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 serious 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, was suddenly overwhelmed by the 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 Newtown were sympathetic).
Since the pioneering studies of 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 distinguished successors, Rudolf Haase and Ernest G. McClain (both of whom are living in retirement). Neo-pythagoreanism is now a significant 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 Kayser, Haase and Joscelyn Godwin, especially his Harmonies of Heaven and Earth (1987). To Godwin's extensive bibliography could be added not only the 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" - of 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 recent writers would describe themselves as pythagoreans, their ideas have significant connections at least with the old tradition; and some of them, especially Sheldrake's, indicate the emergence of new conceptions in philosophy and science. All are symptomatic of an age in which mechanistic and reductionist paradigms are giving way to a wholistic and organic world-view. This new rationality is fundamentally ecological: its impact is felt from metaphysics to everyday manners. The emergent paradigms of the Age of Ecology are already transforming the professions, academic disciplines and human enterprises generally - from the minute study of bird-song and insect music to the envisaged design and management of planet Earth as a single, organic Gesamtkunstwerk.
Central to this global understanding of the world is Teilhard de Chardin's concept of the "Biosphere", which is the very antithesis of Newton's mechanical universe. The pythagorean vision of the living cosmos - as in Plato's "World Soul" - has reappeared in new vitalist theories, such the the Gaia hypothesis of James E. Lovelock (1979). The global 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 archeology, prehistory, anthropology, linguistics and other comparative studies, the unblemished marmoreal idols of Eurocentric scholarship are now revealed in their original gaudy splendour - a Joseph's coat of distinctly oriental hues. While most of Pythagoras' teachings are unquestioanbly Asiatic in origin, Otto Neugebauer probably went too far in his declaration (1937) that "what is called Pythagorean in the Greek tradition had better be called Babylonian"; but in a recent survey, Music and Musicians in Ancient Greece (Anderson 1994), the pythagoreans are reduced to four passing refernces and Pythagoras himself is omitted altogether!
Of the endless innovations that have been ascribed to Pythagoras, few remain credible today: one is the coining of the term "philosophy" which, in any case, must have referred to the dogmatic teachings of a guru rather than genuinely theoretical inquiries like those of Heraclitus and the Eleatics. Our admiration of the Greeks is now tempered by a better understanding of their historical circumstance and indebtedness to other civilisations. Just as Whitehead saw western philosophy as mostly "footnotes to Plato", so modern scholarship cannot deny that most of the doctrines ascribed to Pythagoras were really the achievements of the older high civilisations, particularly of Mespotamia and Egypt.
Many of these ideas had been explored in my General Studies courses at the University of New South Wales, particularly in the "Philosophy of Music" (1974-1988), Australia's first acdemic course on this subject, and, more recently, in courses on "Ancient" and "Modern Rationality" (1988-1995). It was with their arguments and conclusions in mind that I undertook during 1997 a new course at the University, entitled "The Predecessors of Pythagoras", which aimed to examine the origins and analogies of pythagorean traditions in Babylon, Egypt, China and India. Though the survey of relevant literature was far from exhaustive, 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 micrcosm"? And, secondly, how and when was this world-view "mathematised" - that is, refined by or articulated with a system of musical numbers or harmonic ratios which eventually constituted the "harmony of the spheres". Most of the fifteen students had some background in history and philosophy of science but no prior 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).
Lectures concentrated on publications and research areas which might illuminate either or both of these grand issues. Following F.M. Cornford (1952) and Mircea Eliade (c.1964 etc.), it was suggested that there were very ancient precedents for the shamanic powers ascribed to Orpheus, Pythagoras and other pre-Socratics (even Socrates himself). These included the power of healing through music, of communicating with understanding the language of other animals, including birds; the ability to appear in two places at once; and the orphic control of nature itself -the power to "suppress violent winds and hail, and calm storms both on rivers and seas" (Porphyry, Life of Pythagoras). Such marvellous feats have been ascribed to shamans world-wide, even in Aboriginal Australia.
Antipodean precedents to, or analogies with, the cosmology of the northern hemishpere include the dualism of the upper and lower worlds (the former being occupied, as in Christian belief, by the gods and the spirits of the departed); also common to both mythologies is a connecting axis mundi, presumably derived from the path of shamanic ascent. The eminent Australian anthropologist A.P. Elkin himself observed and bravely reported acts of levitation, magical healing and other paranormal feats performed by Aboriginal shamans or "men of high degree" (Elkin 1946/77). Reference was also made to the Aboriginal "songlines", age-old ritual routes - sometimes transcontinental - which structured the landscape with a kind of liturgical or musical geography, crossing tribal and linguistic frontiers with a continuously recognisable chant which functioned like a passport (Chatwin 1987). Although a long way from cosmic harmonics, the songlines could be viewed as an early use of "music" in organising and interpreting the visible world.
While such remote analogies with Greek ideas mostly eluded the white invaders of Australia - and are even resented by some anthropologists today - the following report is highly suggestive:
...Another version of the origin of the Milky Way, current in Queensland, identifies it with the deeds of Priepriggie, an Orpheus-like hero, as famed for his songs and dances as for his hunting. When he sang, the people dance to the rhythm until they dropped with exhaustion, and declared that if Priepriggie wished he could make even the stars dance. One morning when he speared a flying fox, its companions descended upon him in vengeance, carrying him up to the sky. Unable to find him, his people decided to perform his dance hoping for his return, but without him they could not capture the rhythm. Suddenly they heard a sound of singing in the sky. As the rhythm grew louder and more pronounced, the stars, hitherto randomly dispersed, began to dance and arrange themselves in time with Priepriggie's song. Thus the Milky Way serves as a reminder that the tribal hero should be celebrated with traditional songs and dancing" (Haynes 1996).
Given the ancient association of choreography and cosmology, this beautiful Australian legend seems only a short step from the world of Pythagoras (cf. Miller 1986). In his World History of the Dance (1937) Curt Sachs showed how the astral dance, imitating the circular procession of the stars, is found in all continents. Its origins too are lost in prehistory; but the circle is probably the oldest rational form known to and employed by humanity (and it is even danced by other primates). So, if dance mimicry was the original connecting principle between the upper and lower worlds, then the "analogy of the macrocosm and micrcosm" might have had very remote precedents in an age before the invention of writing - perhaps even of speech itself.
Philosophers have rarely contemplated the origins of their subject in Australian prehistory; but, in a recent review of Aboriginal ideas, Robert Lawlor has concluded that indigenous Australia had something like the pythagorean table of opposites and the hippocratic fourfold classification of phenomena into Dry/Hot, Hot/Moist, Dry/Cool and Cool/Moist (Lawlor c.1991). Similarly, there is evidence suggesting that the "analogy of the macrocosm and the microcosm" - perhaps the world's oldest cosmic system - goes back to the myth and ritual of paleolithic cultures, antedating the civilisations of Mesopotamia, Egypt and Greece by thousands of years. Europeans have always regarded Australia as being a long way from the centre of things but, during the last twenty or thirty years, the discoveries of Australian archeology and prehistory are starting to reverse that perspective.
Thus far, we arrived at a tentative answer to our first grand question: the "analogy of the macrocosm and the microcosm" was the classical Greek formulation of a world-view that was prehistoric, most probably paleolithic, in origin. The image of Urania, dancing in the chorus of the Muses, recalls the archaic astral dance which finally achieved concrete form in the ziggurats of Mesopotamia and the pyramids of Egypt.
The second stage, the mathematisation of this cosmology obviously came much later, well after the "neolithic revolution" and the emergence of the first villages and towns made possible by herding and agriculture. It was during the era of civilisation -the culture of cities - that the old mimetic relationship between the larger, universal system and the local, human order was transformed into a precise mathematical analogy: that is, the heavenly order came to be seen as a harmony or attunement controlled by number and the earthly order was accordingly formed on, or adjusted to, the same number system - which was duodecimal, not decimal. This, the oldest known mathematical cosmology, may have been suggested by the use of simple numbers in choreography - primarily the first few integers which are still used to measure rhythm today (1, 2, 3, 4, 6, 8, 9, 12, etc). But it is fairly certain that the "analogy of the macrocosm and microcosm" became the "harmony of the spheres" only when the old mimetic "analogy" was reinterpreted as a system of harmonic proportions shared by macrocosm and microcosm. There may have been an intermediate stage, when the seven visible "planets" (including the Sun and Moon) were identified with musical tones in a "planetary scale". The bold hypothesis that the world was a harmony, a cosmos ordered on specific musical proportions, must have been invented by somebody but the identity and whereabouts of that Asiatic Pythagoras are also lost in time. The evidence points first, to Babylon and, secondly, to Egypt - to the very countries where the historical Pythagoras is said to have studied.
It has long been understood that the kanon or monochord played a central role in the philosophy of Pythagoras and Plato but the early history of the instrument and its use in scientific theory and philosphical speculation are also very poorly documented. Pythagoras, on his death-bed, is said to have recommended the study of the monochord to his disciples; and Plato in effect did the same - if he really was the author of the disputed Epinomis, a kind of appendix to his last dialogue, the Laws. The Epinomis is the only writing in the entire platonic corpus which specifically alludes to the harmonic analogia or tuning module of 6:8::9:12 and, unfortunately, the text is obscure or corrupt at this point (Epinomis 9911-b). This module or system of ratios gives the "divisions of the monochord", the points at which the vibrating string of the monochord can be stopped, with a moveable bridge, to sound the "fixed" or fundamental intervals of the musical scale, the octave (2:1), the fifth (3:2), the fourth (3:4) and the tone (8:9). The integers 6, 8, 9 and 12 are the smallest whole numbers with which the symmetrical system of interlocking ratios - the natural framework of the ancient and modern scale - can be expressed. Just as the Greeks admitted that their lyre was a foreign invention, so they knew that the tuning system of 6:8::9:12 was also imported - presumably from Babylon, where the precise relationship between pitch, string-length and numerical proportion could have been discovered a thousand or even two thousand years before Pythagoras (indeed almost any time during the first three or four millennia of the harp's development).
The oldest book on the monochord and its divisions was written by Euclid (c.300 BC) but the instrument itself was obviously much older. Its early use and significance have now been greatly illuminated by Ernest G. McClain, first with The Myth of Invariance (1976) and then with The Pythagorean Plato (1978). Neither a classical schlar not a mathematician, in the ordinary sense, McClain was a professor of the clarinet at Brooklyn College, New York. Endowed with a rare combination of musical and philosophical intelligence - and a musician's gasp of tuning theory and practice - he went in search of the ancient wisdom, inspired by like-minded colleagues including Ernst Levy and Antonio de Nicola. Following the method of Robert S. Brumbaugh (Plato's Mathematical Imagination, 1954), McClain made an "intellectual breakthrough of the utmost significance" by offering a simple musical explanation of "crucial passages in texts of world literature - the Rg Veda, the Egyptian Book of the Dead, the Bible, Plato - that have defied critics of the separate concerned disciplines" (S. Levarie). His method was not new in principle but his development and application of it produced amazing results.
Taking the numbers used in or derived from monochord tuning, McClain identified their widespread use in numerical allegories, myths and metaphors found in some of the oldest books in the world. For example, when Plato characterised the good man as "living 729 times more pleasantly, and the tyrant more painfully by this same interval" (Republic 587e), he used the number which defines the tritone (the sixth power of three; that is, six fifths above the fundamental tone). Thus the tension between the good man and the tyrant is compared to the worst possible dissonance in the western musical system (Plato's model here, incidentally, is both musical and geometrical). Similarly, McClain has decoded many other musical allegories and discovered the meaning of some incredibly large numbers in Hindu, Greek, Hebrew, Babylonian and Egyptian texts. In The Pythagorean Plato, he applied the method systematically to Plato's numerology and produced a simple, consistent and comprehensive explanation for allegorical texts that had defeated five hundred years of scholarship.
Though really a corollary to his first book, The Pythagorean Plato is much more approachable for the general reader. The introduction explains the basics of tuning theory and the graphic use of the monochord string turned into a tonal circle on which any scale can be represented geometrically. Seven of Plato's numerical allegories are then analysed in detail showing, for example, how Plato's political theory was modelled on musical theory, with the constitutions of Callipolis, Athens, Atlantis and Magnesia corresponding to four different tuning systems.
Plato's musico-political analogies were by no means an idiosyncratic jest: the Greek word "syntagma" can refer to either a political or a musical system, just as the sanskrit "grama" can denote a village or a scale. In Classical and Christian Ideas of World Harmony, written during the 1940s, Leo Spitzer set out to explain the compound meanings of the German "Stimmung" and discovered its relations with a whole gamut of harmonic terms resonating through the European languages (Spitzer 1963). On purely philological grounds, Spitzer divided these terms into two groups: first, those related to "chord", "concord", "accord", etc.; and. secondly, those related to "temperance", "tempo", "temperament", etc. The two groups correspond fairly well to the distinction between tuning by whole numbers and tempering by small adjustments (involving irrational proportions). Spitzer was puzzled by the root meaning of the second group, "a section cut off" - which possibly refers to the act of dividing the monochord string ("sectio canonis"). Of unknown origin, the "tem" words testify to the wide application of the harmonic art and the high antiquity of its vocabulary in the Indo-European languages: words like "temenos" (sacred place), "temple", "time", "template" and "terminus" all involve divisions of space and time based, presumably, on a common mathematics. The association of the musical and the spatial sciences was confirmed by Arpad Szabo in the Beginnings of Greek Mathematics (1978) which shows that all the terms of pre-Euclidean Greek geometry were derived from music or harmonics. For example, "diastema" means an interval, spatial or musical, just as "chord" still has a geometrical as well as a musical meaning. The geometrical representation of an interval as a line terminated by vertical strokes could equally be a picture of the monochord string; and we still tell the time by looking at a monochord circle divided into twelve!
"The Predecessors of Pythagoras" thus took us into a brief survey of temple art and architecture and their connections with the science of music. Temples have always been the grandest microcosmic form that man has built in imitation of the heavenly order. Reviewing a continuous tradition of five millennia, Joseph Campbell has shown how the temple has remained faithful to the old mandala of circle and square - which might be read as a symbolic synthesis of the original paleolithic circle and the neolithic square or cross (the rectilinear form adopted for agriculture and town-planning). The Eyptian hieroglyph for a town is a circle enclosing a St Andrew's cross, with its arms pointing to the minor directions - a perfect representation of the consecrated enclosure that was subdivided to make the four quarters of the town. Campbell points out that the with the rise of the hieratic city-state (c.3500-2500 BC)...
The whole city now (not simply the temple area) is considered as an imitation on earth of the celestial order- a sociological middle csomos, or mesocosm, between the macrocosm of the universe and the microcosm of the individual, making visible their essential form: with the king in the the center (either as sun or as mooon, according to the local cult) and an organization of the palace and the ziggurat; and with a mathematically structured calendar, furthermore, to regulate the seasons of the city's life according to the passages of the sun and moon among the stars; as well as a highly developed system of ritual arts, including an art of rendering audible to human ears the harmony of the visible celestial spheres. It is at this moment tha the art of writing first appears in the world... (The Flight of the Wild Gander)
Campbell emphasises that the same sexagesimal number system was used to represent the "mandala of space" (the circle of 360 degrees) and the "mandala of time" (the almost congruent circle of the year); and he shows how this same sacred mescosm gradually extended from the Nile to Central America. The structure and symbolism of the Egyptian temples have been analysed by John Michell who argues that they incorporate cosmic measures (The Dimensions of Paradise) and by R.A. Schwaller de Lubicz who concluded that the great monument at Luxor actually represents the growth of the human microcosm in extraordinary physiological detail (The Temple in Man). None of these authors, however, succeeds in elucidating the persistent tradition that "architecture is frozen music" - that the canons used in temple design were harmonic. The Renaissance enthusiasm for harmonic proportions in design has not yet been matched by a convincing history of ancient architecture as "applied music": but how else are we to explain Vitruvius' frequent references to music, including the different Greek tuning systems? Lacking more specific texts, we are necessarily restricted to direct measurement of existing edifices, as in Hans Kayser's suggestive analysis of the temples at Paestum (Kayser 1958) and Donald Preziosi's work on the ground-plans of Minoan architecture (Preziosi c.1983).
This difficult question, of course, falls well outside the mainstream of pythagorean studies but it is directly relevant to the harmony of the spheres. If the ancient priests, sages and philosophers were able to discern the musical proportions of the heavenly system, would they not have naturally encoded them in their earthly imitation? A discordance between the macrocosm and the micrcosm seems unthinkable but there is as yet no consensus on the ancient use of musical proportions in sacred architecture.
Of particular importance in this regard is the work of Abraham Seidenberg. In a series of important articles since 1962 in the Archive for History of Exact Sciences, Seidenberg argued that the origins of arithmetic and geometry are to be found in the ritual arts: for example, the oldest precise geometrical operations are found in the sulvasutras, ancient Indian works on altar construction - a liturgical tradition that goes back to the Rg Veda (c.1500BC). Seidenberg's work evoked little response from fellow mathematicians until it was endorsed by no other than B.L. van der Waerden (1983). Seidenberg's findings reveal a surprising association between music and the exact sciences: thus the harmony of the spheres might have been long anticipated in the measured dance of Urania, Muse of Astronomy.
The connection between dance geometry and the mathematical arts is a living reality in India: in The Square and the Circle of the Indian Arts (1983), dance authority Kapila Vatsyayan explores the connections between sacred dance, mandalas and architecture and includes photographs of ritual dances- some possibly as old as the Rg Veda. All of this confirms Lewis Mumford's emphasis on the crucial role of human "biotechnics", the arts of brain and body, in the prehistoric development of tool technology and the constructive arts. For the early Greeks, and all other preliterate cultures, the most important "biotechnics" were the musical arts (the arts of the Muses); and this fact alone might be sufficient to explain the extraordinary value placed on musical numbers in the ancient arts and sciences. If architecture was indeed "frozen music", then one would expect to find the harmony of the spheres reflected in the temples of that era. John Michell has found cosmic numbers in the dimensions of the pyramids but, unfortunately, they do not seem to match McClain's musical numbers!
Such grand issues are hardly ever addressed in ordinary musical theory. Most of the commentaries on ancient Greek music are devoted to harmonics, scales and tuning; and the literature is filled with discussions of textual problems and terminological difficulties which intimidate the general reader, repell the music-lover and frustrate even the most erudite scholar. In pursuing the predecessors of Pythagoras we avoided those intricacies by looking at the Greeks from the other end, so to speak: by viewing them, not as the founding fathers of western art and science but as the heirs of their predecessors in the older civilisations. This longer and larger perspective throws the Greek achievements into sharper focus - however hazy the details might be. It reveals the towering figure of Pythagoras, standing at or near the gateway through which Asiatic ideas and inventions passed into Greece. In his final essay, Pythagoras, Egypt, Sparta, the Dux of the 1997 class, Mr Chad Bochan, argued that the immediate predecessors of Pythagoras were the Spartans - the Greeks who were closest to the Egyptians and the first to imitate their musical system, well before the Athenians.
This reversing of the temporal perspective also suggests that, for all the effort expended on Greek harmonics and tuning theory, very little of their actual practice could have been original. Scholars have long suspected that the diatonic scale had been imported from Asia and superimposed on the native tetrachordal system; but there was little hard evidence to go on. After all the literary exegesis and scholarly debate, new illumination was finally obtained from two highly important archeological discoveries.
The first was the unearthing and decoding of the world's earliest tuning manual, preserved in cuneiform writing on an Old-Babylonian (Akkadian) clay tablet from Ur, dating from the early to middle second millennium BC. Reading this in the light of other tablets with mathematical and musical texts, an interdisciplinary group of scholars made a conjectural interpretation of the tuning instructions, which they demonstrated on a reproduction lyre (modelled on a genuine instrument of a later period). Assuming that the strings were meant to be tuned in a diatonic system, they found that the instructions made musical sense and yielded "specific intervals of the diatonic scale familiar to us as traditional western intervals" (Kilmer et al. 1976). Their brilliant demonstration of "Sounds from Silence" is not, of course, a conclusive proof; but it is a profoundly moving experience to hear our ordinary major scale among those performed on the disc recording. If these scholars are right in surmising that the Babylonians knew the modern diatonic scale, then the contribution of the Greeks to tuning theory was little more than lexicographical! This important discovery serves only to reinforce the tradition that it was the Babylonians who invented the harmony of the spheres.
The second discovery is the most important ever made in the archeology of music. In 1977, 124 musical instruments were found among some 7,000 burial objects in the tomb of the obscure Marquis Yi, who was buried c.433 in Zheng - now Hubei Province, west of Nanjing, in the People's Republic of China. The instruments included 65 bronze bells, forming a well-tuned carillon of five octaves, still in playing order. To everybody's astonishment, the bells produced a very accurate, mostly chromatic scale. Cast in a technique unknown to the west, each bell can sound two clear and distinct musical notes which are much purer than those of western bells - and the sound is obtained from a resonator which is a hundred times lighter than a corresponding western bell! Each bell is inscribed with instructions in gold, explaining the name and function of each note in the scale: a musical Rosetta Stone, no less.
Among those who were most surprised by this find were the Chinese themselves, who were totally unprepared for the discovery of the chromatic scale in China - and very slow to make the bells accessible to the wider scholarly world. Almost all memory of a Chinese chromatic scale had been lost - presumably through the burning of musical books and instruments by the Emperor She Huang-Ti (246 BC). On one of the first recordings on compact disc, the bells perform "Unique Music of Great Antiquity" - which is restricted to the traditional pentatonic scale (China Record Corporation, CCD-89/26, 1989); but the bells are capable of performing tunes in the full diatonic scale, as is shown by later recordings which feature arrangements of western classical music. How long the chromatic scale had been known in China is anybody's guess but the Marquis Yi's's carillon now suggests that there may have been some truth in the legend of the twelve bamboo lüs (pitch-pipes sounding a twelve-note division of the octave), ascribed to the "Yellow Emperor' Huang Ti (2,697 BC). An elaborate description and analysis of these bells is to be found in Chen Cheng Yih (c.1994).
In a preliminary assessment of the Marquis' carillon, Ernest McClain points out that "contemporary fifth century classical Greece, which we are in the habit of venerating, left no artifacts of comparable musical value' (McClain 1985). The bells confirm that "the prevailing diatonic pattern in China as well as India, Greece, and Babylon is that of the C major scale or its inverse (Greek Dorian)" - as argued by McClain in his book published the year before the discovery of these bells (McClain 1976). Thus the bells point to a "tonal cosmology" which anticipates that of Plato and was possibly inherited from Babylon; but, as McClain wisely counsels, the whole wubject needs to be re-examined by an "anthropology educated in the harmoncial sciences of the ancient world', before any firm conclusions can be drawn on the history of tuning theory and the dissemination of musical cosmology.
The writings of the ancient Greeks and their Roman successors remain the foundation of western philosophy and science of music, though much of the theory and practice was certainly derived from earlier cultures. As we come to understand more about the achievements of those predecessors, the actual Greek contribution to musical philosophy and science will seem even more characteristically Hellenic: for, as in most of the arts and sciences they cultivated, the Greeks created a rational theory, invented a systematic vocabulary, ordered the subject with a logical classification and infused the whole with a spirit of inquiry that still inspires us today.
Warren D. Anderson, Music and Musicians in Ancient Greece (Ithaca: Cornell University Press, 1994)
Joseph Campbell, The Flight of the Wild Gander: explorations in the mythological dimension (New York: Viking Press, 1969)
Bruce Chatwin, The Songlines (London: Cape, 1987)
Chen Cheng Yih (ed.), Two-tone Set-Bells of Marquis Yi (Singapore: World Scientific, c1994)
Francis M. Cornford, Principium Sapientiae: the origins of Greek philosophical thought (Cambrdige: Cambridge University Press, 1952)
Mircea Eliade, Shamanism: archaic techniques of ecstasy (Princeton NJ: Princeton University Press, c.1964)
Aldolphus P. Elkin, Aboriginal Men of High Degree, 1946 (2nd ed, St Lucia: University of Queensland, 1977)
Joscelyn Godwin, Harmonies of Heaven and Earth: the spiritual dimensions of music from antiquity to the avant-garde (Rochester Vermont: Inner Traditions, c.1987)
Kenneth S. Guthrie (ed.), The Pythagorean Sourcebook and Library: an anthology of ancient writings which relate to Pythagoras... (Grand Rapids: Phanes Press, 1987)
Rudolf Haase, Geschichte des harmonikalen Pythagoreismus (Wien: Lafite, 1969)
Rudolf Haase, Aufsätze zur harmonikalen Naturphilosophie (Graz: Akademische Druck, 1974)
R. Haynes et al., Explorers of the Southern Sky: a history of Australian astronomy (Cambridge: Cambridge University Press, 1996)
John Hollander, The Untuning of the Sky: ideas of music in English poetry, 1500-1700 (New York: Norton, 1970)
Carl A. Huffman, Philolaus of Croton; pythagorean and presocratic (Cambridge: Cambridge Unviersity Press, 1993).
Hans Kayser, Akroasis: the theory of world harmonics (Boston: Plowshare Press, 1978)
Hans Kayser, Die Harmonie der Welt (Wien: Lafite, 1968)
Hans Kayser, Paestum: die Nomoi der drei altgriechischen Tempel zu Paestum (Heidelberg: Lambert Schneider, 1958)
Anne D. Kilmer et al., Sounds from Silence: recent discoveries in ancient Near Eastern music (Berkeley Calif: Bit Enki, 1976).
Robert Lawlor, Voices of the First Day: awakening in the Aboriginal dreamtime (Rochester. Vermont: Inner Traditions, c.1991)
James E. Lovelock, Gaia: a new look at life on earth (Oxford: Oxford University Press, 1979)
Ernest G. McClain, The Myth of Invariance: the origin of the gods, mathematics and music... (New York: Nicolas Hays, 1976)
Ernest G. McClain, The Pythagorean Plato: prelude to the song itself (New York: Nicolas Hays, 1978)
Ernest G. McClain, The bronze chime bells of the Marquis of Zeng: Babylonian biophysics in Ancient China, Journal of Social and Biological Structures, vol. 8 (1985), pp.147-173
John Michell, The Dimensions of Paradise: the proportions and symbolic numbers of ancient cosmology (London: Thames and Hudson, c.1988)
James L. Miller: Measures of Wisdom: the cosmic dance in classical and christian antiquity (Toronto: University of Toronto, 1986)
Donald Preziosi, Minoan Architectural Design: formation and signification (Berlin: Mouton, c.1983)
Jill Purce, The Mystic Spiral: journey of the soul (London: Thames and Hudson, 1974)
Curt Sachs, World History of the Dance (New York: Norton, 1937)
R.A. Schwaller de Lubicz, The Temple in Man: sacred architecture and the perfect man (New York: Inner Traditions, 1977)
Abraham Seidenberg, The ritual origin of geometry, Archive for History of Exact Sciences vol.1 (1962), pp.488-527
Abraham Seidenberg, The ritual origin of the circle and square, Archive for History of Exact Sciences vol. 25 (1981), pp.269-327
Rupert Sheldrake, The Presence of the Past: morphic resonance and the habits of nature (New York: Times Books, c.1988)
Rupert Sheldrake, The Rebirth of Nature: the greening of science and God (New York: Bantam Books, 1991)
Leo Spitzer, Classical and Christian Ideas of World Harmony: prolegomena to an interpretation of the word "Stimmung" (Baltimore: Johns Hopkins Press, 1963)
Arpad Szabo, The Beginnings of Greek Mathematics (Dordrecht: D. Reidel, c.1978)
B.L. van der Waerden, Geometry and Algebra in Ancient Civilizations (Berlin: Springer-Verlag, 1983)
Kapila Vatsyayan, The Square and the Circle of the Indian Arts (New Delhi: Rolli Books, 1983)