THE MYTH OF INVARIANCE Ernest G. McClain THE MYTH OF INVARIANCE The Origin of the Gods, Mathematics and Music From the Ṛg Veda to Plato Introduction by Siegmund Levarie Edited by Patrick A. Heelan Nicolas-Hays, Inc. York Beach, Maine v First published in 1976 by Nicolas-Hays, Inc. P.O. Box 612 York Beach, ME 03910 This paperback edition, 1984 Distributed exclusively by Samuel Weiser, Inc. York, Beach, ME 03910 Copyright © Ernest G. McClain All rights reserved. No part of this publication may be reproduced, copied, or transmitted in any form or by any means without permission in writing from the publisher. ISBN 0-89254-012-5 Library of Congress Cataloging in Publication Data McClain, Ernest G The myth of invariance. Includes index. 1. Music—Philosophy and aesthetics. 2. Music and mythology. 3. Music—Theory—To 400. 1. Title. ML3800.M15 780'.1 76 -28411 Printed in the United States by Mitchell-Shear, Inc. Ann Arbor, MI This Child is Augusta's Let us with tuneful skill proclaim these generations of the Gods, That one may see them when these hymns are chanted in a future age. Ṛgveda 10.72.1 “It was clear to me for a long time that the origins of science had their deep roots in a particular myth, that of invariance." Giorgio de Santillana Preface to HAMLET'S MILL vii CONTENTS Charts and Tables ix Introduction by Siegmund Levarie xi Acknowledgments xv Glossary of Terms xix 1. Introduction: the problem ; de Nicolás' challenge; a 1 musical hypothesis; procedure 2. Tone Maṇḍala and Sun's Chariot: the Hindu-Greek 9 diatonic scale 3. Musical Generation: God and Mother; Fathers and 19 Sons; Indra and the Aśvin Pair; Indra as Dancer and Lord of the Five Tribes 4. The Tonal Calendar: 360 "like pegs"; Vision in Long 33 Darkness 5. Algebraic Yantras: star hexagon and drum of Śiva; the 43 holy mountain ; origins of geometry 6. Expansion of the Tone-Number Field: the Bṛhaspati 61 and Prajāpati cycles of 3,600 and 216,000 years vii 7. Cosmic Cycles: the four Yugas; Kalpa and Brahmā 73 cycles; the birth of Agni, incarnate savior; the chariot of the gods; the duration of the universe 8. Music and the Calendar: arithmetic and geometric 95 coincidences; Ptolemy's tonal zodiac 9. The Book of Revelation: the meeting of East and West; 107 New Jerusalem; the Choir of 144,000; the Holy City as 12,0003; the Divine Mother; conclusions; postscript on Genesis 10. Babylon and Sumer: the Pythagorean achievement of 129 ancient Semitic cultures; sexagesimal arithmetic; God on the Mountain; Babylonian and Hebrew floods; Old Testament arithmology; Mayan cosmology; Mt. Meru 11. Lost Atlantis: Plato's musical cities; Gilbert Ryle's 161 hypothesis; the Muses' Jest; the Plain of Atlantis; the Islands of Atlantis; Callipolis and Ancient Athens; Egyptian derivation; Egyptian music, Egyptian mathematics; The Book of the Dead; the Underworld; Khepura 12. Conclusions: the myth of invariance; a mathematical 195 perspective; codetta Appendices I Conversion Tables 203 II Multiplication Tables for Numbers 3p5q 205 III Simplified Acoustical Theory for Fretted Instruments 209 viii CHARTS AND TABLES 1. Foundations of Mathematical Harmonics : equivalent representations of the Hindu-Greek scale xxi 2. Maṇḍala of the Single-Wheeled Chariot of the Sun 10 3. The Hindu-Greek Diatonic Scale and its Reciprocal 13 4. The Tonal Matrix : "God" and "Mother" 20 5. The Harmonic Series and its Reciprocal 23 6. The Musical Proportion : 6: 8::9:12 (arithmetic, geometric and harmonic means as "Indra and the Akin Pair") 26 7. A Platonic "Wedding": Indra as Dancer and Lord of the Five Tribes 29 8. The Tonal Calendar: 360 "like pegs" 34 9. Correlation of Hindu and Greek Tuning Systems 38 10. Sacred Stones as Algebraic Yantras: star-hexagon and "the drum of Śiva" 44 11. Musical Yantras 51 2 12. The Bṛhaspati Yantra for 60 and the Prajāpati Yantra 3 for 60 62 13. Maṇḍala of the Thirty-Three Gods in 603 67 14. The Yugas: 432,000, x2, x3 and x4 74 15. Yantras for Kalpa and Brahmā periods : 4,320,000,000 and 8,640,000,000 "years" 77 16. The Birth of Agni in the Flames of Sacrifice 84 17. The Chariot of the Gods, "Fashioned Mentally" 86 18a. "The Duration of the Universe" or "Life of Brahmā": 155,520,000,000,000 'Years" 88 18b. YHWH = 10-5-6-5 91 19. Tonal and Calendrical "commas" 97 20. Approximations to the Tritone = 2 99 21. Yantra for the Precessional Cycle of 25,920 years 101 22. Maṇḍala for the Precessional Cycle of 25,920 years 102 € ix 23. Tone-Numbers for the Precessional "Octave" 103 24. Ptolemy's Tonal Zodiac 104 25. Yantra for the Celestial Choir of 144,000 "male virgins" 110 26. "New Jerusalem" as 12,0003 = 1,728,000,000,000 114 27. Analysis of the Gestation Number, 1260 119 28. Gestation of the Savior 120 28a. Monochord Harmonization of Bible and Kabbala 127 29. Tonal Interpretation of the Sexagesimal System 132 30. The Babylonian Solution to 2 135 31. The Babylonian Pantheon and the Calendrical Scales on Ptolemy's Monochord 137 32. Monochord Reduction€ of the Babylonian King List 138 33. Translation and Interpretation of "Plimpton 322" 139 34. Comparative Cosmology : The Pythagorean Triangles of Plimpton 322 on Ptolemy's Monochord 140 35. God on the Mountain 141 36. Calendrical Yantras 143 37. An Interpretation of Mt. Mashu as Reciprocal Yantras 144 38. The Evolution of Monotheism 145 39. Babylonian and Hebrew Arks 149 40. Harmonization of Babylonian-Biblical Flood Chronology 151 41. A Speculative Old-Testament Yantra : 15,625,000,000 "measures of holy oil" 153 42. A Speculative Mayan Yantra 154 43. Mt. Meru, the "Holy Mountain," c. 200 B.C. 157 44. The Plain of Atlantis 166 45. The Islands of Atlantis 168 46. Hindu-Greek Modular Equivalents 172 47. Nicomachean Table for Numbers 2p3q, the "Army of Ancient Athens" 173 48. Egyptian Mathematical Notation 177 49. The Eye of Horus 178 50. The "Hall of Double Truth" (Maāt) and the Resurrected Osiris 183 51. The Pentatonic Scale as Hieroglyph of the "Underworld" 185 52. Seven Souls of the Sun and Twelve Hours of the Night 188 53. The Boat of Rā in the First Hour of the Night 190 x INTRODUCTION Ernest McClain's book constitutes an intellectual breakthrough of utmost significance. It offers a persuasive explanation of crucial passages in texts of world literature—the Ṛg Veda, the Egyptian Book of the Dead, the Bible, Plato—that have defied critics of the separate concerned disciplines. All these passages deal with numbers. What sounds like mathematical nonsense or literary gibberish has been given life and meaning by McClain's incisive thoughts. The recurrence, moreover, of identical and similar numbers in Babylon, Egypt, Greece, and Palestine confirms ever growing speculations on the historical continuity and direction of a basic spiritual tradition. Dr. McClain's method is simple enough : he recognizes music as the one force capable of projecting a philosophic synthesis. For this approach he is not likely to earn the immediate approval and support of theologians, philosophers, philologists, mathematicians, and others who have become too specialized to view the whole rather than the detail. Nor is he likely to find on his side musicians who consider their art primarily a branch of amusement and musicologists who are instinctively afraid of numbers. Yet his approach, neglected for centuries, is anything but new. He learned it by taking Plato literally. Plato insists on the superior role of music in the education of the whole man. In Phaedrus he writes : "The soul which has seen most of truth shall come to birth as a philosopher, or beauty lover, or fervent musician" (248d). In the Republic he writes: "Education in music is most sovereign, because more than anything else rhythm and harmony find their way to the inmost soul" (3.401d). These statements have generally been interpreted sentimentally. Dr. McClain, by taking Plato's directions literally, found the open-sesame. To give one of many examples : when Plato states that the tyrant is 729 times as bad as the good man, the philologist "simplifies the text" (as Francis Cornford has done in his famous English translation), and the mathematician calls it a "literary license." The trained musician, however, knows that every tone is both a number and a quality, neither one explaining the other one- sidedly, but each corresponding to the other xi exactly. Thus the ratio 1 :2 does not explain the experience of the musical octave any more than the sound of this interval would explain the quantity involved ; but the two facets of tone mutually illuminate each other. Now the number 729 chosen by Plato (Republic 587e) corresponds to the musical quality of the tritone (36 = six fifths above the fundamental), the worst possible dissonance in the musical systems known to Plato and, for that matter, in all Western tonal systems for two thousand years after him. What Plato evaluated by the number 729 was the relation between the good man and the tyrant as that of the greatest possible tension within a civilized system. One sympathizes with the puzzlement of nonmusicians facing such numbers. To appreciate the analogy involved, one must realize (as Plato took for granted) that in antiquity musical symbolism was understood directly by all educated and many less educated people. When in the course of history the role of music as a spiritual force was increasingly sacrificed to that of individual expression or entertainment, the interpretation of once very clear texts suffered. McClain's great accomplishment lies in his recognition of music as the spiritual model par excellence throughout antiquity and in his courageous and self-critical application of this insight to other fields.