Reviewed by: Twentieth-Century Music and Mathematics ed. by Roberto Illiano Paul Lombardi Twentieth-Century Music and Mathematics. Edited by Roberto Illiano. (Music, Science & Technology, vol. 1.) Turnhout: Brepols, 2019. [xxxiii, 382 p. ISBN 9782503585703 (hardcover), $156.] Music examples, illustrations, bibliography, abstracts, biographies, index. Twentieth-Century Music and Mathematics is a compilation of seventeen essays (and an introduction) in English, French, Italian, and Spanish. The first of its two parts, "Twentieth-Century Composers and Mathematics," focuses on specific composers—namely Olivier Messiaen, Iannis Xenakis, Franco Evangelisti, Pierre Boulez, Arvo Pärt, Steve Reich, and Philip Glass (but curiously not Milton Babbitt)—and the last chapter in this part covers a handful of contemporary Spanish composers. The second part, "Music and Mathematics: Theories, Practices, Future," focuses on recent theoretical trends. Although the book has "twentieth century" in its title, its topics are mostly from only the second half of the twentieth century and extend to the beginning of the twenty-first century. It is less of a book on music theory than on the history of music theory, which is by no means a drawback. In the introduction, Massimiliano Locanto explains: My aim, rather, is to put the relationships between mathematics and music into a historical-philosophical context by drawing attention to some crucial moments when the two disciplines adopted similar methodological strategies and were attracted to analogous paradigmatic concepts, thus finding a point of convergence on a fundamental epistemological level. (p. xi) The authors of the essays consider questions about mathematics and musical thought in that they "represent the increasing attraction to mathematical thought as a specific trend of twentieth-century musical modernism" (p. xi). The book in general takes the stance that the composers examined in it are not mathematicians and tend not to directly apply mathematics in their creative processes, but rather that mathematics informs their processes and aesthetics. The book takes a historical-philosophical approach to the mathematics of music. The mathematical details presented are not convoluted, so the layperson will not struggle reading this book. This review does not remark on every author and essay, as every contribution is of comparable merit. Rather, it offers comment on specific essays. Ronald Squibbs and Peter Hoffmann provide two chapters on processes given in Iannis Xenakis's Formalized Music ("Musique formelles," Revue musicale, nos. 253–54 [1963]; expanded English trans. as Formalized Music: Thought and Mathematics in Composition [Bloomington: Indiana University Press, 1971]), the former on stochastic composition and the latter on symbolic composition. Squibbs writes, Xenakis provided a detailed account of the process … in his treatise, Formalized Music, but this account has proven to be difficult to understand by musicians who are not well-trained in mathematics and computer science. … The aim of this modest introduction to stochastic music is to pry that book open a little for musicians (such as myself) who have found Xenakis's description of the process of stochastic composition both tantalizing and difficult to comprehend. (p. 23) Hoffmann also writes, "We cannot be sure that all composers or researchers who have studied Formalized Music really grasped its essence" (p. 41). Both Squibbs and Hoffmann do an excellent job in deciphering the difficult concepts in Formalized Music in their [End Page 640] brief overviews of stochastic and symbolic composition. This volume, however, does not cover Xenakis's strategic composition. C. Catherine Losada has published extensively on the music of Pierre Boulez and, in this volume, shows how mathematical influences inspired Boulez to create "musical systems upon exclusively musical criteria," and define the nature and function of such criteria. In this way, it will yield insight into the music itself, as well as the relationship between the various works in Boulez's oeuvre, thus providing a more nuanced picture of the role of mathematical thinking in Boulez's compositional approach. (p. 91) She outlines how mathematics embodies Boulez's goal to "put contemporary musical thought on a completely and infallibly valid basis" (p. 97, quoting Boulez on Music Today, trans. Susan Bradshaw and Richard Rodney Bennett [Cambridge, MA: Harvard University Press, 1971], 29). Then, she applies her ideas by examining a series of Boulez's compositions from the late 1950s and 1960s. Moving on to...