In the history of music, one finds many different tone systems, the musical scales. From the most ancient times, their origin was related to the mathematical relations between the musical sounds. In this book, the history of the development of the theoretical ideas about the structure of the musical scales is presented at a popular level, in comparison with the history of music. Early scales are described along with various combinatory scales, as well as the well-tempered twelve-tone scale, the basis of contemporary music. The principal elements of the modern theory of pitch scale formation are discussed on the basis of the general psychological theory of activity, information theory, quantum mechanics, and the general hierarchical approach. A new universal musical scale of 19 tones is introduced, and the perspectives of the development of the musical pitch systems are outlined. The book does not require any preliminary acquaintance with the theory of music, and the usual secondary school mathematics is enough to understand the mathematic issues involved. The book might be useful for those who like music and are interested in the applications of mathematics.