Abstract

Objectives: This paper advances a unified analytical perspective on Anatol Vieru's „Book of Modes” and Dan Tudor Vuza's rhythmic canons by explicitly demonstrating that both systems instantiate the same underlying algebraic model: the action of cyclic groups on discrete sets. The original contribution of the study lies in establishing, for the first time, a systematic structural correspondence between Vieru's modal theory and Vuza's rhythmic constructions, thereby extending group-theoretical analysis beyond parameter-specific applications toward a genuinely parameter-neutral framework.

Methods: Drawing on mathematical music theory and transformational analysis, the paper adopts the cyclic group ℤₙ as a standard formal model for pitch-class organization and discrete temporal cycles. Vieru's modes are formalized as subsets of ℤ12 defined by intervallic distributions and invariance under transposition, and are interpreted as orbits under a group action. Vuza canons are analyzed through their strict mathematical definition as aperiodic factorizations of finite cyclic groups, followed by a systematic translation of these algebraic properties into musical terms (canon entries, rhythmic complementation, and temporal unfolding).

Results: The analysis demonstrates that Vieru's modal theory constitutes a coherent system of structural classes with well-defined group-theoretical properties that account for its generative role in composition, independently of tonal or functional hierarchies. In parallel, Vuza's canons are shown to realize, in the temporal domain, the same structural principles of invariance, complementation, and non-periodicity that govern Vieru's pitch organization. The study thereby identifies a shared algebraic logic underlying two bodies of work traditionally discussed in separate scholarly contexts.

Conclusions and Contribution: By articulating an explicit structural equivalence between modal and rhythmic organization, the paper proposes a novel methodological model for cross-parametric analysis in twentieth-century music. It shows that group theory can function not merely as a parameter-specific analytical tool, but as a unifying theoretical framework capable of bridging compositional theory, mathematical construction, and analytical practice. This perspective opens new analytical pathways for understanding structurally oriented repertoires beyond the tonal/post-tonal divide. It situates Eastern European contributions within a broader international discourse on mathematical approaches to music analysis.

Selected Bibliography

Amiot, E. (2024). Une introduction aux mathématiques de la musique. Paris: Calvage & Mounet.
Lewin, D. (2011). Generalized Musical Intervals and Transformations. Oxford: OUP.
Montiel, M., & Peck, R. W. (Eds.). (2019). Mathematical Music Theory. World Scientific.
Vieru, A. (1980). Cartea modurilor. Bucharest: Editura Muzicală.

Vuza, D. T. (1991–1992). Supplementary Sets and Regular Complementary Unending Canons. Perspectives of New Music.