This textbook is a first introduction to mathematics for music theorists, covering basic topics such as sets and functions, universal properties, numbers and recursion, graphs, groups, rings, matrices and modules, continuity, calculus, and gestures. It approaches these abstract themes in a new way: Every concept or theorem is motivated and illustrated by examples from music theory (such as harmony, counterpoint, tuning), composition (e.g., classical combinatorics, dodecaphonic composition), and gestural performance. The book includes many illustrations, and exercises with solutions.

Based on authors' experience teaching mathematics to music theorists Presents ideas in a critical manner, highlighting problems with earlier teaching approaches Includes many illustrations and exercises with solutions Includes supplementary material: sn.pub/extras

Autorentext

Linshujie Zheng is a soprano. She has a master's degree with distinction at Cardiff University. Currently she pursues a doctoral degree in voice at the University of Minnesota. Zheng has received awards and scholarships in China, the UK, and the US. She has performed at the Royal Albert Hall, Wales Millennium centre, and St David's Hall. She also sung principal roles, including Zerlina in Don Giovanni, Genovieve in Suor Angelica, Susanna in Le nozze di Figaro. Zheng not only focuses on the performing arts, she also has a passion for Gesture and reception theory. Guerino Mazzola qualified as a professor in mathematics and in computational science at the University of Zürich. Visiting professor at the ENS in Paris in 2005. Since 2007 professor at the School of Music, University of Minnesota. He developed a Mathematical Music Theory and music software presto and Rubato. 2007-2021 he was the president of the Societyfor Mathematics and Computation in Music. He has published 33 books and 150 papers, 27 jazz CDs, two DVDs, and a classical sonata. His most important book is "The Topos of Music", about mathematical music, performance, and gesture theory.

Inhalt

Part I: Introduction and Short History.- The 'Counterpoint' of Mathematics and Music.- Short History of the Relationship Between Mathematics and Music.- Part II: Sets and Functions.- The Architecture of Sets.- Functions and Relations.- Universal Properties.- Part III: Numbers.- Natural Numbers.- Recursion.- Natural Arithmetic.- Euclid and Normal Forms.- Integers.- Rationals.- Real Numbers.- Roots, Logarithms, and Normal Forms.- Complex Numbers.- Part IV: Graphs and Nerves.- Directed and Undirected Graphs.- Nerves.- Part V: Monoids and Groups.- Monoids.- Groups.- Group Actions, Subgroups, Quotients, and Products.- Permutation Groups.- The Third Torus and Counterpoint.- Coltrane's Giant Steps.- Modulation Theory.- Part VI: Rings and Modules.- Rings and Fields.- Primes.- Matrices.- Modules.- Just Tuning.- Categories.- Part VII: Continuity and Calculus.- Continuity.- Differentiability.- Performance.- Gestures.- Part VIII: Solutions, References, Index.- Solutions of Exercises.- References.- Index.