The book traces the development of the tonal system from ancient Greek times from a mathematical perspective. Topics include the relationship between equal temperament and just intonation, as well as the problem of key characteristics. And why is there actually no 13-tone music? The overtone series of the string, various pipes, and the timpani is derived from the wave equation and the respective boundary conditions. The sound is determined by the initial values, and thus by the artist. Bessel functions are responsible for the timpani. Sound figures and amplitude diagrams calculated by the author illustrate the concepts. Despite all the mathematics, the book bridges the gap to music in a cultural-historical context.