In the third and final book of his iconic piano etudes György Ligeti charts a new path relative to the rest of his musical output, representing a significant arrival in a composer’s oeuvre known for its stylistic transformations. This monograph is the first dedicated study of these capstone works, investigating them through a novel lens of statistical-graphical analysis that illuminates their compositional uniqueness as well as broader questions regarding the perception of stability in musical texture. With nearly 200 graphical illustrations and a detailed commentary, this examination reveals the unique manner in which Ligeti treads between tonality and atonality—a key idea in his late style—and the centrality of processes related to broader scale areas (or “macroharmony”) in articulating structures and narratives. The analytical techniques developed here are a powerful tool for investigating macroharmonic stability that can be applied to a wide range of repertoire beyond these works. This book is intended for graduate-level and professional music theorists, musicologists, performers and mathematicians.

How is the Beatles' "Help!" similar to Stravinsky's "Dance of the Adolescents?" How does Radiohead's "Just" relate to the improvisations of Bill Evans? And how do Chopin's works exploit the non-Euclidean geometry of musical chords? In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an age-old question: what makes music sound good? A Geometry of Music provides an accessible introduction to Tymoczko's revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing relationships among familiar chords and scales, giving readers the tools to translate between the musical and visual realms and revealing surprising degrees of structure in otherwise hard-to-understand pieces. Tymoczko uses this theoretical foundation to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the "common practice" period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. He discusses a host of familiar pieces by a wide range of composers, from Bach to the Beatles, Mozart to Miles Davis, and many in between. A Geometry of Music is accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers' understanding, along with a series of appendices that explore the technical details of this exciting new theory.

Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models--the first examples of the musical spaces of the book's title--depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, diatonic theory, and scale theory. Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques.

In the third and final book of his iconic piano etudes György Ligeti charts a new path relative to the rest of his musical output, representing a significant arrival in a composer's oeuvre known for its stylistic transformations. This monograph is the first dedicated study of these capstone works, investigating them through a novel lens of statistical-graphical analysis that illuminates their compositional uniqueness as well as broader questions regarding the perception of stability in musical texture. With nearly 200 graphical illustrations and a detailed commentary, this examination reveals the unique manner in which Ligeti treads between tonality and atonality-a key idea in his late style-and the centrality of processes related to broader scale areas (or "macroharmony") in articulating structures and narratives. The analytical techniques developed here are a powerful tool for investigating macroharmonic stability that can be applied to a wide range of repertoire beyond these works. This book is intended for graduate-level and professional music theorists, musicologists, performers and mathematicians.

Afterword by Frank O'Hara Morton Feldman (1926-1987) is among the most influential American composers of the 20th Century. While his music is known for its exteme quiet and delicate beauty, Feldman himself was famously large and loud. His writings are both funny and illuminating, not only about his own music but about the entire New York School of painters, poets and composers that coalesced in the 1950s, including his friends Jackson Pollack, Philip Guston, Mark Rothko, Robert Rauschenberg, Frank O Hara, and John Cage.

With contributions by numerous experts

Exploring the application of Bayesian probabilistic modeling techniques to musical issues, including the perception of key and meter.