A detailed study of the well-known, yet poorly understood, music theory of composer Paul Hindemith (1895-1963). The music theory of composer Paul Hindemith (1895-1963), originally entitled Unterweisung im Tonsatz, is well known, yet poorly understood. This book provides a critical engagement with Hindemith's Unterweisung, particularly concerning its relationship to existing acoustic music theories. By examining different Unterweisung-versions, it charts the evolution of Hindemith's use of language and mode of communication, including his reference to polytonality, atonality, Fuxian species counterpoint, and avoidance of existing music for his examples. It also elaborates the source material on which the theory is based, using a reconstruction of Hindemith's personal library. Central to the book is the relationship of Hindemith's Unterweisung to his compositional practice. Hindemith's fascination with the challenges of music theory falls into a middle period in his oeuvre, enabling profitable comparisons with his compositional practice both before and after his theory-making. The book also comprises a detailed discussion of Hindemith's theoretical and compositional legacy. Beginning with an overview of existing polemics, it draws together unpublished materials from the Yale Hindemith Institute with reminiscences from former students to construct an Unterweisung reception history. The book shows that, while many areas of Hindemith's theory have been overtaken by recent interests in music theory that relate to cognition and geometry, his influence has been deeply felt. SIMON DESBRUSLAIS is Lecturer in Music and Director of Performance at the University of Hull and an internationally acclaimed trumpet soloist.