This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book focuses squarely on the content of certain mathematicians' and natural philosophers' engagement with the theory of music, and examines that content in more depth than has previously been attempted. It shows that much of the musical theorizing of the seventeenth century made a distinction between two distinct faculties of hearing, in an attempt to resolve the paradox of the hearing's characteristics. The book discusses Mengoli's work in some detail, a discussion which ultimately sheds light on many other matters. It examines in depth three English theorists, namely John Birchensha, John Wallis and Thomas Salmon, who answered or seemed likely to answer those questions: What is the proper method for the mathematical study of music? What is the proper place of musical performance and musical judgement in such a study?