Part I. Preliminaries: Introduction; 1. Beginnings, and the problem of measurement; Part II. Empirical Harmonics: 2. Empirical harmonics before Aristoxenus; 3. The early empiricists in their cultural and intellectual contexts; 4. Interlude on Aristotle's account of a science and its methods; 5. Aristoxenus: the composition of the Elementa harmonica; 6. Aristoxenus: concepts and methods in Elementa harmonica Book 1; 7. Elementa harmonica Books 2-3: the science reconsidered; 8. Elementa harmonica Book 3 and its missing sequel; 9. Contexts and purposes of Aristoxenus' harmonics; Part III. Mathematical Harmonics: 10. Pythagorean harmonics in the fifth century: Philolaus; 11. Developments in Pythagorean harmonics: Archytas; 12. Plato; 13. Aristotle on the harmonic sciences; 14. Systematising mathematical harmonics: the Sectio canonis; 15. Quantification under attack: Theophrastus' critique; Postscript: The later centuries.