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  • Clouds, Pyramids, and Diamonds:Applying Schrödinger's Equation to Granular Synthesis and Compositional Structure
  • Rajmil Fischman

This article presents the results of research on the generation of musical processes resulting from the application of Schrödinger's Equation for an atomic potential with radial symmetry, a mathematical model of dynamic systems that, like music, may change and develop in time. (For an introductory discussion of Schrödinger's Equation, see Feynman, Leighton, and Sands 1971.) I focus on the creation of musical material and its organization arising from the solutions and implications of the equation. This is applied to asynchronous granular techniques (e.g., Truax 1990b; De Poli, Piccialli, and Roads 1991; Di Scipio 1994), as these are ideally suited for stochastic processing of musical material and allow the manipulation of multi-dimensional timbral characteristics. Work carried out consisted of three main parts: the development of relevant theoretical principles and algorithms for the generation of music, their implementation as compositional software, and their realization as a composition that demonstrates their musical validity and viability.


The relationship between scientific disciplines and music is not new in Western culture: suffice it to mention Boethius's concept of "music of the spheres," the medieval quadrivium, the relationship between talea and color in isorhythmic music of the 14th century, and rules of counterpoint regarding temporal proportions between voices in Fux's Gradus ad Parnassum in the 18th century. During the 20th century, a renewed link between science and music began to emerge in composition, from the relatively straightforward principles of serialism to Schillinger's compositional theory (Schillinger 1948), the complex constructs realized by Xenakis in works such as Achorripsis and Nomos Alpha (Xenakis 1992), the compositional strategies of Koenig (Koenig 1970a and 1970b; Laske 1981) and Hiller (Hiller 1981), and the use of stochastic processes by Truax (1982) and others (e.g., Jones 1981; Lyon 1995; Ross 1995; Hoffman 2000). Theoretical and technical advances in information technology propitiated further exploration in areas such as generative grammars (Holzman 1981; Beyls 1991); self-similar, chaotic, and more general dynamic systems (Bolognesi 1983; Dodge 1988; Pressing 1988; Truax 1990a; Di Scipio 1991; Bidlack 1992); and more general algorithmic approaches (Maia et al. 1999; McAlpine, Miranda, and Gerard 1999).

The aim of the present work is to investigate and experiment with the possibilities offered in this area by an equation that contributed to a deep change in our conception of the world, following this path to its ultimate conclusion (i.e., the musical composition). In this sense, this study presents an additional link between the thought processes that lead to the mathematical abstraction of reality (i.e., models of the physical world) and those involved in the creation of musical works. These processes mirror each other in the sense that the former construct meaning out of the observation of the physical world, and the latter transform meaning into a physical trace, namely, a piece of music.

The current project develops previous work on the derivation of organic structure and generation of material offered by Schrödinger's Equation (Bain 1990; Fischman 2003), which preserves musical [End Page 47] logic and coherence. While previous research by the author was restricted to the derivation of structure in a composition for a single instrument (harpsichord) and to the generation of pitch material, the goal of the present study was to derive a strategy for the creation of structure and musical material for a medium which allows greater complexity, includes parameters other than pitch, and provides the basis for the application of similar strategies to other media. Thus, the electroacoustic domain, with a specific focus on asynchronous granular techniques, was chosen, because it allows a significant degree of complexity, and it allows thorough investigation of the possibilities inherent in Schrödinger's Equation for the manipulation of multi-dimensional timbral characteristics, with pitch included as one of these dimensions (Smalley 1986, 1997). In particular, granular techniques are ideally suited for stochastic processing of musical material (Lorrain 1980; Xenakis 1992). Furthermore, a work for tape can provide the immediate aural feedback necessary to evaluate the results.

The research methodology adopted during the project consisted of three main parts. First, I developed...


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