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Mathematical Soundscapes:
Architecture, Music, and Geometry
A Case Study of Metastaseis by Iannis Xenakis

Abstract

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This paper offers a comprehensive technical analysis of Iannis Xenakis’s orchestral composition Metastaseis (1953–54), investigating its deep interconnection with architectural structures and mathematical formulations. As both an architect and composer, Xenakis exemplified an interdisciplinary creative approach, merging his engineering background with compositional innovation. The study traces Xenakis’s trajectory from his architectural work at Le Corbusier’s studio—particularly the Philips Pavilion for Expo 58—to his development of a compositional language grounded in geometry, probability theory, and mathematical physics.

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The analysis begins with an overview of Xenakis’s background in engineering, architecture, and music, highlighting his pivotal mentorship under Olivier Messiaen, who encouraged Xenakis to fuse his scientific expertise with musical creativity. The paper then explores the mathematical and structural design of the Philips Pavilion, constructed from nine interwoven hyperbolic paraboloids (“hypars”), and demonstrates how these architectural elements directly informed the formal and sonic architecture of Metastaseis.

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Through a detailed structural and pitch analysis, the paper shows how Xenakis encoded the geometry of hyperbolic paraboloids into the work’s orchestration. Particularly, it focuses on the opening (bars 1–35) and closing (bars 305–345) glissando sections, where individual string parts trace divergent and convergent pitch trajectories, creating sonic analogues to the curves of architectural hypars. Using original graphical models and referencing Xenakis’s own sketches, the study demonstrates how pitch trajectories in glissandi reflect the parametric equations of hyperbolic surfaces, with each instrument functioning as a spatial vector in a dynamic sound structure.

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Furthermore, the paper deciphers Xenakis’s application of coordinate systems and semitone calculations to shape pitch material, mapping the zero point (unison G) and its multidirectional expansions onto a three-dimensional sonic form. This mathematically grounded pitch architecture is analyzed using the formula Z=y2−x2, and key interpolated pitch values are plotted as acoustic realizations of spatial data points.

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Ultimately, this article frames Metastaseis not only as a seminal orchestral work but also as a pioneering realization of architectural acoustics and mathematical modeling in music. It positions Xenakis’s work as a landmark in interdisciplinary composition, where the boundaries between visual structure and sonic form are dissolved through rigorous formal abstraction.

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The Mathematics of Chance:
Indeterminacy, John Cage’s Anarchist Response to Serialism

Abstract

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This paper explores the philosophical and compositional foundations of John Cage’s concept of indeterminacy, positioning it as a deliberate and radical response to the serialist paradigm that dominated mid-twentieth-century avant-garde music. While serialism—represented by composers such as Boulez, Stockhausen, Webern, and Babbitt—sought to impose total control over all musical parameters through pre-determined series, Cage instead advocated for the relinquishment of authorial intent, embracing unpredictability, performer freedom, and aleatoric structures.

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The study begins by outlining the key tenets of total serialism: pitch, rhythm, dynamics, articulation, and timbre are serialized into fixed matrices, leaving little room for deviation or spontaneity. Cage’s early compositional experience under Arnold Schoenberg exposed him to this rigid framework, but also highlighted his discomfort with harmonic determinism and his lack of affinity for traditional structural thinking. Rather than continuing in this system, Cage turned toward indeterminacy, developing methods to remove his own preferences and intentions from the compositional process.

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Central to Cage’s practice is the distinction between chance music and indeterminate music. The former involves the use of stochastic processes (e.g., coin flips, dice throws, the I Ching) during composition, whereas the latter transfers decision-making power to the performer, allowing each performance to manifest uniquely. The paper examines Cage’s Music of Changes (1951) for solo piano as a pivotal example of chance-based composition, where all musical parameters were derived using the I Ching. By contrast, his later works—including Concert for Piano and Orchestra (1957)—adopt ambiguous notation to ensure variability across performances, enacting true indeterminacy.

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The paper also discusses how Cage’s philosophical transformation—heavily influenced by Zen Buddhism, Eastern metaphysics, and Indian philosophy—shaped his approach to composition. Zen principles such as spontaneity, non-attachment, circularity of thought, and the dissolution of subject-object boundaries align with Cage’s aesthetics of sound, silence, and open-form music. His turn away from Western linear dualism (e.g., order vs. chaos, structure vs. freedom) is framed through the lens of Eastern cyclical cosmology, in which the extremes of determinacy and indeterminacy are not opposites but converging points on a continuous circle.

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Ultimately, the paper argues that Cage’s indeterminate works were not merely experimental novelties but a coherent and philosophical response to the prescriptive rationalism of serial music. By erasing the hierarchical control of the composer and reimagining the role of sound itself, Cage forged a new musical ontology—one rooted in presence, randomness, and non-intention. In doing so, he bridged the most rigidly structured music of his era with its conceptual antithesis, demonstrating that the total control of serialism and the total freedom of indeterminacy may exist as reciprocal reflections within the same artistic continuum.

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The Lorenz attractor, a deterministic system exhibiting chaotic behavior, visually mirrors Cage’s indeterminate music—where underlying rules generate seemingly unpredictable results.

I Ching Hexagram Binary Tree, representing the 64 hexagrams as used by John Cage in Music of Changes. Each hexagram is labeled using the classical Yin (⚋) and Yang (⚊) lines

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