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


*

Afterword

       609

Partial-Tone Logarithms

       612

Great Table of Ratios

       621

The Author’s Publications

       630