquy from Monteverdi's Lamento della Ninfa, McClary's descriptive language reflects the situation.
This obsessive quality is created musically through an unvarying cycle of four bass notes that seem to progress rationally . . . only to double back inevitably to starting position. Against this backdrop, the nymph's vocal lines sometimes acquiesce, sometimes struggle—though always in vain. It is a mark of Monteverdi's powers of imagination that we never hear the ostinato bass the same way twice: it is constantly being reinflected by the nymph's dramatic moments of resistance (her dissonant refusals of cadences implied in the bass, the futile attempts at cadencing . . . ) or of sudden collapse.27
Admittedly, the tone of such language is in keeping with McClary's general theme of madness and is a logical byproduct of the traditional association of disorder with irrationality. However she recognizes and describes the anomalous nature of sexual passion in Bizet's Carmen in equally chaotically inflected terms.
The energy in the opera is, of course, located in the musical characterization of Carmen herself: she is the dissonant Other who is necessary for the motivation and sustaining of the plot. . . . Carmen's music is further marked by its chromatic excesses. Her melodic lines tease and taunt, forcing the attention to dwell on the moment—on the erogenous zones of her inflected melodies. . . . While there is never any question of her tonal or melodic orientation in this phrase, her erratic means of descending through the tetrachord . . . reveals her as a "master" of seductive rhetoric.28
I should stress at this point that I find McClary's reading of an underlying sexually excessive subtext in Carmen's character to be entirely compelling, though there could also be other potentially important metaphors at work. Many of the important moments of her criticism and interpretation, however, rest on instances of relative instability and superficial disorder within the text, how "dementia is delineated musically through repetitive, ornamental, or chromatic excess."29 McClary's readings of these instances (and indeed it would seem true of most narrative subtexts) attempt to provide or recognize potentially significant explanations of various degrees of textual anomalies. In other words they recognize order, meaning, and form in otherwise superficially meaningless and disordered events.
As described in his book, Music as Cultural Practice, Lawrence Kramer's "hermeneutic windows" provides another example of what I am suggesting. His description of the process of locating these "windows," including a quote from Clifford Geertz's, The Interpretation of Cultures, is rife with images of the recognition and ordering of moments of textual disorder.
Hermeneutic windows tend to be located where the object of interpretation appears—or can be made to appear—explicitly problematical. Interpretation takes flight from breaking points, which usually means from points of under- or over-determination: on the one hand, a gap, a lack, a missing connection; on the other, a surplus pattern, an extra repetition, an excessive connection. In some cases our effort to turn these breaking points into sources of understanding may involve no more than reflection on the explicit expressive acts that are particular to the object. . . . The goal of this process, at least its ideal goal, resembles what anthropologist Clifford Geertz calls a "thick description": an account of "a multiplicity of complex, strange, irregular, and inexplicit, and which [we] must contrive somehow first to grasp and then to render." [emphasis mine]30
Kramer, like McClary, recognizes the importance of investigating excessive, missing, or otherwise superficially unordered and problematic points within the object of interpretation.
Other instances of the recognition of potentially chaotic moments in other interpretive approaches are not hard to find. The work of Carolyn Abbate provides another notable example. In her book, Unsung Voices, Abbate attempts to describe and locate latent narrating voices within music. Once again, her focus is on instances of irregularity or disorder within a work, which she describes as "disjunctions."
The narrating voice . . . is not merely an instrumental imitation of singing, but rather is marked by multiple disjunctions with the music surrounding it. These disjunctions, their forms and signs . . . change from work to work; they are fugitive. They exist on many levels. [Abbate's emphasis]31
Later, in the same chapter, Abbate further articulates her interpretive approach.
In my own interpretations (and my own narrative behavior) I will interpret music as narrating only rarely. It is not narrative, but possesses moments of narration, moments that can be identified by their bizarre and disruptive effect. Such moments seem like voices from elsewhere, speaking (singing) in a fashion we recognize precisely because it is idiosyncratic.32
Each of these critics is dealing with a repertory of works that operates using similar cultural mechanisms and a similar musical language of the common practice period. Nonetheless each of the authors quoted above concentrates on disruptive, excessive or disjunctive moments in music as a jumping off point for the positing of an ordered subtext of some kind or other. In other words, they recognize that order is often hidden at the most chaotic and complex moments in a musical score.
Perhaps nowhere in recent musicology have the principles of chaotic theory been more readily apparent than in Rose Rosengard Subotnik's book Developing Variations. Early in the first chapter Subotnik develops two categories of ideological approach and philosophic orientation; the metaphysical "Continentalist" view (to which Subotnik herself subscribes) is placed in opposition to the empiricist "Anglo-American" view.33 Underlying the empiricist "Anglo-American" orientation is the concept of ordered determinism, where meaning and value of a musical work are potentially objectively determinable, able to be extracted from some manifestation of deep order within the work. This is a type of order which, it might additionally be claimed, must govern not only the work in question, but also the culture, society and physical universe in which it was created. The direct antithesis of this view underlies the metaphysical orientation of the Continentalists who believe that "all people . . . operate with a comprehensive, though not necessarily conscious, immutable or irrefutable view of how things are."34 In other words they would adhere to a type of subjective meaning, the validity of which is dependant on the critic's ability to summon enough circumstantial evidence, usually taking the form of metaphors connecting the text and the critic's sense of its meaning, to convince a public or academic jury of the reasonableness of their claim. To some extent adherents of this view also see themselves as oppressed by a sense of determinism, inherent in the Anglo-American orientation, that attempts to repress the potential for (at least the appearance of) random subjective meanings. In other words, if order is good, then disorder is bad. That disorder has traditionally been regarded as a negative value seems in large part due to the dominance of binary logic which pervades Western thinking.
Other critical approaches to music also manifest various aspects of chaotic paradigms. In his book Style and Music, for example, Leonard Meyer advocates the theory that style change results from countless interactions between individual composers, performers and audience over time.35 As such, I would argue that his theory tacitly recognizes stylistic change as the product of complex, chaotic system that continually organizes itself. In similar fashion, virtual music communities, such as those engendered by file-sharing services such as Napster and Kaaza, also exhibit chaotic, complex behavior, and are essentially self-organizing structures. Notions such as the "web of culture," postmodern pastiche and the internet, also reflect aspects of chaotic systems whereby superficial chaos masks deep complex order.
Contemporary popular culture is particularly rife with examples of self-ordering systems. For example spontaneous organization is a feature of many dance floor communities. Consider the following passage from Jeremy Gilbert and Ewan Pearson who evoke the work of Gilles Deleuze to describe rave and acid house dance floor communities precisely in such terms.
A quasi-utopian view of rave is suggested by the heterogeneous chemical alliance formed by its participants. In this respect, acid house and the rave movement it spawned represented instead what Deleuze has termed singularity, an example of systematic self-organization when previously disconnected elements reach a point at which they begin to co-operate (or oscillate) as a larger morality . . . such agglomerations are difficult to predict and, importantly, are transient . . . 36
On the dance floor multiple, localized random subjectivities thus often organize into a larger collective. Such systems are also found in various biomusicological organizations—such as in the cultural rhizonomic patterns as articulated in the work of Deleuze and Guattari: "The rhizome is an acentered, nonhierarchical, nonsignifying system without a General and without an organizing memory or central automation, defined solely by a circulation of states."37 The system described here also closely resembles Nils Wallin's biomusicological suggestion that musical patterns, or "Nucleations [,] are the result of fluctuations between deterministic versus indeterministic, stochastic aspects of tonal structure as interpreted by the nervous system. . . ."38 The complexities implied in the fluctuation between deterministic and indeterministic systems of the dance floor, or of a nervous system relating to how we receive music, are closely aligned with the deep order described by chaos theory.
The trend, evident in much contemporary popular music, of reverting to old analog technology and obsolete electronic instruments, or "low fidelity," also has chaotic implications. The reliance on such "technological brokenness" has been read as a metaphor for racial and gender expression.39 Indeed the intentional misuse of technology by many musicians, such as Hip Hop musicians using a turntable as a source of sound production rather than reproduction, is a manifestation of this notion. In some sense such misuses of technology are analogous to a loss of order, and represent the intentional creation of chaotic aural surfaces that masque deeper social cultural meaning. In the case of turntable "scratching" this meaning might be interpreted as the conscious desecration of and resistance to the trappings of traditional modes of white technological power.
Beyond Metaphor?
It may be argued that chaos and chaos theory merely exist as another subtextual metaphor within music. As such it appears no different than any other metaphor-based interpretation of music and thus a mere continuation or extension of some subliminal quest for determinism. I have so far only suggested chaos theory as a possible repertory of tropes that have the potential to illuminate a musical text.
Given the constraints of language, it is impossible ever to transcend the mitigative role of metaphor and narrative in achieving any sense of communicable understanding or meaning. Nonetheless it seems important to find more rigorous connections. In attempting to go beyond metaphor and establish a more concrete connection between literature and science William Paulson, in his essay, "Literature, Complexity, Interdisciplinarity," states:
Literary works exhibit the complexity of emergent systems possessing singular, context-dependent constraints and forms. Although texts are made of language, the passage from linguistic structure to textual effect can not be described with anything like the regularity or predictability found in, say, grammatical description of sentences. . . . We suppose for example, that a poem presents itself to a reader as a complex system of relations. . . . The reader brings to her assimilation of the poem a knowledge of the linguistic codes; if she is an experienced reader of poetry, she also brings some general sense of where and how to look for further relations that make up the poem. But she does not begin with precise, operational knowledge of how all these different phenomena will interact and thereby contribute to the poem's effect, exactly how they will combine to produce what she will call the poem's meaning. In other words, the reader does not initially posses all of the codes needed to understand the poem, so that some of its variety is uncoded, or in other words, is noise.40
The situation Paulson describes is obviously similar to, if not the same as, listening to music. The act of listening, like reading, is an example of a complex often performative dynamic system. When we first listen to a difficult or complex musical work, a symphony for example, there are often many elements of it that remain as unassimilated information and that, consequently, we process only as noise (literally and figuratively) rather than meaningful information. If we were to listen to the work again, informed by the first hearing, more of the work is able to be processed as information, thus we listen at a higher level of complexity. It is precisely the presence of chaos, or noise that, in some sense, forces the system to reorganize itself at an elevated level of complexity.
Among other applications the paradigm of chaos allows us to begin to understand the subtle fluctuations of musical style change. As a compositional strategy, for example, chaos played a particularly important role in the 1950s and 1960s and the emancipatory freedom from traditional forms of musical order. Exhibited in genres as diverse as avant garde art music, free jazz and psychedelic and even punk rock, the paradigm of chaos can be read as an active reflection of cultural change—a disorder which enabled new liberatory orders to arise.41 The seemingly noisy dissonance of rock 'n' roll, for example, released popular music from previous tin pan alley and jazz conventions. In turn the sonic experimentalism of bands like the Beatles and Pink Floyd, at least in part, freed popular music from the need to conform to conventional verse-chorus structures and associated three-minute and thirty-second song format of fifties and sixties pop music. The extended free form jam sessions, often associated with the psychedelic San Francisco sound of bands like the Grateful Dead and Jefferson Airplane can also be grounded in this type of liberatory chaos. Audiences at these events also often engaged in a simultaneous chaotic "freeing" of mind and body through ingesting hallucinogenic drugs and ecstatic freeform dancing. Similarly the chaotic surface anarchy of punk emancipated popular music from need for complex structures associated with more bombastic styles of hard and progressive rock of the late seventies, and also from the formulaic production of disco. These brief examples underline the role of chaos in shaping popular music, and indeed culture in general. Viewed in this way the paradigm of chaos thus offers a potentially powerful challenge to our common understanding of modern and postmodern cultural construction and stylistic evolution.
Complexity and chaos can be viewed as a type of evolutionary strategy which enables new strategies for creativity. Some chaos theorists have, in fact, speculated that the entire creative potential of the universe is due to some naturally preferred strategies of self organization. Chaos theorist Joseph Ford, for example, parodies Einstein's famous maxim in support of such a proposition. "God plays dice with the universe. But they're loaded dice."42 However chaos theorists have stated their case, there seems to be an underlying commonality of belief that, ironically, chaos could represent a profound underlying unity of nature. The result of viewing the world as a potentially limited series of creative strategies is that the world simultaneously imitates and changes itself with every new creation. It is a type of infinite feedback loop, whereby art and any other type of cultural production, including science, progressively becomes more and more complex. The relatively chaotic postmodernist state in which music, and art in general, finds itself would then represent a state which is both creatively liberating yet also capable of instilling fear of the demise of culture as we know it. It would appear to be no accident that chaos and chaos theory have gained prominence within the framework of a postmodern context.
Among other fields of scientific or artistic endeavor, the complexity of postmodernism is mirrored in music which is itself a product of an infinitely complex system of cultural, societal, and biological negotiations. Chaos theory, and its encompassing multidisciplinarity, offers a paradigm with which to understand potentially deeper patterns in the apparently chaotic stylistic fragmentations of the postmodern era. Do apparent surface stylistic differences between composers evince a deeper order? Is there a type of underlying natural selection that occurs in some musical compositions, and compositional strategies, that are more adaptive to survive? Might we be able to understand or even predict the rise of musical trends or fads that currently seem to randomly and spontaneously arise? These are just some of the questions that chaos theory could allow us insight into. Many critical and analytical approaches to music already employ aspects of chaos theory and practical applications of chaos theory are already available and operable. A comprehensive practical application is more elusive though we would do well merely to view moments of musical dislocation, confusion, and apparent chaos as potential analytic windows to deeper meaning.43
The recognition of the extensive presence of chaos in music analysis and music criticism may serve as an aid in understanding the complex links which bind, rather than oppose, traditional interpretive and analytic approaches. Indeed given the recent divides between, seemingly, positivistic theoretical analysis and more relativistic "new" critical theories of musicology we might remind ourselves and our students that both positions can potentially be reconciled through an understanding of chaos theory. Particularly important is the notion that music does not exist in an autonomous vacuum, but rather that it is part of a larger dynamic system of interrelated activities and subjectivities. Through an understanding of chaotics we can recognize that, though creativity exhibits an infinite variety of forms, there is often deeper order in effect. As such, chaos, as distinct from the truly random, is among the most moving and meaningful forms of human musical expression. Seemingly chaotic moments in music often embody the very qualities that we think of as uniquely human—an intelligence and sense of community that comes only from the highest form of sympathy or deep order. Chaos thus leads us to consider music in the context of a larger dynamic system of cultural identities that constitutes the human condition.
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Pressing, Jeff. "Nonlinear Maps as Generators of Musical Design," Computer Music Journal 12/2 (1988), 35-46.
Prigogine, Ilya and Isabelle Stengers, Order Out of Chaos: Man's New Dialogue with Nature. New York: Bantam, 1984.
Schenker, Heinrich. Free Composition: (Der Freie Satz) Volume III of New Musical Theories and Fantasies. Ernst Oster trans. New York: Longman, 1979.
Steinitz, Richard. "Music, Math, and Chaos," The Musical Times 137:1837 (March 1996), 14-21.
. "The Dynamics of Disorder," The Musical Times 137:1839 (May 1996), 7-14.
. "Weeping and Wailing," The Musical Times 137:1841 (August 1996), 17-22.
Subotnik, Rose Rosengard. Developing Variations: Style and Ideology in Western Music. Minneapolis: University of Minnesota Press, 1991.
Wallin, Nils L. Biomusicology: Neurophysical, Neuropsychological and Evolutionary Perspectives on the Origins and Purposes of Music. Stuyvesant N.Y.: Pendragon Press, 1991.
1See for example Prigogine and Stengers, Order Out of Chaos; and Hayles ed., Complex Dynamics in Literature and Science.
2See for example Crilly, "The Roots of Chaos," 193.
3In particular see Lansdown, "Chaos Design and Creativity," 211-24; Argyros, A Blessed Rage for Order; and Hayles ed., Complex Dynamics in Literature and Science.
4Lochhead, "Hearing Chaos," 210-46. Of less importance to the current discussion but of no less interest are a series of articles by Richard Steinitz that discuss the relationship of various aspects of chaos theory in the music of György Ligeti: "Music, Math, and Chaos," 14-21; "The Dynamics of Disorder," 7-14; "Weeping and Wailing," 17-22. Also see Attinello, "Signifying Chaos," 84-110. Also of related interest is Burrows, "A Dynamical Systems Perspective on Music," 529-45.
5Lochhead, "Hearing Chaos," 210.
6Davies, The Cosmic Blueprint, 23. Also quoted in Argyros, A Blessed Rage for Order, 236.
7Briggs and Peat, The Turbulent Mirror, 24.
8Perhaps the most important and pioneering research in this aspect of chaos theory has been done by Ilya Prigogine and Isabelle Stengers. See Order Out of Chaos.
9Argyros, A Blessed Rage for Order, 250.
10It is interesting to note that there appear to be relatively few instances of fully linear systems existing in the physical world. The fact that most science and mathematics, if not also cultural theory, is predicated on and consumed by the quest for linear systems seems somewhat problematic though understandable, given the apparent absolute nature of such inquiry. The complexity of non-linear systems seems to have provided an obvious impediment to their study—hence the relatively proportional rise in chaos and non-linear dynamic theory and advances in computer technology.
11Argyros, A Blessed Rage for Order, 252.
12This idea is drawn from a discussion by Argyros. Ibid., 345-46.
13Crilly, "The Roots of Chaos," 1.
14Ibid.
15Analyses based on organism metaphors, and compositions which are to some degree based on the idea of a germinating cell, such as the major third which begins and permeates Beethoven's Fifth Symphony, are extremely common in music. An overview of the history of this approach to composition and analysis is provided in Bonds, Wordless Rhetoric.
16Schenker, Free Composition, 18.
17Ibid., xxiii.
18It is ironic, though not unexpected, given Schenker's dogmatic insistence on the absolute presence and form of the fundamental structure. It seems somehow less than surprising that a fundamental structure can be universally prized out of tonal, common practice works.
19Baudrillard, Simulations, 1-4, 23-26.
20Burrows discusses a similar concept of performance and perception in "A Dynamical Systems Perspective on Music."
21I would argue that fundamentally aleatoric, i.e., randomly generated, music, such as much of that produced by John Cage, falls outside the realm of chaos theory. This is due to the fact that, if such music is truly random (many open works are considered random but exhibit some underlying aspect of order, i.e., Earl Brown's Available Forms I), it must necessarily transcend any inherent order. Such music will not, however, transcend the influence of chaos in its interpretation. Indeed, if one believes that all music is, at least in some part, the product of non-linear dynamic cultural and societal systems, complete with various attendant cultural attractors, then higher level chaos is, in effect, governing the production of such purportedly aleatoric works. This is to say that aleatoric compositions are probably only truly random in concept rather than in practice. It is a little known fact that even computerized random number generators are usually limited to a randomness sample of approximately 4,000,000—large enough to be considered statistically random.
22I have previously described the terminal point of a musical work as analogous to a stable attractor, implying that music was ultimately a predicable dynamic system which, from this singular point of view, it is. The form and content within and between individual works, nonetheless, exhibit aspects of a non-linear system. In their potential for bounded infinite variation, recurrent formal and compositional paradigms can be viewed as analogous to strange attractors.
23See Orlov, "Temporal Dimensions of the Musical Experience," 368-78.
24Judy Lochhead addresses this aspect of chaos theory and identifies four descriptive categories of compositional chaos: ordering to create unpredictability, ordering to create an aural analogue of chaos, chaos as a creative potential, and libratory chaos. Lochhead, "Hearing Chaos," 215, 216.
25See, for example McNabb, "Dreamsong: The Composition," 36-53; Dodge, "Profile: A Musical Fractal," 10-14; Bolognesi, "Automatic Composition," 25-36; Pressing, "Nonlinear Maps as Generators of Musical Design," 35-46; Degazio, "Musical Aspects of Fractal Geometry," 435-41.
26I would argue that the presence of chaos within a work (i.e., the conscious employment of chaotic procedure to generate a work), though operating on a different plane, is relatively irrelevant to the presence of potential chaos, or infinitely variegated interpretations, external to the work. This is to say that the composer can never completely determine the outcome or meaning of a composition, whether chaotically generated or not.
27McClary, Feminine Endings, 87.
28Ibid., 57, 58.
29Ibid., 81.
30Kramer, Music as Cultural Practice, 12-13. Kramer's quotation of Geertz is taken from Geertz, The Interpretation of Cultures, 10.
31Abbate, Unsung Voices, 19.
32Ibid., 29.
33Subotnik, Developing Variations, 4.
34Ibid.
35Meyer, Style and Music.
36Gilbert and Pearson, Discographies, 28.
37Deleuze and Guattari, A Thousand Plateaus, 21.
38Wallin, Biomusicology, 323.
39This concept was introduced in Joseph Auner's paper "Technological Brokenness" at the American Musicological Society meeting, Toronto, 2000.
40Paulson, "Literature, Complexity, Interdisciplinarity," 47-48.
41Judy Lochhead refers to a similar concept of "Liberatory Chaos." See Lochhead, "Hearing Chaos," 234-36.
42As quoted in Gleick, Chaos, 314. Also in Argyros, A Blessed Rage for Order, 259.
43David Burrows attempts what he calls a "Processual Analysis" of Bach's Sarabande from the Sixth Unaccompanied Cello Suite in D Major (BWV 1012) that relies on recognizing aspects of music, particularly "the flow of immediacies . . . that makes up the performance," as a dynamic system. See Burrows, "A Dynamical Systems Perspective on Music," 530, 534.