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 Reductio€ n 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 1 1 Introduction THE PROBLEM Plato's later dialogues abound in mathematical allegories. Timaeus begins with a very long one, Statesman contains a short one, the Republic has three, and both Critias and Laws are permeated with them from beginning to end. When Plato died in 347 B.C. his pupils and friends immediately began to argue about these mathematical constructions and about Plato's purpose in using them for models of souls, cities, and the planetary system. By the beginning of the Christian era much of Plato's mathematics had become a riddle. Many rivals clamored for recognition as the “single harmony” Socrates heard from the planets.1 A certain number which he confidently proclaimed “sovereign” in political theory was labelled “numero Platonis obscurius” by Cicero (c. 100 A.D.), with the hearty concurrence of later scholars; an interpretation which Nicomachus promised at about this time was either lost or never written.2 By the fifth century A.D., Proclus, one of the last to head the Platonic Academy, could not pretend to understand Plato's arithmetic, although he was astute enough to label as spurious a then popular interpretation of the Timaeus “World-Soul.” Down through history Plato's mathematical allegories defied Platonists either to reconstruct his arithmetic or to find in it the implications he claimed for it. The twentieth century opened with a vigorous new effort. In 1901 James Adam searched for the common-sense implications of the allegories of the Republic, rejecting as spurious most of the interpretation of the intervening millenia, and setting a new standard of rigor in remaining respectful of Plato's precise words.3 In 1928 A.E. Taylor performed the same service for Timaeus, gleaning from his effort one principle which—had he known how to apply it— might have saved him from the failure which plagued his predecessors: Plato, he wrote, “demands perfect symmetry.”4 In 1937 Francis Cornford, perhaps the most prestigious of 20th century Platonists, inadvertently took a step backwards. Cornford concluded that the difficulties which arise in abstracting a planetary system from Plato's musical arithmetic in 2 Timaeus are due to a metal “armillary sphere” which the Academy possessed (and an unworkable one, at that): “Plato probably had it before him as he wrote.”5 In 1945, in his translation of the Republic, Cornford not only omitted “the extremely obscure description” of Socrates' “sovereign number,” but he also allowed himself to “simplify the text” of the tyrant's allegory.6 Then, in 1954, Robert Brumbaugh opened a whole new vista on Plato's mathematics.