Chapter Four. Pythagorean Harmonic Ratios from the Octave to the Comma by Continuous Subtraction

Chapters in this book

1. Frontmatter i
2. Contents vii
3. Acknowledgments ix
4. Note to the Reader xi
5. Introduction 1
6. Part One. The Ancient Greek Tradition in Practice and Theory
7. Chapter One. The Ancient Harmoniai 11
8. Chapter Two. The Tonoi 24
9. Chapter Three. Alypian Notation 42
10. Part Two. Mathematical Foundations
11. Chapter Four. Pythagorean Harmonic Ratios from the Octave to the Comma by Continuous Subtraction 53
12. Chapter Five. Boethius’s Error in the De institutione musica, 4.6 56
13. Chapter Six. Aristoxenus’s Proof That the Perfect Fourth Is the Sum of Two Tones and a Semitone 60
14. Chapter Seven. Aristoxenus’s Anticipation of the Logarithmic Logic of Musical Cognition 63
15. Chapter Eight. The Three Mathematical Means in the Theories of Euclid, Boethius, Glarean, and Zarlino 68
16. Chapter Nine. Guido and the Monochord 77
17. Part Three. Emerging Theories of the Ecclesiastical Modes
18. Chapter Ten. Transposition and the Doctrine of Modal Affinity 85
19. Chapter Eleven. The Misunderstood Confinalis 92
20. Chapter Twelve. Reading the First Quidam of the Alia Musica 111
21. Chapter Thirteen. The Prologus in tonarium of Bern of Reichenau: A Translation 131
22. Chapter Fourteen. Reading Hermannus 149
23. Chapter Fifteen. Idealist and Empirical Perspectives in Theories of the Ecclesiastical Modes 162
24. Glossary of Terms 171
25. Notes 177
26. Bibliography 213
27. Index 219