This book examines, for the first time, the scientific procedures devised by Ptolemy (second century AD) for investigating the structures underlying musical melody, a project that he conceived as closely related to astronomy. Ptolemy's account of his methods is unusually explicit, and he pursues them faithfully. By providing an analysis of Ptolemy's sophisticated theoretical apparatus, his strategies for integrating theory with observation, and his meticulous instructions for the design and conduct of experimental tests, the book offers historians of science a new starting-point for wider studies of ancient scientific method.