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1. Introduction 2. Reason and perception 3. Pitch and quantity 4. The ratios of the concords: (1) The Pythagoreans 5. The ratios of the concords: (2) Ptolemy's hupotheseis 6. Critique of Aristoxenian principles and conclusions 7. Ptolemy on the harmonic divisions of his predecessors 8. Melodic intervals: hupotheseis, derivations and adjustments 9. Larger systems: modulations in music and in method 10. The instruments 11. The tests 12. Harmonics in a wider perspective.