Summary Music and mathematics have been connected since ancient times. During the twentieth century, however, many composers consciously started using many mathematical concepts, algebraic operations and theorems as bases for their creative processes. The first part of this volume deals with the relationship between music and mathematics in the music of composers such as Olivier Messiaen, Iannis Xenakis, Franco Evangelisti, Pierre Boulez, Arvo P rt, Steve Reich and Philip Glass. A chapter is then dedicated to Spanish composers of the last forty years (Francisco Guerrero, Alberto Posadas, Jos Mar a S nchez-Verd , Iluminada P rez Frutos, Nur a Gimenez Comas, Helga Arias and Jos L pez Montes). The theme continues in the second part of the book through the examination of prominent theories (Neo-Riemannian theory, diatonic set theory, theory of musical kaleidocycles), the use of diagrams and charts in music, the algorithmic evolution of music, contemporary compositional practices inspired by mathematical concepts; it arrives at studies on double canons and trichords. We publish a philosophical study in order to ask what the future of the relationship between music and mathematics will be at the beginning of the 21st century. Includes an introduction by Massimiliano Locanto. TABLE OF CONTENTS Roberto Illiano Preface Massimiliano Locanto Music Composition, Mathematics, and the Modernist Legacy Twentieth-Century Composers and Mathematics Athanase Papadopoulos Messiaen et les math matiques Ronald Squibbs Xenakis's Method of Stochastic Composition: A Brief Introduction Peter Hoffmann My Music Makes no Revolution : Thoughts on the Role of Mathematics in the Work of Iannis Xenakis Alessandro Mastropietro Simboli matematici nei titoli delle composizioni di Franco Evangelisti C. Catherine Losada Boulez and Mathematics Andrew Shenton 1 + 1 = 1: The Elengant Mathematics of Arvo P rt's Tintinnabulation Joel Haack Mathematical Considerations of the Rhythmic Patterns in the Music of Steve Reich Richard Cohn Glass Graphs Pedro Ord ez Eslava Ars combinatoria: po ticas del n mero en la composici n espa ola contempor nea (1980-2018). Breve antolog a inacabada Music and Mathematics: Theories, Practices, Future Henry Klumpenhouwer Neo-Riemannian Theory Franck Jedrzejewski Graphes et diagrammes, l'en-de de la composition Angelo Orcalli L'evoluzione algoritmica della musica. Omaggio a Jean-Claude Risset Jos L. Besada Where Science-Based Music Comes From: Some Cognitive Remarks upon Contemporary Compositional Practices Inspired by Mathematical Concepts, Operations, and Objects Richard Hermann On Near Maximally Even Set-Classes Guerino Mazzola The Role of Mathematics for Music in Theory, Composition, and Performance Luigi Verdi Due esempi di Chim(us)ica Dodici canoni tricordali e loro combinazioni. Caleidocicli e canoni ritmici Fran ois Nicolas Du partage moderne/contemporain en math matiques et en musique la lumi re de Fernando Zalamea (Philosophie syst matique de la math matique contemporaine) Abstracts Biographies Index of Names