The Pythagorean Plato

204 Pages·2010·4.32 MB·English

The Pythagorean Plato
The
Pythagorean Plato
Prelude to the Song Itself
Ernest G. McClain
Nicolas-Hays, Inc.
York Beach, Maine
First published in 1978 by Nicolas-Hays,
Inc.
Box 612
York Beach, Maine 03910
First paperback edition 1984 Copyright © Ernest
G. McClain
ISBN 0-89254-010-9
Library of Congress Catalog Card Number 77-13355
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
photocopy, recording, or any information storage and retrieval system,
without permission in writing from the publisher.
Distributed by
Samuel Weiser, Inc.
Box 612
York Beach, Maine 03910
Printed in the United States by Mitchell-Shear, Ann
Arbor, Michigan
to Ernst Levy
musician, philosopher, friend
The consideration of which numbers are concordant and which not, and why in each case . . .
don't we know that all of this is a prelude to the song itself, . .. the song itself that dialectic
performs?
(Republic 531c-532a)
Contents
Acknowledgements vii
Charts and Tables ix
1. Introduction 1
2. The Marriage Allegory (Republic) 17
3. The Tyrant's Allegory (Republic) 33
4. The Myth of Er (Republic) 41
5. The Creation of the World-Soul (Timaeus) 57
6. Ancient Athens (Critias) 71
7. Atlantis (Critias) 77
8. Magnesia (Laws) 97
9. Plato's Musical Trigonometry 117
10. Conclusions 127
Appendices
I. Historical Commentary 135
II. Conversion Tables 165
III. Tone Numbers in the Marriage Allegory 167
IV. Introduction to the Monochord 169
Footnotes 177
Index 189
Acknowledgements
My book develops certain musical insights of Ernst Levy and Siegmund
Levarie into Pythagoreanism in general and Plato in particular, and owes its
existence to their encouragement and assistance over a period of many years.
My further collaboration since 1974 with Antonio T. de Nicolás, reported in
The Myth of Invariance, uncovered the musical-mathematical foundations of
this material within the much older Hindu sacred writings and permitted Plato's
story to be told with greater confidence and in fewer words than I had thought
possible. Throughout this work Richard Sacksteder has been a wise and patient
counselor on mathematics, and Malcolm Brown has contributed a fund of
insights into ancient Greek philosophy. I profited greatly from comments and
criticism by Robert S. Brumbaugh, Edwin Davis, Vera Lachman, Patrick
Milburn, Wendell Mordy, John Rouse, Robert Sanders, Trevor J. Saunders,
Harvey Wheeler, and Francis D. Wormuth, who read all or parts of various
earlier versions of the manuscript. Robert Lawlor inspired the discovery of the
material in Chapter 9. This final version of the book has been read in its entirety
by Malcolm Brown, Siegmund Levarie, Antonio T. de Nicolás, and Richard
Sacksteder and incorporates a multitude of their suggestions. Sins of
commission and omission, however, must be charged solely to the author, who
never found it possible to do justice to all the insights of his advisors, each of
whom could add very much to the story told here. Neil Litt, my editor, has
made the production of my book a pleasure.
The author and publisher gratefully acknowledge permission to quote
extensively from the following translations of Plato's dialogues:
The Republic of Plato, translated with notes and an interpretive essay by
Allan Bloom, © 1968 by Allan Bloom, Basic Books, Inc., Publishers, New
York.
Plato: The Laws, translated by Trevor J. Saunders, © Trevor J.
Saunders, 1970, Penguin Classics, Harmondsworth.
Timaeus and Critias, A. E. Taylor, Methuen and Co. Ltd., London,
1929.
Plato's Cosmology, Francis M. Cornford, Humanities Press Inc., New
Jersey, and Routledge & Kegan Paul, Ltd., London, 1937.
Robert J. Brumbaugh has kindly consented to the reproduction of his
maps of Atlantis and Magnesia (cf. figs. 32 and 57) from his work, Plato's
Mathematical Imagination, (Bloomington: Indiana University Press, 1954, and
New York: Kraus Reprint Corporation, 1968).
The following journals have graciously permitted the use of material
which first appeared as articles:
Main Currents in Modern Thought, “Plato's Musical Cosmology,” vol.
30, no. 1, Sept.—Oct. 1973, revised here as chapters 1 and 6.
Journal of Music Theory, “Musical Marriages in Plato's Republic,” vol.
18.2, Fall 1974, revised here as chapter 2.
Music and Man, “A New Look at Plato's Timaeus,” vol 1, no. 4, 1975,
revised here as chapter 5; and “Thirty Seven Musical Guardian's in
Plato's Laws,” vol. 3, no. 1, 1977, revised here as
chapter 8.
Interdisciplina, “The Tyrant's Allegory,” vol. 1, no. 3, Spring, 1976,
revised here as chapter 3.
The Mathematics Teacher, “Pythagorean Paper Folding: A Study in
Tuning and Temperament,” vol. 63, no. 3, March 1970, revised here as
appendix IV.
Charts and Tables
1. The Equal-Tempered Scale 4
2. Circular Projection of the Musical Proportion 11
3. Cycles of Barrenness 20
4. Equivalent Matrices for Just Tuning 22
5. A Platonic Wedding 24
6. The Chromatic Scale as a Tonal Calendar 26
7. Diameters of the Square 28
8. Numbers 2p3q5r ≤ 12,960,000 in Logarithmic Arrays 29
9. The Elimination of Bastard and Slavish Pleasures 36
10. The Cube of Nine as 2 38
11. The Dorian Tetrachord and Its Reciprocal 39
12. The Demonic Plain 44
13. Tonal Symmetry of the Demonic Plain 44
14. Cyclic Re€du ction of the Demonic Plain 45
15. Er's Five-Day Journey to Heaven 45
16. The Spindle of Necessity 49
17. Coordination of Colors and Tones 51
18. Rotation and Counter-Rotation of the Model 52
19. Coordination of Tones and Planets 53
20. The Thrones of Necessity and the Fates 54
21. Portions of the World-Soul “Exponed in One Row” 61
22. The Lambda Pattern of Nicomachus 63
23. Splitting the Fabric Lengthwise 65
24. Plato's Cross (Chi = X) 65
25. Circles of the Same and the Different 66
26. Plato's Cross With the Means Inserted 67
27. An Interpretation of Plato's Diagonals 67
28. Motion of the Circle of the Same 68
29. The Two Springs, 4:3 and 5:4 81
30. The Islands of Atlantis 85
31. The Plain of Atlantis 90
32. Robert Brumbaugh's Map of Magnesia 100
33. Arithmetical Computation of the Eighteen Guardians 102
34. Eighteen “Parent” Guardians as a Tone-circle 104
35. Determination of the Nineteen New Arrivals 107
36. The Thirty-Seven Guardians of Magnesia 108
37. The Guardians as a Sequence of Superparticular Ratios 110
38. Guardians as Smallest Integers 112
39. Platonic Guardians as a Tuning System 114
40. Trigonometric Functions of the 37 Guardians 118
41. Symmetric Trigonometric Functions within 432,000 120
42. Consecutive Guardians as Generators of Consecutive
Pythagorean Triangles 123
43. Tonal Interpretation of Plimpton 322 124
44. The Construction of Pythagorean Triangles 125
45. The Greater Perfect System of the Sectio Canonis 142
46. The Multiplication Table for 10 × 10 146
47. The Pythagorean Table of von Thimus 148
48. Ptolemy's Tonal Zodiac 151
49. Monochord Analysis of Archytas' Tunings 152
50. The Ancient Greek Modes of Aristides Quintilianus 155
51. Dorian as the Comprehensive Mode 156
52. Historical Solutions of the Timaeus Scale 158
53. Philolaus' Division of the Wholetone into 9 Commas 161
54. The Monochord 170
55. Pythagorean Tuning, Just Tuning, and Equal Temperament
on the Monochord 174
56. The Tyrant's Number 180
57. Canals on the Atlantean Plain 183