Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Figures -- Preface -- Part I Preliminaries -- Introduction -- The agenda of greek harmonics -- A note on the 'perfect systems' -- A note on the arrangement of this book -- 1 Beginnings, and the problem of measurement -- Musical intervals as linear distances -- Musical intervals as ratios -- The two systems of measurement compared -- Part II Empirical harmonics -- 2 Empirical harmonics before Aristoxenus -- The evidence of plato -- The evidence of aristoxenus -- (a) The harmonikoi and the enharmonic genus -- (b) The harmonikoi and their diagrams -- (c) Eratocles' systematisation of the 'ancient harmoniai' -- (d) Empirical studies of the tonoi -- (e) Harmonics and the study of instruments -- (f ) Harmonics and melodic notation -- 3 The early empiricists in their cultural and intellectual contexts -- Harmonic theorists in the world of the sophists -- Harmonic theorists as practical musicians -- The purposes of early empirical harmonics -- 4 Interlude on Aristotle's account of a science and its methods -- 5 Aristoxenus: the composition of the Elementa harmonica -- Aristoxenus' life and writings -- The structure of the elementa harmonica -- (a)A preliminary survey -- (b)The programmes announced in Books iand ii -- (c)Relations between Book iiiand Books iand ii -- (d)The first treatise and its revisions in the second -- The evidence of porphyry -- 6 Aristoxenus: concepts and methods in Elementa harmonica Book i -- The scope and limits of harmonics -- The movement of the singing voice -- Formal objectives and procedures -- The nature of melos -- 7 Elementa harmonica Books ii- iii: the science reconsidered -- Three approaches contrasted -- Perception, thought and memory -- The role of non-quantitative discriminations, and the concept of dynamis.

The reflections of book ii on apodeixis, 'demonstration' -- 8 Elementa harmonica Book iiiand its missing sequel -- Two specimen theorems -- How the collection of theorems is arranged -- Three major problems -- Tonoi, modulation, and the missing continuation to book iii -- 9 Contexts and purposes of Aristoxenus' harmonics -- Harmonics and musical composition -- Harmonics and musical criticism -- Technical knowledge and practical analysis -- Technical knowledge and critical judgement -- Critical judgement, ethos and the 'appropriate' -- Evaluative judgement, ethos and 'ethics' -- Part III Mathematical harmonics -- 10 Pythagorean harmonics in the fifth century: Philolaus -- Fragment 6a: preliminary analysis -- The evidence of boethius -- Boethius and fragment 6a -- The musical structure of philolaus' attunement -- Harmonics and cosmology -- 11 Developments in Pythagorean harmonics: Archytas -- Archytas and ptolemy's 'principles of reason' -- Archytas' divisions of the tetrachord: mathematical principles and musical observations -- The three mathematical means -- Archytas' theorem on the division of epimoric ratios -- Harmonics, physical acoustics and musical practice -- 12 Plato -- Musical ethics in the republic and the laws -- The philosophical context of the republic's discussion of harmonics -- Harmonics in the republic -- Harmonics in the timaeus: the soul of the universe -- The timaeus on harmonics and human psychology -- Harmonics in the ivory tower -- 13 Aristotle on the harmonic sciences -- An aristotelian fragment on pythagorean harmonics -- Aristotle's own uses of mathematical harmonics -- Aristotle on the methodology of harmonic science -- Conclusions -- 14 Systematising mathematical harmonics: the Sectio canonis -- The sectio canonis in ptolemy and porphyry -- The introduction to the treatise -- Propositions 1-9: the mathematical groundwork.

Propositions 10-13: the transition to harmonics -- Propositions 14-18: inequalities and controversial conclusions -- Propositions 19-20: the division of the kanon -- Closing reflections on the sectio and its target readership -- 15 Quantification under attack: Theophrastus' critique -- Arguments against mathematical theorists: the first phase -- Arguments against mathematical theorists: the second phase -- An argument against aristoxenus? -- Theophrastus' exposition of his positive views -- Postscript: the later centuries -- Bibliography -- Index of proper names -- General index.