Contents

Volume I

 Translation Society Preface i Translator’s & Editor’s Preface v Foreword vii Introduction xii

A. The Theorems of the Tone-Number

 §1.   The Monochord 1 §2.   Tone-Value and Tone-Number 3 §3.   Tone-Value 5 §4.   Tone-Number 9 §5.   String Lengths – Wavelength 11 §6.   Vibration-Numbers = Frequencies 13 §7.   The Reciprocity of Waves = String Length and Frequency         (Space and Time) 14 §8.   Rhythm and Periodicity 19 §9.   Resonance 28 §10. Interference 32 §11. Haptic – Ektypic – Prototype – Value-Form – Theorem,         Step – Being - Value 41 §12. Coincidences of the Theorems of Tone-Number and Their         Value-Formal Ambivalences 43

B. The Theorems of the Tone Series

 §13. The Overtone Series 46 §14. The Undertone Series 54 §15. The Whole Numbers 57 §16. The Quanta 60 §17. The Intervals 67 §18. Harmonic Logarithms 80 §19. Convergence – Divergence And Perspective – Equidistance 98

C. The Theorems of the Tone-Groups

I.       The Partial-Tone Diagram

 §20. The Partial-Tone Coordinates 114 §21. The Rationing of Partial-Tone Coordinates 122 §22. Major and Minor Series of the Partial-Tone Coordinates 131 §23. Polarity 148 §24. The Equal-Tone Lines 157 §25. Value-Forms for 0/0, 1/1, and the Equal-Tone Lines 176 §26. Index and Generator 187 §27. Parabola, Hyperbola, Ellipse 195 §28. The Proportions 204 §29. Harmonic Proportions in Architecture 227 §30. Trinity 247

II.      The Arrangements

 §31. Variation Models 254 §32. Combination Models 267 §33. Polar Diagrams 287 §34. Tone-Spirals and Tone Curves 304 §35. The Complete Partial-Tone Diagram 317 §36. Logarithmic Arrangements 326 §37. Tone-Space 338 §38. Sound-Images 359

Volume II

D. Selections

 §39. Scales 378 §40. Chords 416 §41. Ektypics for Melodies and Chords 433 §42. Cadence 444 §43. Counterpoint 446 §44. Directions 454 §45. Interval Powers and Constants 457 §46. Symmetry 468 §47. Space and Time 474 §48. Enharmonics 482 §49. Tempering 490 §50. Number Symbolism 494 §51. Tolerance: a Dialogue (and Review) 520 §52. Hierarchy 531 §53. Norm 539 §54. Harmonic Cosmogony 554

E. History

 §55. Summary of the History of Harmonics 594

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 Afterword 609 Partial-Tone Logarithms 612 Great Table of Ratios 621 The Author’s Publications 630