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.
Prof. Dr. Guerino Mazzola earned his Ph.D. in Mathematics from Zurich University. He wrote the groundbreaking book The Topos of Music in 2002, its formal language and models are used by leading researchers in Europe, India, Japan, and North America and have become a foundation of music software design. Prof. Mazzola has an appointment as professor in the School of Music at the College of Liberal Arts, University of Minnesota.
Maria Caterina Mannone is completing her Ph.D. in the School of Music of the University of Minnesota.Yan Pang Clark is completing her Ph.D. in the School of Music of the University of Minnesota.
"The authors try to develop a discourse full of pleasure and fun
that in every moment motivates concepts, methods, and results by
their musical significance-a narrative that inspires the reader to
create musical thoughts and actions. ... The book contains many
interesting musical and mathematical examples and exercises, and
the last part of the book is devoted to the solutions of exercises.
The book has also an interesting list of references for further
studies in this field." (Peyman Nasehpour, Mathematical Reviews,
"This textbook in mathematics for music theorists introduces topics such as sets and functions, algebraic structures including groups, rings, matrices and modules, and more. The book includes many illustrations, online sample music files, and exercises with solutions. ... Concepts are motivated and supported by examples from composition music theory." (Tom Schulte, MAA Reviews, maa.org, February, 2017)