Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Myth of Invariance: Sound, Music, and Recurrent Events and Structures


From  Philosophy and Science of Music in Ancient Greece:
The Predecessors of Pythagoras and their Contribution

 One of the ironies of twentieth-century thought is that the final dethronement of Pythagoras as a ‘father’ of western science and philosophy and the ‘inventor’ of music and mathematics should be accompanied by a world-wide revival of Pythagorean research and speculation. During the seventeenth century, the ‘harmony of the spheres’, which had remained an article of faith until the age of Shakespeare and even Louis XIV [Isherwood 1973, Ch. 1], was suddenly overwhelmed by the mighty discoveries of Kepler and Newton; but this traumatic ‘Untuning of the Sky’ [Hollander 1970] did not entirely obliterate the Pythagorean tradition (to which both Kepler and Newton were sympathetic).

Since the pioneering studies of Thomas Taylor (1758-1835), Antoine Fabre d’Olivet (1767-1825) and Albert von Thimus (1806-1878), there has been a steady renewal of interest in the old science of harmonics, culminating in the work of Hans Kayser (1891-1964) and his two most influential successors, Rudolf Haase and Ernest G. McClain (both of whom are living in retirement). Neo-pythagoreanism is now a conspicuous feature of post-modern philosophy and science: the revival of musica speculativa, part of a larger resurgence of neo-classicism, is well represented in the writings of Joscelyn Godwin [Godwin 1987, 1993, etc.]. To his extensive bibliographies could be added not only impressive results of recent mainstream research into Pythagoras and the Pythagoreans, e.g., Huffman [1993], but also the publications of several ‘alternative’ thinkers, including the French-American composer, music theorist, and astrologer, Dane Rudhyar, the French ‘neo-astrologer’ Michel Gauquelin, the English numerologist John Michell, and the English geneticist Rupert Sheldrake. Sheldrake’s notion of ‘morphic resonance’ — forms resonating in Nature’s memory — is a very Pythagorean-Platonic alternative to mechanistic causality. His wife, Jill Purce, is a music therapist [Purce 1974]; so both sides of the Pythagorean tradition — the ‘hard’ and the ‘soft’ sciences — are here reunited in the work of one family.

Though hardly any of these writers would describe themselves as Pythagoreans, their ideas have important connections with the old tradition and all are symptomatic of a new era in the history of thought when mechanistic and reductionist paradigms are giving way to a holistic and organic world-view. This emergent rationality is fundamentally ecological and its impact is being felt from metaphysics to everyday manners. The new paradigms of the Age of Ecology are already transforming the professions, sciences, arts, academic disciplines, and human enterprises generally — from the minute study of bird-song and insect music to the utopian vision of planet Earth designed and managed as a single, organic Gesamtkunstwerk [Pont 1997].

Central to this new understanding of the world is the concept of the ‘Biosphere’, which is the very antithesis of Newton’s mechanistic universe [Teilhard de Chardin 1955]. Thus the Pythagorean vision of the living cosmos — or Plato’s ‘World Soul’ — has reappeared in new vitalist theories, including the Gaia hypothesis of James E. Lovelock [1979]. The modern world-view and its vast astronomical time-frame have changed our conception of humanity itself, if only in recognising our evolutionary affiliations with, and biological dependence on, other species in the terrestrial ecosystem. And it has also transformed the idea of the ‘humanities’: never again can they be taught as just a narrow study of the ‘classical’ texts or litterae humaniores of Greece, Rome, and the Renaissance. No longer can the ancient Greeks be contemplated, in museum-like isolation, as perfect models of everything European.

With the growth of modern archaeology, prehistory, anthropology, linguistics, and other comparative studies, the marmoreal idols of Eurocentric scholarship are now revealed in something like their original gaudy splendour — a Joseph’s coat of distinctly oriental hues. Most of the discoveries traditionally ascribed to Pythagoras were Asiatic in origin; and, in a recent survey, Music and Musicians in Ancient Greece [Anderson 1994], the Pythagoreans have been reduced to four passing references and Pythagoras himself is omitted altogether!

The innovations still plausibly credited to the historical Pythagoras include the coining of the terms ‘philosophy’ and ‘theory’ which, in his case, must have referred to the dogmatic teachings and pre-scientific wisdom of a guru rather than genuinely theoretical inquiries like those of Heraclitus and the Eleatics. Pythagoras was also credited with inventing the term Kosmos, but the idea of the beautiful world-order (above and below) must surely have been Egyptian in origin [Cf. Plato, Laws II, 656a-657b]. Our admiration of the Greeks is now tempered by a better understanding of their true historical circumstance and actual indebtedness to other civilisations [Cf. Bernal 1987].

Just as Whitehead saw western philosophy as ‘a series of footnotes to Plato’, so modern scholarship has established that most of the doctrines traditionally ascribed to Pythagoras were really the contributions of the older high civilisations, particularly of Mespotamia and Egypt.[1] The rise and dissemination of these perennially influential doctrines remains one of the most formidable problems for the historian of ideas.

Many of these ideas had already been explored in my General Studies courses at the University of New South Wales, particularly in ‘The Philosophy of Music’ (Australia’s first academic course on the subject, 1974-1988) and, more recently, in shorter courses on ‘Ancient Rationality’ and ‘Modern Rationality’ (1988-1995). It was with their arguments and conclusions in mind that I undertook during 1997 my last course at the University, entitled ‘The Predecessors of Pythagoras’. This aimed to examine the origins and analogies of Pythagorean traditions in Mesopotamia, Egypt, China, and India. The lectures contained little that was new and the literature survey was, unavoidably, far from exhaustive; but, even so, the course had the unintended effect of changing the lecturer’s point of view — and, indeed, his whole approach to Greek philosophy and science of music.

Instead of burdening the class with the meagre texts of the early Pythagorean school and the interminable difficulties of their interpretation, lectures took a broad view of ancient history and prehistory, in an attempt to answer two very large and necessarily speculative questions: first, what might have been the origins of the famous ‘analogy of the macrocosm and the microcosm’? And, second, how and when was this world-view ‘mathematised’? — that is, when was it precisely articulated with a system of musical numbers or harmonic ratios that eventually constituted the ‘harmony of the spheres’? Most of the fifteen students had some background in history and philosophy of science but no prior musical knowledge was required for the course and readings had to be confined to material available in English. The only set text was The Pythagorean Sourcebook and Library [Guthrie 1987].


Key People:

  • Hans Kayser
  • Ernest G McClain
  • Jay Kappraff
  • Barabara Hero
  • John Goldman
  • Guy Beck
  • Joscelyn Godwin
  • Joachim-Ernst Berendt
  • Antonio De Nicolas
  • Richard Heath
  • Subhash Kak
  • Rachel Wells Hall
  • Amba Kulkarni
  • Leon Crickmore
  • Pingala
  • Rudolf Haase


Key Concepts:

  • Scale Invariance
  • Time Invariance
  • Cycles
  • Fractals
  • Power Laws
  • Periodicity
  • Multi-scales
  • Octave Invariance
  • Octave Equivalence
  • Self Similarity
  • Pattern Formation
  • Harmonics
  • Repeating Patterns
  • Repeating Events
  • Attractors
  • Basin of Attraction
  • Prosody
  • Vedic Sanskrit Meters
  • Sounds, Numbers, Akshara
  • Ratios and Proportions
  • Harmonic Mean
  • Geometric Mean
  • Arthematic Mean
  • Symmetry



Key Sources of Research:


Ancient Harmonic Law

Jay Kappraff


Click to access bridges2007-311.pdf


Ancient Harmonic Law (version 2)

Jay Kappraff


Click to access report0809-19.pdf


Philosophy and Science of Music in Ancient Greece:
The Predecessors of Pythagoras and their Contribution

Graham Pont


The Lost Harmonic Law of the Bible

Jay Kappraff

Click to access bridges2006-481.pdf



Sound: A Basis for Universal Structure in Ancient and Modern Cosmology

Jung Hee Choi


Click to access Sound_JHC_v2013.pdf


Beck, Guy L.

Sonic theology: Hinduism and sacred sound.

Motilal Banarsidass Publ., 1995.


Beck, Guy L.

Sacred sound: Experiencing music in world religions.

Wilfrid Laurier Univ. Press, 2006.


Beck, Guy L.

Sonic Liturgy: Ritual and Music in Hindu Tradition.

Univ of South Carolina Press, 2012.


“Vedic perspective of Sound—Science and Spirituality.”

Prasad, M. G.

Click to access mgpSoundArticles.pdf


Conch-shells, bells, and gongs in Hindu temples

Marehalli Prasad

Click to access 54b3ebc70cf2318f0f969b1f.pdf


Acoustical studies on conch shells.

M.G. Prasad and B. Rajavel


On the role of acoustics in the Vedic Hindu tradition and philosophy.

M. G. Prasad and B. Rajavel


Conch‐shell and bamboo flute: Spiritual and musical expressions of acoustics in Vedic Hinduism

M. G. Prasad


Philosophical and cultural perspectives on acoustics in Vedic Hinduism

M. G. Prasad


Vedic chanting and vowel intrinsic pitch: Evidence from an ancient source

T. V. Ananthapadmanabha1, Kim Silverman1 and M. G. Prasad


“Perspectives on sound in Sanskrit literature on natural philosophy.”

Prasad, M. G.

The Journal of the Acoustical Society of America 85.S1 (1989): S147-S147.


Akasha (Space) and Shabda (Sound): Vedic and Acoustical perspectives

Prasad, M. G.

Click to access workshop2009_speakermgp5.pdf

The Golden Mean and the Physics of Aesthetics


Early Indian Music

Subhash Kak

Click to access manila.pdf


Hero, B. F.

“Design in nature from Pythagoras to Helmholtz to the Cantor musical array.”

WIT Transactions on Ecology and the Environment 57 (2002).

Click to access DN02021FU.pdf


Hero, Barbara, and Robert Miller Foulkrod.

“The Lambdoma matrix and harmonic intervals.”

IEEE engineering in medicine and biology magazine 18.2 (1999): 61-73.

Click to access Lambdoma%20Matrix%20Harmonic%20Intervals.pdf


Goldman, Jonathan.

Healing sounds: The power of harmonics.

Inner Traditions/Bear & Co, 2002.


Heath, Richard.

Matrix of Creation: Sacred Geometry in the Realm of the Planets.

Inner Traditions/Bear & Co, 2004.


Heath, Richard.

Sacred Number and the Origins of Civilization: The Unfolding of History Through the Mystery of Number.

Inner Traditions/Bear & Co, 2006.


Heath, Richard.

Precessional Time and the Evolution of Consciousness: How Stories Create the World.

Inner Traditions/Bear & Co, 2011.


A re-valuation of the ancient science of harmonics


Psychology of Music 31.4 (2003): 391-403.


Crickmore, Leon.

“New Light on the Babylonian Tonal System.”

CONEA 2008: Proceedings of the International Conference of Near Eastern Archaeomusicology, held at the British Museum, December 4-6. Vol. 24. 2008.


Crickmore, Leon.

“A possible Mesopotamian origin for Plato’s World Soul.”

Hermathena 186 (2009): 5-23.


Crickmore, Leon.

“The musicality of Plato.”

Hermathena 180 (2006): 19-43.


Crickmore, Leon.

“A New Hypothesis for the Construction and Tuning of Babylonian Musical Scales.”

Journal of Ancient Civilizations (2007): 006.


Recursion and Combinatorial Mathematics in Chandashastra

Amba Kulkarni

Click to access 0703658.pdf



Math for Poets and Drummers

Rachel Wells Hall


The Sound of Numbers

A Tour of Mathematical Music Theory

Rachel Wells Hall



Bhattacharya, Aryya.

“Vāc–Its Ontological Status and Importance in Prayers and Rituals of Śakti Oriented Tantric Tradition.”

The Polish Journal of the Arts and Culture. New Series 2015.1 (2015) (2015): 7-22.


Godwin, Joscelyn.

The Harmony of the Spheres: The Pythagorean Tradition in Music.

Inner Traditions/Bear & Co, 1992.


Breath, Voice, and Embodiment of the Metaphysical in Hindu Tradition

Gabe C. Alfieri

Click to access Breath_Voice_Embodiment.pdf


Franklin, Ellen, and Donna Carey.

“From Galaxies to Cells: Bridging Science, Sound Vibration and Consiousness Through the Music of the Spheres.”

Subtle Energies & Energy Medicine Journal Archives 16.3 (2005)


McClain, Ernest G.

“The bronze chime bells of the Marquis of Zeng: Babylonian biophysics in Ancient China.”

Journal of Social and Biological Structures 8.2 (1985): 147-173


McClain, Ernest G.

“The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato.”



McClain, Ernest G., and Ming Shui Hung.

“Chinese cyclic tunings in late antiquity.”

Ethnomusicology 23.2 (1979): 205-224


McClain, Ernest G.

“Structure in the ancient wisdom literature: The holy mountain.”

Journal of social and biological structures 5.3 (1982): 233-248



“Pythagorean Paper Folding: A Study in Tuning and Temperament.”

The Mathematics Teacher 63.3 (1970): 233-237


McClain, Ernest G.

“Musical Theory and Ancient Cosmology.”

The World and I (1994): 371-391.


McClAIN, Ernest G.

“The pythagorean Plato: prelude to the song itself.

York Beach, Mine: Nicolas-Hays.” (1978).


Pont, Graham.

“Plato’s philosophy of dance.”

Dance, Spectacle, and the Body Politick, 1250–1750 (2008): 267-281.


McClain, Ernest G.

“The ancient Chinese “calendrical” pitchpipes.”

The Journal of the Acoustical Society of America 80.S1 (1986): S101-S101.


The World Is Sound: Nada Brahma
Music and the Landscape of Consciousness

By Joachim-Ernst Berendt


De Nicolás, Antonio T.

Meditations through the Rig Veda: Four-dimensional man.

iUniverse, 2003.


De Nicolás, Antonio T.


iUniverse, 2003.

Author: Mayank Chaturvedi

You can contact me using this email mchatur at the rate of AOL.COM. My professional profile is on

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