A Theory of Sets and Harmonic Complexity in Just Intonation and its Equal Temperament Approximations in the Chamber Music of Georg Friedrich Haas

Abstract

Just Intonation (JI), with its foundation in the harmonic series, resists many of the analytical tools developed for music in equal temperament (ET). Ratio-based approaches offer precision in describing JI’s intervallic relationships but struggle to account for larger-scale harmonic organization, while set-theoretic models developed for ET rarely engage with JI’s rational tuning systems. In this dissertation, I develop a set-theoretic framework for JI that remains attuned to its acoustic and perceptual realities while expanding the analytical vocabulary available for microtonal music. I examine two principal traditions in contemporary JI composition: extended JI, which applies systematic and transformational approaches to rational tuning, and spectral practices that explore the fluid interactions between JI and ET. These analytical tools are applied to a diverse repertoire, ranging from works that rigorously adhere to JI structures to those that blur the boundaries between tuning systems.The dissertation is divided into two parts, each addressing distinct but interrelated aspects of JI and its applications. Part 1, “A Theory of Sets and Harmonic Complexity in Just Intonation,” introduces the theoretical framework and analytical tools underpinning the study. It comprises three chapters: Chapter 1 provides a general introduction, Chapter 2 defines the key concepts and tools, and Chapter 3 demonstrates their application in analyzing contemporary compositions employing JI. Part 2, “Equal Temperament Approximations and Harmonic Complexity in the Chamber Music of Georg Friedrich Haas,” builds on the methods and tools from Part 1 to explore Haas’s integration of JI and ET in his music. This part includes four chapters: Chapter 4 introduces Haas’s chamber music up to 2015, highlighting his distinctive style and techniques. Chapter 5 focuses on what I call static harmony in Haas’s works, while Chapter 6 examines his use of progressive harmony. Chapter 7 concludes Part 2 by discussing large-scale structure in select chamber works and offering a summary of the dissertation with directions for future research.