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This book was produced by Human Bridges. Michael Hudson has devoted his career to the study of debt.
Introduction
Whistles, flutes, and tambourines made of bone and skin are among the earliest artifacts found in Ice Age caves. These instruments would have made up an archaic orchestra if they were tuned together. But they could only be played solo or atonally until a symphonic harmony was developed on the basis of a common scale and temperament.
Music was omnipresent in public gatherings, above all in conjunction with dance. The choreography of New Year rituals imitated the celestial periodicities and orientation of the sun, moon, and planets, culminating in a ritual combat between tribal moieties representing solar and lunar forces. In time this combat evolved into musical competitions staged alongside athletic contests.
Statues and portraits shaped the images of rulers to fit a geometric and indeed, a numerological canon of proportions for the human body. Reigns and genealogy were cosmologized by assigning important “generative” numbers to lifespans, begetting ages, periods of rule, and related chronology.
Music and dance likewise were imbued with cosmology through analogies involving number symbolism. A knowledge of harmonic ratios was necessary to put the proper stops in stringed instruments, to make holes in metal and wooden wind instruments, and to cast bells. These ratios of length were related to musical tones. The first five overtones included the octave (2/1), fifth (3/2), and third (4/3). From these intervals the scale of seven notes (A to G) and, altogether, 12 semitones could be built up.
The mathematical characteristics of musical scales were perceived to share some striking parallels with the major calendrical numbers. Just as the year has 12 months, so the diatonic scale has 12 tones. And just as the Near Eastern Semitic week had seven days, so there are seven notes in each modal scale. Finally, just as the calendrical year had a “dissonance” between its lunar and solar periodicities, so the musical scale generated dissonances by tuning via the circle of fifths as compared to building it up by thirds. Thus, just as the lunar year had to be adjusted to the solarized year, so the musical scale had to be tempered.
In this tempering ancient philosophers found a metaphor for society’s members subordinating their individual egoism so as to work together in concert. However, Pythagorean philosophizing over “geometric” and “arithmetic” tuning was used to rationalize inequalities among society’s economic classes:
- “Art denaturalizes nature in order to raise it to a higher, or at least a different, plane.” — Curt Sachs,[1] The Rise of Music in the Ancient World: East and West (1943: p. 31)
- “Numbers and combinations of numbers are used in the sacred writings to convey instruction under a figurative guise, and ignorance of numbers often shuts out the reader from this instruction.” — St. Augustine,[2] De doctrina christiana, II.xvi.25
This chapter reviews the cosmological principles underlying four types of archaic art. I begin with the visual arts: sculpture, painting, and stone friezes. I then discuss “origin” epics such as the Sumerian King List and the biblical Patriarch List. I review dancing in order to show how archaic choreography imitated the starry kosmos in various ways. Finally, I discuss the role of music in the classical educational curriculum, above all the mathematics of tuning the scale and its social metaphors.
To show how far the cosmological foundation of ancient societies extended, I will trace some connections between the calendar, music, and art on the one hand, and social structuring on the other. As recently as a couple of generations ago, scholars hardly would have expected to find that the canons of proportions used by Greek and Near Eastern sculptors would share a common denominator with the interest rates of these regions. And at first glance the mathematics of tuning the musical scale would not seem to have much in common with the timing of debt cancellations or rationalizing disparities in wealth. Yet a historian of musical tempering, Ernest McClain[3] (1976: p. 98), has pointed out that the biblical Jubilee Year called for in Leviticus, to be celebrated every 50 years to put the economic world put back in order, is established at the precise calendrical counterpart to the point where the musical scale must be put back in tune by a 50:49 “comma.” This adjustment is a micro-interval necessary to temper the scale so as to establish an inner harmony of tones. In McClain’s view, the celestial proportions of musical tuning worldly financial adjustments thus went hand in hand, at least for the world planned by the biblical authors when they returned from Babylonia in the fifth century BC, imbued with Mesopotamian ways of cosmologizing public policy.
Such parallelisms show how important the cosmological subtext was for shaping worldly life so as to reflect a schematized calendrical, numerological, and musical kosmos. Understanding these connections will throw light both on ancient artistic creativity and on the structuring of social life. Economic policy was conceived as aesthetic an endeavor as was the calendrical and its associated measurement systems.
Although these connections between calendrical, artistic, and economic cosmology may seem circuitous to modern observers, the ancient worldview found it logical to extend parallels between calendar-making and musical tuning into such realms as the geometric proportions underlying much ancient art, the algebraic essence of king lists, and kindred cosmo-historical compositions, above all those associated with Creation rituals and periodic New Year renewals of the social kosmos.
To understand the cosmological underpinnings of archaic art, one might well begin with Book 8 of Aristotle’s Politics.[4] This work contains one of the best distinctions between the guiding spirit of archaic art and its subsequent trivialization into merely escapist play.
Aristotle made clear in Chapters 3 and 5 of Book 8 that the word “mousike” had much broader and philosophic connotations in his day than its modern cognate “music.” “Most men nowadays,” he says, “take part in music for the sake of the pleasure it gives; but originally it was included in education on the ground that our own nature itself… wants to be able not merely to work properly but also to be at leisure in the right way.”
By “the right way” he meant music and art which elevated its audience, not music as mere recreation. He granted that “Play has its uses, but they belong rather to the sphere of work,” being basically a recovery from toil. “Amusement is for the purpose of relaxation, and relaxation must necessarily be pleasant, since it is a kind of cure for the ills we suffer in working hard.”[5] But it is not “the good life,” which Aristotle defined as being “a state attained not by those at work but by those at leisure.”[6] Play is no more creative (in the deep cosmological sense) than a hobby is art. Rock and roll is not Beethoven, and in Chapter 5 of Book 8 Aristotle emphasized that the sort of music that is only “for our amusement and refreshment, like taking a nap or having a drink,” is not of serious importance, although it may be “pleasant and help us forget our worries, as Euripides says” (Bacchae[7] 381).
One of the most striking aspects of ancient writings on music was their dismissal of actual performance. The musical curriculum concentrated on tuning the scale, not on performing tunes. A cultured person would appreciate hearing music or contemplating the cosmological significance of musical temperament, but not actually performing music. What was deemed most important was to understand the ratios of the musical intervals and to relate them to the philosophy of musical temperament and its social metaphors. “The poets do not depict Zeus as playing the lyre and singing in person,” pointed out Aristotle[8] (1339b). “In fact we call the performers ‘technicians’ and think that a man should not perform except for his own amusement or when he has had a good deal to drink.” Thus, like his contemporaries, Aristotle dismissed musicianship as being a mere craft (“techne”), and as such, unbecoming to men of leisure.
The Ceremonial Setting for Archaic Art, Dance, and Music
The most important ancient artworks were ceremonial, and the paradigmatic ceremonial occasion was the New Year re-creation of order. The objective of artistic, musical, and choreographic creativity was to reflect the ordering of society as part of renewing the kosmos. Artworks, music, and literary “origin” epics, including king lists and related genealogies of the gods or “founding fathers,” reinforced and also shaped society’s cosmological skeleton.
Archaic art was surrealistic in treating its subjects in a cosmological way yet appearing realistic at first glance. This treatment of mundane structures was cosmologically grounded, rather than literally realistic or arbitrarily “spontaneous.” This hardly should be surprising in view of the fact that most surviving artifacts come from ritual contexts, and antiquity’s major public ceremonies were kosmos-renewing. The most important artworks were part of the re-creation process (as distinct from mere recreation).
Much of what formerly was viewed merely as decoration has recently been found to have a symbolic significance. To think that the markings found on Ice Age bones, neolithic bowls and other pottery, or Bronze Age statues of rulers are simply random (or, if they are indeed ordered, that they reflect purely decorative design elements) is to overlook the genius of archaic artists in achieving a seamless integration of cognitive content and realistic treatment.
The cognitive content of archaic artworks is so vast a subject that I will focus here on just one dimension: the numerical proportions of sculptured images of the human body.
With regard to the distinction between cosmological and individualistic art, the first modern idea that should be put aside is that of art for art’s sake. The long pedigree of cosmological iconography began deep in the Ice Age, and continued through the neolithic and Bronze Age, at least for ritualistic ceremonial art (which seems to represent the great majority of surviving artifacts). Only at the end of antiquity did a literal realism emerge, an individualistic representation of art patrons as subjects for sculpture and other forms of art that hitherto had focused on ritual cosmology, deities, or rulers.
Nearly all surviving artifacts come from burials or caves, traditional ritual contexts. At the Russian site of Sungir, for instance, hundreds of beads appear to have been sewn onto clothing and a cap, and the interred body is wearing ivory bracelets. Marshack (1990: p. 20) observed that such clothing “would probably not have been worn in the activity of hunting, butchering or daily domestic activity.” He added that “The assumption that ivory beads, bracelets and head bands evidence the beginnings of an awareness of ‘self’ is Eurocentric.” The “self” is still merged into the kosmos.
For portable art, the practice of covering images such as the Willendorf “Venus” with red ocher may be more than merely decorative. The red color may have symbolized the blood of life. Whatever was being signified, red ocher coloring predates the Cro-Magnons, being found as early as 100,000 BC in Neanderthal burials (Marshack 1990: p. 3).
Art became more cognitive by the late Ice Age. Marshack has shown the extent to which images from the Magdalenian period (15,000–10,000 BC) through the neolithic “were essentially ‘time factored,’ that is they were not simply ‘art,’ but were made at the right time, to be used at the proper moment, in the proper way, and for proper cultural reasons. … In fact symbolic carvings may in some measure have been made so they could be used ritually and ceremonially.” Cave art probably helped heighten and even coordinate seasonal rituals. “The ritual and symbolic use of the caves found in the Franco-Cantabrian area may have been ‘scheduled’ to that calendar frame, along with the manufacture and use of different classes of imagery and the performance of different types of rituals both inside and outside of cave sanctuaries” (Marshack 1990: p. 9).
From this sort of calendrical content deep in the Ice Age, Bronze Age art came to depict (“reenact”?) a more abstract cognitive content.
Cosmological Public Art Contrasted With Individualistic and Commercial Art
Bronze Age ceremonial art was composed in an epoch before rival economic policies became a factor in shaping cosmological systems. The social function of artworks, and hence the creative focus of artists as public workers, was to reflect eternal cosmological proportions. This meant subordinating their individual identity to timeless models. The creativity of individual artists was subject to a common cosmological canon.
This was literally a reenactment of creation.
What kind of creation? This question brings us back to the ultimate purpose of art. What was being expressed was a re-creation not of the artist’s personal experience but something more universal—the kosmos. Antiquity’s major public ceremonies were kosmos-renewing, and it was at these ceremonies that music was most pronounced. The celestial movements of the planets were emulated in song, dance, and instrumental playing (harps or lyres and flutes).
Chapter 16 of Athenaeus’s Banquet of the Sophists dealt largely with the role of music in banquets and related occasions. He reported that:
- “while most persons devote this art [of music] to social gatherings for the sake of correcting conduct and of general usefulness, the ancients went further and included in their customs and laws the singing of praises to the gods by all who attended feasts, in order that our dignity and sobriety might be retained through their help. For, since the songs are sung in concert, if discourse on the gods has been added it dignifies the mood of every one.”[9]
By this Athenaeus meant that:
- “by stripping off a man’s gloominess, music produces good-temper and gladness becoming to a gentleman, wherefore Homer introduced the gods, in the first part of the Iliad, making use of music. For after their quarrel over Achilles [Iliad[10] I.493–594] they spent the time continually listening ‘to the beautiful phorminx [lyre] that Apollo held, and to the Muses who sang responsively with beautiful voice’ [Iliad[11] I.603–604]. For that was bound to stop their bickerings and faction.”[12]
Athenaeus cited Homer (Odyssey[13] 8.99) to the effect that the gods made the lyre “the companion of the feast.”[14]
We know from the biblical story of Daniel and the fiery furnace that statues were “born,” and imbued by their artists with life and political power in a ritual public presentation much like newborn children.
An early description of ancient music occurs in the biblical story of Daniel, in Chapter 3 where Shadrach, Meshach, and Abednego were thrown into the fiery furnace when they refused to bow down to the golden image made by the Babylonian King Nebuchadnezzar. As was normal in the New Year dedications of new temples or major additions to them, Nebuchadnezzar “summoned the satraps, prefects, governors, advisers, treasurers, judges, magistrates, and all the other provincial officials to come to the dedication of the image he had set up.”[15] So they all convened and stood before it. “Then the herald loudly proclaimed, ‘This is what you are commanded to do, O peoples, nations, and men of every language: As soon as you hear the sound of the horn, flute, zither, lyre, harp, pipe, and all kinds of music, you must fall down and worship the image of gold that King Nebuchadnezzar has set up. Whoever does not fall down and worship will immediately be thrown into a blazing furnace.’”[16]
Public Status of Artists
Most artists were public servants (Greek demiourgoi, literally “workers for the demos”), and as such dependent on the civil state, palace, or temple for their livelihood. Some were outright slaves.[17]
Supervising them were various officials such as the choregos (originally in charge of the chorus, later the person who financed it), and the aesymnetes who beat the time for the choral dances. (I will discuss these officials in greater detail below.)
They were Greek demiourgoi, from the many blind men who were trained as musicians to the metic foreign “guest workers.”
Some of the most important Greek playwrights were public officials at the top of society. Sophocles twice was elected strategos (“commander in chief,” or “strategist”) of Athens, and treasurer for the Athenian empire. Euripedes was sufficiently political to be exiled. I suppose that the modern analogy would be for an American business leader or politician to write a Broadway play.
In any event, while Greek tragedies and comedies still speak to modern audiences through universal themes, their content was highly political. Aristophanes’s The Clouds and Plutus provide nearly as informative a discussion of moneylending as Aristotle’s Politics.
The Public Character of Music
“In the Indo-European languages” noted the classicist Norman Brown[18] (1947: pp. 29f.), “words meaning ‘song,’ as well as words meaning ‘speech,’ are commonly derived from roots meaning ‘loud sound’… Furthermore, these roots commonly have connotations of magic. Philologists have therefore concluded that the origins of song and poetry lie in the intoned formulae of magical incantations…: certain words spoken or chanted, certain ceremonial actions, and an officiating minister of the ceremony.” He related the etymology of the Greek word for “herald” (“keros”) “to Latin carmen, a ‘song,’ and the Sanskrit karuh, to ‘sing,’ and karus, a ‘bard.’” In Homer, both bards and heralds had “excellence of voice.”[19]
We are dealing here with an epoch in which each profession had its distinct song, and the tune conveyed a specific message, much as salesmen, scissor grinders, and other craftsmen until quite recently went down the streets singing their own professional songs.
Some hint of the role of “singing heralds” was provided by Schneider[20] (1957: p. 40) in describing contemporary practice in hunter-gatherer societies: “It is also customary for the musicians to lead a village community when it pays an official call on a neighbouring community and the latter presents the singers and instrumentalists with gifts.” One reason for this role probably was the musician’s role as “praise singer.” This is the original meaning of the Greek muse Polyhymnia, whose name derives from “hymnos” (English “hymn”), a song of praise, especially for newlyweds. Another reason for the prominent role of musicians as intermediaries, according to Schneider (1957: p. 3), was that “The agreement of sounds is always a symbol of identity or at least of mutual understanding. In the Solomon Islands, when an invitation is sent to a neighbouring tribe it is customary to send the measurements of the tribal panpipes so that the guests can tune theirs beforehand, thus ensuring the greatest agreement in the mutual musical greeting.” The resulting political duet signified the mutual desire for harmonious relations between the two communities. The general idea was reported by Athenaeus[21] (XIV.627): “Many of the barbarians also conduct diplomatic negotiations to the accompaniment of flutes and cithara to soften the hearts of their opponents.”
Art as Harmony and Measure
Precise arithmetic and geometric measurements were the keys to artistic harmony. The etymology of “harmony” is informative: It stems from Greek “arthron,” meaning “joint” (and hence the root of “arthritis”).
“Order” equaled “arthron,” “joint.” “Joining” had a cosmological and indeed, numerological dimension.
Some philologists (e.g., Shipley[22] 1984: pp. 16ff.) went so far as to derive Latin “ordos” from Greek “arthron” (“joint,” whence the word “arthritis”).
“Arthron” is the root for “art” and “arrangement,” “articulate,” and “arithmetic.” The idea of “joining” or “fitting” is related to that of “ligature” and “religion” (from Latin “ligare,” “to bind”); both have the meaning “to connect.”
In time the “first order” took on the connotation of “best” (Greek “aristos”), and also “arete,” “virtue.”
The other set of words has to do with the “rations” distributed to public dependents. They thus came in “ratio.”
This goes beyond merely counting, to comparing things once they have been counted. The first mental categories thus were numeric: Things were ranked by similar qualities, e.g., 12 months and notes of the scale.
We live in romantic times. Art has become individualistic, either a private effort for the artist or produced for the commercial private market. Whatever its motivation, it is supposed to be something novel. The artist often stands in a position of outsiderness or even of antagonism toward society.
This was not the archaic view of artists and their art. To be sure, shamans probably were idiosyncratic, but even their work had a strong cosmological content. The function of archaic artworks was not to be new but to re-create the traditional and universal. Behind the notes lay a scale that reflected the mathematical order of the kosmos, promoting harmony and order.
Most archaic cosmology expressed itself in precise numeric relationships, e.g., in the standardized proportions of the human body followed by sculptors and painters, in the composition of the Sumerian King List and the biblical Patriarch List, in the number of strings in ancient musical instruments, and most important of all, in the scales to which they were tuned.
Antiquity’s quantifying spirit was quite different from that of modern times. Today, there is a faddish belief that whatever can be expressed in quantitative terms is “objective” and hence probably true. But wherever we find numbers in the most archaic times, we are liable to find cosmological symbolism at work—in music and architecture, sculpture, and even what might seem at first glance to be history.
History was more art than literal annals. When the earliest protohistorical records appeared in the third millennium BC, they were standardized along cosmological lines in a similar way to public art and music. This chapter therefore examines the Sumerian King List and the biblical Patriarch List as examples of cosmological literary expression, not as literal truth.
Sculpture and painting likewise regressed their subjects into a standardized canon that had a cosmological foundation. In an age when most art was ceremonial, this meant that the subjects beginning with rulers as the primordial artistic subjects were standardized. Looking at Mesopotamian royal statues, one finds a sameness over the centuries. They seem realistic at first glance, but on closer examination this impression turns out to be only illusory.
The most famous set of statues are those of the ruler Gudea of Lagash c. 2100 BC. Every naturally rendered detail has a symbolic meaning, which is spelled out in the texts engraved on these statues.
Investigators have long analyzed the symbolism of public architecture—the Mesopotamian ziggurats, their orientation, number of stages (seven), and colors (corresponding to those of the planets).
If music is frozen architecture, then one may call the musical scale “frozen astronomy.” The seven strings of Apollo’s lyre symbolized the five planets, sun, and moon. There are some remarkable parallels, including the seven notes of the normal scale (denoted by the first seven letters of our alphabet, A through G).
Music
Speaking of primitive peoples, the ethnomusicologist Marius Schneider[23] (1957: pp. 38f., 4) observed that music marks the flow of everyday life, being a form of communication in its own right. “All the rites relating to birth, circumcision, marriage, hunting, war, weather, medicine, and death are permeated with musical elements” (p. 39). In short, music—along with the group-meals described in the next chapter, The Distributive Justice of Group Feasts and Banquets—provided a basic element of all the rites of passage. Children heard cradle songs, sang game songs, and learned riddle and proverb songs, while adults sang spontaneously to give validity to their sentiments, to placate nature or cast spells, or simply to use a kind of prefabricated emotion. “Primitive man sings to call out, to play, to mock, to greet someone, to give thanks at the end of a meal. Many of the songs are improvised. Songs can even be sung before a judge; if two men quarrel or a couple want a divorce the contestants will plead alternatively in words and more or less improvised songs” (p. 38). It is as if music validates the idea being expressed, or makes it somehow either objective or formal. Plowmen “address their horses or oxen in song” (p. 38).
Athenaeus (XIV.618) noted that many professions had their own work-songs: a mill-song (himaios) for grinding flour, an ioulos for the wool-spinners, and so forth. Schneider added that even marriages were accompanied by group-songs during the wedding night, as were the funerals with their choral prayer-songs. In sum, if an event was public, it generally was accompanied by music.
As the mathematics of tuning came to be discovered in archaic times, music became a key to the kosmos taking its place alongside the calendar as a basic model of society.
Today, schoolchildren are taught to sing tunes in unison, and perhaps to participate in rhythm sections, but they take for granted the tuning of the musical scale. The principles by which the scale evolved are neglected. Why include its history in the curriculum, now that even temperament has been established?
Yet musical tuning of the scale once was deemed a poetic subject worthy of the highest mythology.
One of the most striking aspects of ancient writings on music was their dismissal of actual performance. What was deemed most important was to understand the ratios and to relate them to the kosmos.[24] Thus, despite the fact that the educational curriculum revolved around “music,” what it neglected was the actual art of performance. It concentrated on the tuning of the scale, not on performing tunes. What was emphasized was the theory and philosophy of musical temperament and its social metaphors.
When it came to poetry, most important was its sound, rather than its written form as it is today. See, for example, the work of Marie-Louise von Franz on the idea that music was intuitive, not seen or perceived quantitatively, but having a numerical base. Learning this foundation answered many riddles.
Some Mathematical Parallels Between Music and the Calendar
The seven notes of the scale equated to the seven movable bodies in the heavens. The numerological symbolism of the time described the number eight as the “rebirth” number, probably because it started a new octave or, in Semitic cultures, a new seven-day week.
Also of a periodic nature in both music and calendar-keeping was the number 12. It signified the octave’s 12 tones, and also the 12 months of the year. And, as noted in Alphanumeric Notation and the Calendrical-Musical Kosmos, music and astronomy may have shared similar symbols for their notation.
Plutarch stated in Moralia: “The Chaldeans say that Spring stands to Autumn in the relation of a Fourth, to Winter in the relation of a Fifth, and to Summer in the relation of an Octave.” And Curt Sachs also pointed out that “the ancient Chinese also viewed the Spring as distant from the Autumn by a Fourth and from the Winter by Fifth.”[25] But they viewed the spring as distant from the summer by a second, not by an octave as the Babylonians did.
This suggested a 12-spoked wheel for tones, as for the zodiac. But it was different.
There were five tetrachords, and five “extra” calendrical days in the year (bridging the gap between the 360-day public-sector administrative calendar and the actual 365-day solar year).
Music was omnipresent in archaic public life. Archaeologists have found whistles and percussion instruments in Ice Age caves, including thighbone trumpets used as ocarina-like instruments.
Wind instruments seem to have begun with the horns and bones of animals. (We still call the brass instruments “horns,” and Latin “corni” provides the root for the modern “cornet.”)
In the Bronze Age musical historians first encountered stringed instruments, and it is here that mathematical computations must have begun to shape the musical scale. Reconstructions of Bronze Age musical compositions and even the instruments and notation remain controversial, but the basic ratios are clear enough. The monochord—a single cord run over a ruled measure—elaborated the scale’s mathematical proportions.
Music was performed at all the major festivals, as well as at private banquets (see The Distributive Justice of Group Feasts and Banquets). But by the same token, in archaic Greece “Music would be heard as widely as poetry (indeed, at this time it existed only as an accompaniment to poetry), and in Greece it must have acted as a unifying force” (Snodgrass[26] 1980: p. 177). Recall from Athenaeus, as was stated earlier, “Many of the barbarians also conduct diplomatic negotiations to the accompaniment of flutes and cithara to soften the hearts of their opponents” (Athenaeus, translated by Charles Burton Gulick,[27] Vol. VI [1937], p. 383).
One of the most striking aspects of ancient writings on music was their dismissal of actual performance. What was deemed most important was to understand the ratios and to relate them to the kosmos. Many performers were public dependents, above all blind men.[17]
Others were slaves, and then there were the Greek flute-girls. (For an idea of the disparaging of practical music see Boethius, who wrote at the beginning of the sixth century AD, Principles of Music.) (See Curt Sachs,[28] The Rise of Music in the Ancient World: East and West [1943].) Sound was made on the stringed instruments by rubbing, that is, friction. The primordial creative friction was of course that of sex.
Also sexual were the number-relations of the musical string-lengths. (These lengths were the inverse of the frequencies of the musical tones as we would measure in vibrations per second.)
McClain[29] (1976: pp. 19f.) pointed out that the 2:1 interval can only generate octaves: CC’cc’ and so forth. But it cannot “fill out” these octaves with tones. For that, “male” intervals are necessary, beginning with 3. This is basic Cosmology 101.
The ratio 3:2 creates the most basic musical interval next to the octave: the fifth. This is the second overtone (Table 5.1). In our example it represents the G over the first C.
The octave and fifth were the only “perfect consonances” recognized in the Dark Ages.
When the Greeks used the term “harmony” (’armonia), they did not refer to the harmonizing arrangements of modern songs and orchestras, but to the acoustics of tuning the scale. As a matter of fact, Greek music ignored “harmony” in the modern sense. It was melodic and multivoiced but not aiming at the triad chords that have marked post–Renaissance European music down to the present day.
Musical theory for the ancients consisted of studying the ratios of the major musical intervals. The key instrument for this acoustic experimentation was the monochord. Henderson[30] (1957: p. 341) pointed out that “Music, though practically ruled by the voice, was theoretically analysed in terms of the stretched string, which yields the words syntonos (taut) for high pitch and aneimenos (slack) for low, the nomenclature of notes form the plucking fingers, and some basic features of the notations.” And significantly, “Astronomy remained a regular branch of harmonics”—a spirit that would culminate in Kepler in modern times.
This notion of cosmic harmonics was the central idea of Plato’s Timaeus, as well as the books on harmonics by the astronomers Aristoxenus and Ptolemy.
Mathematics of Tuning the Scale
In the beginning was chaos: undifferentiated sound. Musical order emerged as the pitch continuum was divided into discrete tones. Sound came into being.
But the archaic concept of musical creation was not “primitive.” Rather than simply sound emerging from silence (like the rustling of the waters of creation, of leaves of vegetation, of the animals created, of the weather and thunder and lightning), there emerged a purer sound—one resolvable to mathematical order.
From a single primal tone were generated harmonic overtones. To the ancients, the cosmos was being put in tune.
The most basic musical ratio is that of the first overtone: the octave (from C to c; see Table 5.1). Let us set C at “30” (or 360, so that we may use convenient calendrical notation—which is very likely what the ancients did, at least according to Ernest McClain[31] and others. If we take this tone, and halve the length of the string, we get the octave. (Pitch is inversely proportional to string length.) Mathematically, if middle C is 360, then the higher C’ is 720. And high c is twice again as high, 1,440.
Today, modern orchestras tune their instruments to tuning forks whose rate of vibration is measured in cycles per second. The ancients had no way of measuring cycles per second. What they could measure was string-lengths.
For this purpose they used a monochord. Why just one cord rather than, say, the four strings we find on modern violins, violas, cellos, and basses? The answer is that measures of length are only one variable. As every musician knows, the strings of modern string instruments are all the same length—but the lower pitches are made of thicker material, and are also tuned less tautly.
The word “tuning” (“tenir”) comes from “pull,” “stretch”; to a string player, the pitch of his or her instrument is a function of how tautly he or she turns the strings on the pegs.
Pictures of musical instruments on ancient Greek vases and walls show that matters were much the same back then. The Greek harps and lyres were strung with similarly long strings, whose tautness and probably thickness varied. This was done by ear, with the musician tuning each string to a particular interval from the basic reference tone, much as done by modern string players.
To reduce all this to basic principles of length (and hence pitch), only a single string could be used—one which did not vary in thickness or tautness. It was divided by a movable bridge, which was moved back and forth along a canon—a ruled scale extending the length of the string.
This scale could assign any given number to the long “low” note. The number 2 could be doubled, or 30, or 360.
Very early on, harmonic theorists must have decided to use a calendrical measurement of length. In the first place, as Measures, Rules, and Prices has described, most archaic measures of length had a calendrical basis, for the first carefully measured buildings were temples modeled on the calendrical kosmos. But there was an even better reason to use a calendrical reference point, and that was the set of striking parallels between musical harmonics and the worldly calendar.
We can go on doubling the original tonic indefinitely, by successively dividing the string length in half. But that will produce only “empty” octaves (C, C’, c, c’, c”, etc., in a ratio of 1:2:4:8:16:32:64:128:256 and so forth).
Was the number 1 conceived of as unrelated to other numbers, a case of “itself” (a parthenogenic number)?
Was number 2 thought of as a “female number”?
Were “male” numbers represented by odd numbers, beginning with 3?
To fill out the scale we need a series of fifths. This is in nature the second overtone one hears over low C: the G. And musicians soon found that this note could be produced by the ratio 3:2. To archaic cosmologists, this represented the first “male” number divided by the female number.
The first fifth—from C to G—is thus 3:2. But to go up another fifth, we must multiply the first 3:2 by another 3:2. In other words we must square it. This produces 9:4—the note D. It relates to C’ as 9:8, and this fraction thus represents the whole tone—C to D.
We thus see that to add one fifth to another, we must multiply their ratios. And this is what is meant by “geometric” growth.
In our epoch we are familiar with rates of growth. We measure gross national product (GNP) and other economic time series by rates of growth, not by absolute amounts. We compute doubling times: How many years does it take capital to reproduce itself? In how many years will the earth’s population double (or a herd of animals, or any other economic stock)? These geometric rates of growth are best illustrated on a log scale.
The word “logarithm” was coined from the Greek roots “logos” (“logic,” “law”) and “arith” (“arithmetic”).
A whole tone scale can be made by proceeding by whole tones. On the piano, we would get the series C, D, E, F♯, G♯, A♯ (=B♭), then back to C. But we would not get the other six tones. To do this, we must continue tuning by fifths.
For ease of visualizing this on the black keys of a piano, let us begin by a fifth below middle C: F. This gives us what musicians call the “circle of fifths,” consisting of F, C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯—and then, a fifth above A♯, E♯. (Alternatively, if we begin with C, we will end up with B♯.)
However, as every pianist knows, there really is no E♯ (or B♯). That is F. We would seem to have arrived back at our starting point.
But in fact, we have not done so, for a very good reason. To return to a higher octave of our starting F (or C), we would need some doubling, that is, power of 2 in the denominator of whatever fractional system we are using. Its powers are 9, 27, 81, 243, and so forth, with the power always ending in an odd number—a 9, a 7, a 3, or a 1. Thus, as we reach the circle’s return at the seventh octave, we get a slight dissonance, an out-of-tuneness. Since antiquity, musicians have called this a “comma” (Greek “komma”). This represents the ratio of tuning by odd-numbered intervals relative to even ones.
Ancient harmonic theorists represented this by the following “fork”: 1, 3, 9 on the left, and 2, 4, 8, 16, etc., on the right.
Suppose we try to tune the scale by thirds, e.g., C, E, G♯, C. We are back at the tonic. Yet here again we encounter the same problem: The third is 5/4. And (5/4)2 is 25/16. We thus are dealing once again with powers of 5 (odd number) divided by sequences of even-numbered powers of 2. We soon encounter another type of comma.
First came the calendar. Then came music. I think that its importance is twofold. On the one hand is the fact that music does indeed touch emotions. It indeed plays a major role in human life. But as its principles were refined—as instruments were tuned, and symphonic music developed by the Bronze Age—archaic intellectuals made a remarkable discovery: Tuning the musical scale was in many respects like establishing the calendar.
Not only were there immediate similarities between the seven tones in the scale and the seven movable bodies of the heavens, and even between the 12 tones in each scale and the 12 months of the year, but there was something else: Just as the actual calendar was slightly “out of tune,” so was the process of tuning the musical scale. Setting the intervals by fifths (3:2 ratios) produced one mathematical-tonal series, doing so by thirds (5:4 or 6:5 for the major and minor third respectively) another.
Calendar-makers solved this problem by setting a year-end “time out of time.” After all, they hardly could take an average of 360 days and lengthen each day. The sun rose and set 365 1/4 times each calendrical year as measured by the sun, and there was nothing to do but add some extra days.
But musicians found a different solution available—indeed, dictated. They hardly could add an “extra note” to the gamut of octaves. (If they had, there would be two separate keys for G♯ and A♭ in one of the higher octaves.) Musical tuners did what calendar-makers could not do: Having drawn an analogy comparing musical tones and octaves to the days and months of the year, they then adjusted each tone slightly, so as to temper each tone by just enough so that there would be no “large” disparities developing among the “later generations” of tones such as G♯/A♭. At least, this was the “equal temperament” system developed thousands of years before Bach took the great step toward re-establishing the tendency with his Well-Tempered Clavier in 1722.
I am afraid that a rather sophisticated mathematics is needed to understand the complexities involved in tuning the musical scale in antiquity. A discussion of these complexities is necessary not only to understand how ancient mythmakers mathematized their kosmos, but to proceed to the central thread of this chapter: How the mathematics of musical temperament (and by extension, the visual art and what I call cosmo-historical narrative) served as a social and indeed, highly political analogy. By the time of classical antiquity we find the Platonic followers of Pythagoras using the mathematics of musical temperament to rationalize economic and social inequities. Disparities in wealth and status were rationalized by saying that there were two kinds of equality—one for the democracy, and one for the aristocracy. In other words, this meant an equality of proportion rather than of tones.
This kind of discussion will appear quite alien and arcane to modern audiences. Today, musical tuning is relegated to specialists in acoustics and hardly taught to students at any time in their education. And probably the least likely to study the subject are students of philosophy. Yet music formed the very crux of ancient training.
Plato’s Timaeus was the major work bequeathed by antiquity to the Dark Ages/the first millennium and a half of our era. The dialogue begins with Socrates counting, “One, two, three…”
Not only classical Greece but also in India the Rg Veda dealt with the cosmology of musical temperament. But unlike the western European tradition, it proscribed arguments over tuning the scale. (And partly as a result, henceforth, Hindu music went its own way, as did its astronomy, calendar-making, and financial and economic development.)
McClain[32] (1976: p. 7) quoted Schneider[33] (1957: p. 45) to the effect that for the historian of culture, “sound represents the original substance of the world.” Indian tradition emphasized the “luminous nature of sound” in the similarity between “svar” (“light”) and “svara” (“sound”).
If light ruled the day, then music ruled the night. As such, it was associated with the moon (as the day was the realm of the sun). In this fact we find much of the early association of musical numerology with that of the major lunar numbers. Thus the early lyres had seven strings.
But just as calendars became solarized, so did the mathematics of tuning the scale. And just as the solar calendar rounded off the dimensions of the year, so did musical temperament.
[Table 5.1]
Generation of Notes by Threes Versus by Twos and Fours
Defining “commas”: 71/72 (= 6 x 12). McClain[34] 1978: p. 38: “The comma we have defined exceeds the octave 1:2 by about the same ratio by which Plato’s year of 365 days exceeds his calendar base of 360 (12 months of 30 days each), a problem he solves in Laws in the Egyptian manner by adding five extra ‘election’ days at the end. The comma of 73:74 and the ratio 360:365 = 72:73 show the same discrepancy in the soul as in ‘days and nights and months and years.’ [Republic[35] 587b–588a.]” He added that “It is amusing that Philolaus, the teacher of Plato’s Pythagorean friend Archytas, counted 364 1/2 days in a year so that his days and nights together totalled 729, a number essential to Pythagorean musicology, however badly it fitted the facts of calendar making.”
Thus (per McClain[36] 1978: p. 33): A tyrant suffers “exactly 729 times” as much as a philosopher, “king of himself” (Republic[37] 572b). This reflects the Pythagorean comma, 531,441 (= 7,292): 524,288 (= 219 = 86 = 5,122), all numbers 2p being Platonic tone circles.
As McClain[38] wrote, “[E]ven numbers which define the octave matrix are ‘female,’ [while the] odd numbers which fill this matrix with ‘tone-children’ are ‘male’” (McClain 1976: p. 4).
The Hindu drum of Shiva which fills the world with life (and sound) may be understood as the intersection of powers of two with powers of 3, or sequences of octaves and fifths.
A “comma” is “a small disagreement between two different definitions of a tone”; e.g., the Pythagorean, diaschisma, diesis (McClain[39] 1976: glossary).
The Pythagorean comma has a ratio of 524,288:531,441, or “about 24 cents,” according to McClain (1976: glossary).[40] This is the difference which “arises between the reference pitch and the twelfth tone tuned by pure fifths of 2:3 or fourths of 3:4 (numerically, between 312 and the nearest power of 2)” (McClain[41] 1976: glossary).
What McClain[42] (1976: p. 119) called the “Jubilee” comma, 50:49, requires a mathematical base 1,260:630 to incorporate both this ratio and the “lunar” diesis of 125:128.
Modern historiography finds that while the debt cancellation of Leviticus—the deror law—has roots in the Assyrian andurārum practice and earlier Sumerian and Old Babylonian amargi and misharum acts, the mathematics of the postexilic compilers of the Torah drew on Babylonian mathematics they had learned during the Exile.
In this manner Neo-Babylonian esoteric mathematical cosmology was incorporated into Jewish cosmography of the Torah. The Jubilee Year “cured” the dissonance and restored harmony in the social continuum of the Hebrews, just as adjusting the notes to moderate the Jubilee comma restored harmony for the musical scale.
I point the interested reader to the works of McClain.[43] He may have gone too far in trying to find in the archaic Vedic, Mesopotamian, and Greek mathematics of musical temperament a conscious analog to the 25,920-year precession of the equinoxes. But his basic premise of parallels being drawn between adjusting the calendar and tuning the musical scale were certainly correct and “in the spirit” of ancient higher wisdom.
The three Fates were Clotho, Atropos, and Lachesis. “The interval CE in Pythagorean tuning gives a C that is too low, an E that is too high: hence Clotho’s and Atropos’s adjustments. Lachesis’s task is to make A♭ and G♯ coincide, adjusting one with each hand. ‘Plato saw the necessity of temperament for systems meant to function in harmony, be they musical scales, planetary orbits, or communities of just men’” (Godwin 1983: pp. 298f., quoting McClain[44] 1977: p. 55).
In the Republic[45] (587c), noted McClain[46] (1976: p. xi), when Plato stated that the tyrant is 729 times as bad as the good man, some translators (e.g., Francis Macdonald Cornford[47]) “simplif[ied]” the text. But to the musician trained in acoustics, the number 729 has a particular meaning: It “corresponds to the… tritone (36 = six fifths above the fundamental), the worst possible dissonance in the musical systems known to Plato.” He thus was referring to the tyrant as creating tension and dissonance within an otherwise harmonious society. See McClain[48] 1978: p. 38.
McClain[49] (1976: pp. 102–103) discussed his Chart 23: “The gap which remains between a♭ and g♯ at the bottom of the circle is now narrowed to a diaschisma worth about 20 cents or 3/10 x 20 = 6 degrees, wondrously close to the 5 1/4-day shortage between the ancient calendar base of 360 days and the true solar year… The ratio results from the reciprocal meanings of 45:32…”
He went so far as to try and make sense out of the numerology found in the Bible’s closing Book of Revelation (historically a pitfall which one enters only at the risk of losing one’s sanity).
[See this query for the section: The Role of the Square Root of Two]
Summary: ‘Natural’ Tuning Versus Equal Temperament
McClain (1978: p. 3) set out to show that “not only are all of Plato’s mathematical allegories capable of a musical analysis—one which makes sense out of every step in his arithmetic—but all of his allegories taken together prove to be a unified treatise on the musical scale so that each one throws light on the others.”
To begin with, every division by 2 (for an octave) is a kind of “cyclic identity,” and thus is assigned the same letter name in modern notation—low C, middle C, high C, and so forth; or C, C’ c, c’, c”, etc.
The basic subdivisions are the fifth (2:3), the fourth (3:4), the major third (4:5), minor third (5:6), and finally the whole tone (9:8).
According to McClain[50] (1978: p. 3), Ernst Levy pointed out that Socrates’s “sovereign” number defined “what musicians know as Just tuning—idealized, but impracticable on account of its endless complexity—and that the political disaster Socrates foretold must therefore have been related to the difficulty musicians have always known.”
McClain[51] (1976: p. 163): “Ernst Levy pointed out that the ratio 4:3 produces tones linked by perfect fourths and fifths, that ‘mating’ with 5 produces pure musical thirds, and that the ‘unmusical children’ Socrates predicted from his formula must therefore be those plagued by the notorious commas which make ‘Just tuning’ impractical and motivate the eventual adoption of some form of ‘temperament.’”
[See this query for the section: Social Analogies Based on Musical Temperament]
Ruler/Noise/Harmony
Citing Plato’s Republic (531ac), McClain[52] (1978: p. 4) pointed out that musical-mathematical autocrats accused the humanists of “put[ting] ears before the intelligence.”
Actually, inasmuch as the argument between oligarchy and democracy was one of proportions—of distributive justice as far as wealth and debts were concerned—the difference was boiled down to ratios: the mean versus the average. McClain[53] (1978: p. 131) boiled down the dispute as follows: The oligarchic model took “sight and its criteria as its primary organizer of sensation,” while the democratic party took sound as the major criterion.
McClain[54] (1978: pp. 140–141) pointed out that Aristotle’s pupil Aristoxenus, “The Musician,” endorsed his teacher’s conception of a “relaxed proportion” as against Socrates and Plato.
McClain[55] (1978: p. 19) stated that Plato theorized “that even the best aristocracy will degenerate in time through a timocracy [rule by the rich], oligarchy [rule of the few], and democracy [rule of the many].” Similarly, “Any tuning which uses the ‘perfect’ ratios of integers—and Socrates’ system uses 1:2:3:4:5:6[—]will degenerate unless the number of tones is rigidly limited. A series of perfect fifths 2:3, for instance, slightly larger than 7/12 of the octave, could agree with the octave series only if some higher power of 2 agreed with some higher power of 3, an obvious impossibility since the first series is even and the second is odd.”
Overarching both parties was the fact that cosmological considerations were indeed put over “the ear.” At least our epoch finds the two most striking modes the major and minor. These are called the Ionian and Aeolian, respectively. (Beginning from C on the “white keys” of the piano, the Greek modes are Ionian, Dorian, Phrygian [with its small second step between E and F], Lydian [with its dissonant augmented fourth FB], Mixolydian, and Aeolian [beginning on a].[56] There really is no mode beginning on b, for a minor second and fifth [BF] is universally perceived as being too dissonant.)
Nowadays, almost all our tonal music is written either in the major or minor modes (Ionian and Aeolian). Why then did the Greeks prefer the Dorian mode? Proponents of a value-free music might say “To each his own,” and chalk the matter up to subjective preference. But there seems to have been another element at play, one which was actually more visual or intellectual than musical. That is the fact that the Dorian mode had a striking visual/intellectual character: It was palindromic (or, as McClain put it, “reciprocal”). It had the same sequence of intervals whether going up or down the scale.
Pythagorean Theory
As ancient economies grew more commercialized and prosperous, the role of music followed suit. One of the most notable anecdotes (reflections of this trend) comes from the wealthy city of Sybaris (whence our modern term “sybaritic” for luxurious), in southeastern Italy/Sicily. Like most ancient cities, the aristocracy was synonymous with the cavalry, thanks to the fact that every cavalryman had to equip himself with his own horse, and have the leisure to practice group tactics.
Apart from fighting, the major context engaged in by the Sybarites was that of conspicuous consumption, and to impress their fellow men. They trained their horses to march not only to the rhythms of battle but also to perform other tasks of ingenuity to the sound of the flute.
Athenaeus[57] (XII.3–41) reported that “To such a point had they carried their luxurious refinement that they had even trained their horses to dance at their feasts to the accompaniment of pipes. Now the people of Croton knew this when they made war on the Sybarites, as Aristotle records in his account of their Constitution.” Apparently some Sybarite had insulted (probably made unwelcome advances on) one of the flute-players, who resolved to avenge the insult by playing the tune to the Crotonites. At a signal in the battle all the Crotonite pipers played the melody to which the horses were accustomed, whereupon they rose on their hind legs, throwing off their riders, and so caused an easy victory for Croton.
This last is the editor’s summary from Julius Africanus, Cesti 293. Athenaeus says simply, the Crotonites “struck up the dance tune for the horses; for they had with them pipers in military uniform.”[58]
So appealing was this story of the ingenuity—but also inappropriate behavior—of horses that it was told of a number of cities. Charon of Lampsacus (Annals II, reported in Athenaeus[59] loc. cit.) wrote that the town of Cardia was besieged by the neighboring Bisaltians. The latter were led by a man named Naris, who had been sold as a child to a Cardian and become a barber. While working in his shop, he heard much talk of an oracle warning that one day the Bisaltians would attack Cardia. He managed to escape back to his native town, and used the stories to muster enough support to be appointed leader. He knew that “All the Cardians had schooled their horses to dance at their drinking parties to the accompaniment of the pipes. Rising on their hind legs and, as it were, gesticulating with their front feet, they would dance, being thoroughly accustomed to the pipe melodies. Knowing this fact, Naris purchased a flute-girl from Cardia, and on her arrival in Bisaltia she taught many pipers; accordingly he set out with them to attack Cardia. And when the battle was on, he gave orders to play all the pipe melodies which the Cardian horses knew. And when the horses heard the piping, they stood on their hind legs and began to dance; but since the whole strength of the Cardians lay in their cavalry, they were beaten in this way.”[60]
When the Spartans went to war, says Athenaeus (XIV.631), their soldiers sang marching songs as they marched in time to the music.
But in time music and dance became more Dionysian in character, “for the dancers carry Bacchic wands in place of spears,” and hurled stalks of fennel at each other.
Plato was a Pythagorean, and like him an elitist.
Farrington[61] (1939 [1946]: p. 32) noted that Plato condemned the study of arithmetic and even mechanics “as having egalitarian tendencies” and thus being “a danger to the soul.” In his Laws (IV.757) Plato asserted that the legendary Lycurgus “banished the study of arithmetic from Sparta, as being democratic and popular in its effect, and to have introduced geometry, as being better suited to a sober oligarchy and constitutional monarchy. For arithmetic, by its employment of number, distributes things equally; geometry, by the employment of proportion, distributes things according to merit. Geometry is therefore not a source of confusion in the State, but has in it a notable principle of distinction between good men and bad, who are awarded their portions not by weight or lot, but by the difference between vice and virtue. This, the geometrical, is the system of proportion which God applies to affairs.”[62] The deity “protects and maintains the distribution of things according to merit, determining it geometrically, that is in accordance with proportion and law.”[63] (Plato went so far as to insist that “Dike and Nemesis… [teach] us that we ought to regard justice as equality, but not equality as justice.”[64]
Democratic Theory
The democrats never responded with a temperament theory of their own. What they did was simply to maintain the tradition of Nemesis, the fair “divider” and avenger of hubris and arrogance of the wealthy.
Dance
James Miller[65] (1986, Chapter 2), a historian of archaic cosmological choreography, described the views of Philo’s synthesis of Platonism and Judaism with regard to the moral and ethical implications of a truly cosmic understanding of music and dance. In his treatise “On the Virtues” (72–76), Philo[66] imagined the final moments in the life of Moses, in touch with the dancing movements of the kosmos. “[T]hough inhabiting a perishable body,” Moses “trained his soul in the ways of the Muses, following the example of the sun and moon and the wholly sacred chorus of the other stars, and attuning himself to the divine instrument, the heavens and the whole cosmos.”
Commenting on this passage, Miller[67] (1986: p. 73) wrote that “The Mosaic choreia is more than a display of universal rationality. It is essentially a religious movement, a… ‘wholly sacred chorus’ expressing the love felt by man and nature for the transcendent Ruler of the universe.”
Yet at the same time, “Astronomy, music, mathematics, and all the branches of philosophy were not ends in themselves, …but simply preparations for leading the life of virtue and piety which allowed the human spirit to leap into the Kingdom of Heaven.” Philo thus “imagined the Lord as a Socratic teacher.” In his essay “On the Change of Names”[68] (72–73) Philo asked:
- “For what purpose do you investigate the choral dances and revolutions of the stars? Why have you leapt from earth up to the region of the ether? Is your purpose merely to busy yourself idly with what is there? And what great advantage may be gained from all that idle labor? How does it serve to purge pleasure, to overthrow lust, to suppress grief and fear? What surgery has it for passions which rock and confound the soul? For just as trees are useless if they bear no fruit, so also is the study of nature useless if it does not lead to the acquisition of virtue.”[69]
Contemplation of the heavens and the ability to reflect their celestial movements and proportions in dance and music “were the same self-control, sobriety, and freedom from disruptive passions that the Athenian sought to inculcate in the citizens of his utopian society through the study of astronomy, music, and choral dancing,” summarized Miller[70] (1986: p. 74).
If man were created after the image of God, it was not as a corporeal image, “since God is not human in appearance and the human body is not godlike. No, the likeness meant by Moses pertains to the mind as ruler of the soul, for the one archetypal Mind of the universe was the model after which the mind in every individual was patterned. … The mind is unseen but sees all things itself, and while it reveals the essences of all other things, it keeps its own essence hidden. … After it has been raised on the wing and has observed the air and all its properties, it is borne still higher to the ether and to the revolutions of the heavens and is whirled round with the choral dances of the planets and the fixed stars in accordance with the laws of perfect music, following the love of wisdom which guides its footsteps.” (Philo,[71] “On the Creation,” 69–71, discussed in Miller[72] 1986: pp. 56f.).
Art was thus a reenactment of creation, literally a re-creation (not “recreation,” as it would later degenerate into).
The essence of astronomical movements—the dance of the planets, and hence the highest musical order—was their regularity (once again our “reg” idea), “a predictable cycle of changes, itself unchanging.”[73] Miller[74] (1986: p. 57) added: “the stability and uniformity of the patterns of motions detected in the heavens revealed the existence of an archetypal Mind governing both the psychological and physical dynamics of the temporal cosmos.” It thus reinforced a faith in authority—on earth as well as in the heavens.
“In Plato’s longest and probably last dialogue, the Laws,” wrote Miller[75] (1986: p. 52), he “went so far as to equate choreia with paideia, arguing that the art of choral dance could not be mastered without a full understanding of the mathematical principles of music, the structure of the human body, the dynamics of the human soul, the celestial models for earthly dance, the political benefits of training a society to move together and to obey the commands of a choragus, the divine origins of harmony, in short, everything that might be learned in a comprehensive philosophical education.” In Plato’s Laws[76] (II 672e) the idealized Athenian stranger stated that “the art of choral dance as a whole (choreia) is identical with education as a whole (paideia); and the part of it pertaining to the voice consists of rhythms and harmonies.”
In Epinomis[77] (982c–e) Plato stated that “as proof that the stars and the whole moving system of the heavens possess intelligence, mankind ought to consider the fact that the stars always do the same things and have done so for an amazingly long time. Because they are carrying out what was planned long ago, they do not alter their plans now this way, now that, sometimes doing one thing and sometimes another, wandering and changing their orbits.” This is why “the nature of the stars is the fairest to behold, for they dance the fairest and most magnificent procession and choral dance of all the choruses in the world…” Along these lines, Miller[78] (1986: pp. 41, 5) noted that “Images of harmony… invite us to participate in a reality we normally ignore and draw us back into the primitive immediacy of sensory experience to commune with a divinity that shapes our ends.” But they may be authoritarian (viz. the Pythagoreans): “Images of harmony… tend to silence debates, for they are essentially undebatable.”
Plato imagined that the best government was the most stable one. (This is why he has been accused of authoritarianism, most notably by K.R. Popper[79] in The Open Society and Its Enemies.) “The priests of Egypt were the wisest of law-makers, Plato contended, because they turned the dance into the stable foundation of their culture by outlawing changes in their choral ceremonies and identifying their ritual movements with the unchanging government of the gods” (Miller[80] p. 5, quoting Plato’s Laws[81] [II 656–657]).
If man was created in the image of the gods—and if it was art and paideia that led him to perceive this godliness—then the gods themselves were astralized. The major expression of this view occurs in Plato’s Timaeus[82] (see for instance 40a–d, in Miller[83] p. 19).
As Werner Jaeger put it in Miller[84] (1986, p. 14), “The chorus… was the high school of early Greece, long before there were teachers of poetry. … It was not for nothing that the institution of chorodidaskalia preserved in its name the word which means ‘instruction.’”
Athenaeus (XIV.632) pointed out that Demetrius of Byzantium (On Poetry, IV) noted that the Greeks “used to employ the term choregi, not, as today, of the men who hired the choruses, but of those who led the chorus, as the etymology of the word denotes.”[85]
Miller[86] (1986: p. 14) wrote:
- “To be ‘apaideutos’ was to be ignorant, boorish, uneducated, uncultured, uncritical, undisciplined, unsocialized, intellectually unformed, morally disordered, deprived of consoling religious and artistic traditions, pitifully unaware of the perfecting (if not exactly perfect) design of the cosmos, and thus wholly outside the perfectible sphere of classical Greek civilization. … The meaning of the adjective ‘achoreutos’ [without choreia] was no less insulting. If a person was not trained in the fine and public art of ‘choreia,’ which blended the arts of poetry and song with the visible rhythms of the dance, he was bound to be out of step with society and incapable of graduating into responsible adulthood.”
Apollo and Dionysus were “the supreme chorodidaskaloi or ‘dance instructors’ of the Olympian universe.”[87] In sum, to be achoreutos was to be “a soul perilously out of tune with the harmonious powers governing the cycles of nature. … everyone knew from Homer and Hesiod there was no god or goddess on Olympus who was not an accomplished dancer except poor Hephaestus, who was a cripple. … a danceless man was ungodly. … To be considered cultured one must be fully trained in the dance.”[88]
The leader of the choral dancers who “beat time” became the name for an official (aesymnetes) because it was equal measure. Thus, rhythm expressed the same idea as harmony.
But measured became unmeasured, as economic inequality crept into society. I will now discuss the cognitive content of archaic artworks.
The Cosmology of Sculptural Proportions
The round proportions found in archaic sculpture and painting were not dictated by nature. Decimalized cultures (Egypt, Crete, and Greece) had decimalized proportions of how the human body was depicted, just as they had decimalized interest rates. Sexagesimal Near Eastern cultures had sexagesimal bodily proportions, interest rates, and so forth.
Guitty Azarpay[89] (1990) compared the canons of human proportion as reflected in Mesopotamian and Egyptian art from the third millennium BC onward. What formerly appeared to be a spontaneous and naturalistic practice of blocking out proportions turned out to embody abstract underlying numerical relations. The idea was made most explicit in late antiquity. For instance, in Book III of his treatise On Architecture, the Roman writer Vitruvius compared the proportions of temples to those of the Roman body, following a decimal base. As Azarpay[90] (1990, pp. 95–96) summarized, Vitruvius stated that “Nature has so planned the human body… that the face from the chin to hairline, is a tenth of the total body height’ (III.i.4). He then described the ideal proportions of the human body as the ‘perfect 10’… the sum of 1 + 2 + 3 + 4, a Pythagorean concept that Vitruvius contrasted with the ‘perfect 6.’ The sum of 1 + 2 + 3, the ‘perfect 6’ was important in the sexagesimal system of numeration, which prevailed in the ancient Near East through the Achaemenid period. The decimal system of the Greeks was Egyptian in origin. … Consequently, if Egypt provided the model for the proportions of Archaic Greek sculpture of the seventh to early sixth century BC, it was the ancient Near East that directly inspired the Achaemenid Persian canon.”
Along similar lines in the Renaissance, Albrecht Dürer divided six-foot male and female bodies into 10 face-lengths (or alternatively, eight head-lengths).[91] Azarpay found a long pedigree for this kind of conceptualization, but along the lines of the 60-based system and its associated 360-day year. A glazed-brick panel from the Susa palace of the Persian ruler Darius exemplifies a head measuring 3/18 of the body length.[92] The face is 2/18, equal to 2/3 of the total head length.[93] Azarpay concluded that “art is modeled on art, not on life,”[94] for it ultimately was based on numerological proportions. The fraction of 1/18 seems to signify 36, hence the 360-day civil administrative year. What makes these regional differences in sculptural proportions so important is that they neatly parallel the charging of interest according to the local unit fraction: 1/60th per month in Mesopotamia, and 1/10th annually in Greece, whose dekate was influenced by the Egyptian decimal system, as discussed in Measures, Rules, and Prices. Ultimately at work is a common denominator with sufficiently deep cultural roots to include the rendering of the human body on the one hand, and administered interest rates on the other.
As a case in point, among the most famous Sumerian statues are those of Gudea, the ruler of Lagash c. 2100 BC. Measures, Rules, and Prices has discussed their inscriptions and measuring rules from two of his seated statues. With regard to the numerous standing statues, the cosmological content of archaic art is nowhere better exemplified. And the changing descriptions of these statues in modern times reflect the evolution of modern views on archaic art and creativity.
Writing more than a century ago, the archaeologist Hermann Hilprecht[95] (1903: pp. 236f.) found that these statues were “remarkable for their unity of style and technique… through the simplicity and correctness of their attitude and through the reality and power of their expression. … the Chaldean artist endeavors to express real life and to imitate nature within certain limits set by the peculiar material, ‘the routine of the studios and the rules of sacred etiquette.’ The swelling of the muscles of the right arm, the delicately carved nails of the fingers, the expressive details of the feet firmly resting on the ground… betray a remarkable gift of observation…”
This view exemplified the danger of jumping to the conclusion that just because an ancient artifact appears realistic or decorative, it has no particular cognitive content. For more recently, Azarpay[96] (1990) has examined these statues and found them to be based on a canon of sexagesimal proportions. “The composite shows the overall height of the figure expressed in six multiples of the length of the forearm (here measured from elbow to wrist). So each statue is six ‘cubits’ tall, with the length of the cubit specific to the statue.” Azarpay further found that dividing the human body into six equal parts corresponded to its six natural bends: the neck, elbow, waist, groin, knee, and ankles. Gudea’s statues accordingly were divided into six segments: “l) Head: crown to chin; 2) upper torso: chin to elbow, vertical distance; 3) lower torso: elbow to hips…; 4) upper legs: hips to knees; 5) lower legs: knees to hemline; and 6) ankles to base: hemline to baseline.” These proportions showed that what at first seemed to be an individual character of Gudea (and by extension, other rulers) actually was regressed to formal geometric standards.
The art historian Irene Winter (1989) has elaborated some of the nonnumerical symbolism reflected in Gudea’s 20 surviving statues. There are so many of them, and they resemble each other so closely, as to suggest at first glance that they were true likenesses, perhaps reflecting Gudea’s egoism. However, Winter cited five iconographic qualities indicating that these statues were unlikely to be actual portraits. They embodied stereotypical royal qualities in a seemingly realistic guise, just as their sexagesimal proportions appeared naturalistic at first glance.
One of the most striking characteristics of Gudea’s statues was their large-muscled arms. (Hilprecht[97] [1903: p. 385] described similar well-muscled contemporary statues from Ur.) However, rather than being a realistic portrait of Gudea, this characteristic reflected the fact that “strong” was a typical epithet for rulers (“strongmen”). Gudea’s Statue B (iii:1213) stated that his lord Ningirsu gave him heroic strength. Another royal epithet was “outstandingness,” e.g., in a crowd, from whom deities traditionally selected Sumerian rulers. Gudea’s statues depicted him as being larger than life and broad-chested, reflecting the Sumerian word for “ruler,” “lugal,” “big-man.”
Gudea’s statues also had large eyes, apparently to signify his close attentiveness to the city-god Ningirsu. (To be all-seeing was an epithet for the sun-god Anu, while the moon-god Nannu saw at night.) Similarly outstanding were Gudea’s wide ears, signifying that he was “listening hard” to Ningirsu’s advice (Cyl. A, i:12).
Although individual personality was first attested historically among rulers and the best-placed families, there was strong pressure for rulers to conform to archetypes. On the artistic plane, rulers were themselves ruled by these conventions.
Stated another way, the Bronze Age appeared not to have been a ripe epoch to exercise one’s individualism, regardless of social status. Whereas many modern European chieftains and rulers gave their own physical measurements—notably their foot and cubit—to their local communities as Measures, Rules, and Prices noted, just the reverse was the case in archaic times. Cosmologized proportions determined the way in which rulers were represented in sculpture and paintings.
This cosmological regression followed in large part from the ritualistic context for such art. Rulers were supposed to be perfectly formed men, having large eyes, ears, and bodies, whose physical dimensions followed a strict canon of proportions reflecting their culture’s numerical system. The genius of Bronze Age art consists precisely in the seeming realism of these cosmographic representations.
Leaders and Music
Archaic musicians and artists learned calendrical and numerological proportions, while the compilers of the Sumerian King List and biblical Patriarch List wrote what may be called “ceremonial history” placing rulers and patriarchs into an ordered temporal continuum.
Such modular proportioning gave history a seemingly objective cosmological dimension, the Bronze Age version of the Emperor of Ten Thousand Years or the Thousand-Year Reich.
Social Setting for Art
Aristotle (Politics VIII.3 at 1337b) stated that high culture required leisure time for contemplation, and hence wealth (much as to be a member of the aristocratic cavalry, one needed enough wealth to spend one’s time training with one’s horse). But there also was a danger here: The ruling class might become conservative.
Hence, the rationale for associating art and music (harmony) in this chapter.
What once was a cosmological realism was left with only a surface realism—appearances without traces of the old cosmology of proportions or other ritual cosmology which underlay more archaic art going back to the Ice Age.
Plato’s Laws (700–701a) reflected a disparagement of modern tendencies in music that form a model for all subsequent conservatism. In archaic times, he pointed out, each musical form had its particular place. “Knowledge and informed judgment penalized disobedience. There were no whistles, unmusical mob-noises, or clapping for applause. … But later, an unmusical anarchy was led by poets who had natural talent, but were ignorant of the laws of music. Overintoxicated with love of pleasure, they mixed their drinks—dirges with hymns, paeans with dithyrambs—and imitated aulos-music in their kitharoedic song. Through foolishness they deceived themselves into thinking that there was no right or wrong way in music—that it was to be judged good or bad by the pleasure it gave. By their works and their theories they infected the masses with the presumption to think themselves adequate judges. So our theaters, once silent, grew vocal, and aristocracy of music gave way to a pernicious theatrocracy.” What a wonderful word, “theatrocracy.”
Aristoxenus (according to Athenaeus[98] XIV.632) decried the decadence of his times (already over two thousand years ago): “Now that our theatres have become utterly barbarized and this prostituted music has moved on into a state of grave corruption,” he accused; few people recalled the old musical arts. Athenaeus added his own view that “It happened that in ancient times the Greeks were music-lovers; but later, with the breakdown of order, when practically all the ancient customs fell into decay, this devotion to principle ceased, and debased fashions in music came to light, wherein every one who practised them substituted effeminacy for gentleness, and license and looseness for moderation. What is more, this fashion will doubtless be carried further if some one does not bring the music of our forebears once more to open practice.”[99]
Capsule History of Tuning Through Bach and the Moderns
According to McClain[100] (1976: p. 112), in the 16th century AD “the new ‘chordal’ harmonies… made the ‘human number 5’ (in the triad ratios of 4:5:6) a factor to be reckoned with.” The result was called “Just tuning.” “[I]ts plague of commas” made music a cacophony, until Bach helped right matters with his Well-Tempered Clavier in 1722. (It would take another century for truly even temperament to establish itself throughout Europe and its overseas culture area.)
Note: The Chinese had used the number five to come up with pentatonic music. Thus, there were many ways of applying any given cosmology to history, calendar-making, etc.
The Dissolution of Dance and Music (Etymologically)
James Miller[101] (1986: p. 17) explained his disappointment at discovering that “choreia,” via Latin “chorea,” survives in modern English (according to the Oxford English Dictionary) “as a medical term signifying ‘a convulsive disorder… characterized by irregular involuntary contractions of the muscles, especially of the face and arms,’” much as “arthrum”—the root of “joint,” “harmony,” and “art”—survives as “arthritis.” “The tragic fate of my noble Greek word forced me to question my naïve assumption that the mighty themes of order and harmony could be conveyed through the centuries on a verbal vehicle so frail and easily overturned.”
Also, consider how “aesymnetes” changed meaning from “master of the dance” to “ruler.”
Music, Art, Archaeology, Math, and the Calendar
In reconstructing the principles which guided Bronze Age artists, all we really have to go on are the physical traces of artworks. In the first rank are statues, wall paintings or stone friezes, and pottery. The latter evolved from bowls (often decorated with cruciform patterns, perhaps to represent the calendrical crossing points at which the bowls were used), to large Greek kraters, bowls to mix wine and water, with ceremonial or other festive subjects showing musicians—some musical instruments.
Archaeologists have even recovered some musical instruments, perhaps most famously the great harps from the tombs of Ur and other Bronze Age sites. But alas, merely finding a harp does not tell us either how it was played or how it was tuned. We must reconstruct this, both from what we know of the tuning systems as described in contemporary texts, and by knowing what was within the capacity of Bronze Age acoustic-masters.
Musical intervals are tuned by taking square roots. We know that Babylonian mathematics had this. In fact, music may have been a major application—perhaps the major application—of square roots, and of geometric progressions (such as the circle of fifths used in generating the musical scale).
Just how far the musical cosmology of antiquity went is still open to conjecture. What has made it so debatable is the remarkable fact—a fact of nature—that the calendar has many parallels with musical temperament.
The question is, how early in history was this fact discovered? For many generations, scholars have read archaic creation myths as astro-mythology. More recently, other scholars have begun to read these as musico-mythology. When we have 12 gods, and a new one comes to take the place of an older one, is this perhaps a new “note” coming into the musical scale (the circle of fifths)? Or is this reconstruction an anachronistic modernism?
Some music-mythology historians have gone so far as to imagine that archaic myths reflect an understanding of the precession of the equinoxes, and also of highly sophisticated tunings by “commas,” or microtonal pitch corrections necessary to put the scale in order. In modern times these tonal corrections involve tempering each tone by the 12th square root of 2, that is, 12√2. How early was this “even tempering” known? Could it have been so important as to play a role in archaic mythology?
While this question cannot be answered, what is indeed known is that various forms of tempering were necessary for complex orchestral music to be performed without dissonance, and that various principles of tempering were discovered in classical antiquity. This tempering has been associated in the popular mind above all with the name of Pythagoras, although historians have ascertained that it actually was created by his followers, especially Archytas of Tarentum, who wrote early in the fourth century BC.
Discussing the complexities of musical tuning and the calendar would merely be indulging in a quasi-occult body of striking coincidences. It would be a Kabbalistic exercise. Why would we be interested in retracing such Kabbalism, except for antiquarian ends or in search of some forgotten secret wisdom?
The answer is not merely that music was indeed made into the mathematized archaic kosmos alongside calendrical proportions. Throughout The Creation of Order I have defined the kosmos as being one of society as well as of objective nature.
Throughout the Bronze Age, the kosmos was not a debatable issue. There was room for many different types of kosmos-building, but they were a matter more of individuality than of social and economic policy. It is toward the end of the Bronze Age, c. 1400 BC, that we find ‘apiru leaders descending on Canaan and promising local populations that they would cancel the debts and redistribute the lands—“eat the rich”—if they defected to the side of the ‘apiru mountaineers.
Populations were divided along political lines only in the epoch of democratic-oligarchic tensions from the late seventh century BC onward in classical Greece and Italy[102] as popular tyrants unseated and exiled the old landed aristocracies in many Greek cities, a revolutionary fervor only staved off by major reforms such as the Lycurgan legislation in Sparta and that of Solon in Athens c. 594 BC.[103]
At first sight, these political divisions would not seem to have much to do with the arts. This would seem to be the case above all as the arts grew most abstract, culminating in music. For the visual arts, we might expect to see rulers and other political leaders depicted with the iconography of the gods, in the kind of iconography found, say, in the French Revolution.
Yet to the ancients the most highly politicized art was music. If music was the head of Plato’s quadrivium taught at his Academy at Athens, it was above all because its mathematics contained the principles of an economic and social metaphor well understood by archaic oligarchs and their democratic adversaries.
If music was the center of the academic curriculum, it also was the nominal center around which oligarchic “Pythagorean” clubs were formed in the gymnasia, the city centers around which Greek aristocrats concentrated.[104] What gave music this political role was the metaphor of harmonic (“geometric”) proportion by which inequities of wealth and status were supposed to make men truly equal in proportion to their virtue (which meant, in typical circular reasoning of equilibrium economics ever since, simply wealth). As W.A. Oldfather[105] (1938) put it: “Pythagoras was the founder of the very theory of aristocracy and a chief mover in the oligarchic reaction against democracy. … The spiritual affinity between these principles and those of certain modern totalitarian states would seem to be fairly obvious.” Here we find the ideas of “rule” and “order” taking an authoritarian coloring.
Pythagoreanism and Ancient Greek Politics and Economics
The classic study of Pythagorean oligarchic principles is Edwin L. Minar Jr.’s 1942 monograph on Early Pythagorean Politics in Practice and Theory.[106] He traced how the theory of harmonic ratios gave a rationale to aristocracies threatened by democratic reformers.
By “Pythagoreans” is meant the late followers of Pythagoras, above all those of Tarentum early in the fourth century BC. It always is of the Pythagorean school—not the man himself—that Aristotle and other writers speak.
Burkert[107] (1972) has shown how Pythagoras was one of the vehicles (along with Thales) from Asia Minor (the Ionian home of most early Greek philosophy, on the western shore of what is now Turkey) who conveyed Babylonian learning to Greece. He therefore stood in a long line of transmitters of Near Eastern knowledge to the Aegean and Italy—a process begun in the 14th century BC when the Mycenaean Greeks concentrated their trade on the entrepôt of Ugarit, and continued in the 10th and ninth century BC when Phoenicians and Syrians took the lead in reviving trade throughout the Aegean and bringing it to Italy on the island of Pithekoussai (modern Ischia), off the coast of the bay of Naples at the mouth of the Tiber.
Many (most?) of the biographic details of Pythagoras as written in classical times were made up for political/ideological purposes. He is said to have been born around 569 BC,[108] and to have fled Samos when the tyrant Polycrates took over in 540 BC. But as Burkert[109] (1972) pointed out, this latter detail may have been invented to explain away the embarrassing association of later Pythagoreans with tyrants, from Croton to Syracuse and the Athenian oligarchy which ruled brutally. (Most of the Seven Sages were tyrants. Such was the mentality of the archaic Greeks.)
Burkert plausibly reconstructed Pythagoras as a shaman, preoccupied with ritual and the otherworld, and with the doctrine of reincarnation—the transmigration of the soul (metempsychosis).[110] As such, the Pythagorean tradition would seem to have originated along much the same lines as Orphism. Both were concerned with travels to the celestial underworld, recalling the journey of Inanna, Ereshkigal, and Dumuzid, Grecianized into Demeter and Persephone, and also into Dionysian rituals. However, where the democratic Athenian tyrant Peisistratus sponsored Orphism, and its allied Dionysian mystic cults were worshipped primarily by the poor and by the lower orders outside of the aristocratic lineages (Minar[111] 1942: p. 127), Pythagoreanism was elaborated and refined over the centuries mainly by upper-class oligarchs.
Whereas Polycrates was reported by Herodotus[112] (3.39ff.) to have exiled the leading aristocrats and supported industry, the handicrafts, and the navy—and hence economic opportunities for the lower orders—the Pythagoreans found their major sponsors to be the ruling class. This was above all the case in the Greek colony of Croton, where Pythagoras was reputed to have settled. (He was also supposed to have been active in Metapontum. And Burkert[113] 1972: p. 113 noted that these two south Italian cities above all others were noted for their worship of Apollo, apparently a transplant from Asia Minor.) Tarentum was the other center of Pythagoreanism, and it was there that the sect survived longest.
One of the common themes of the period is how philosopher kings tended to act like tyrants whenever they managed to gain power. The Athenian dictatorship of the 400 at the end of the fifth century BC is a case in point. But one of the first counterrevolutions was that of Croton.
Croton went to war with Sybaris. On this topic, Athenaeus[114] (XII.3–41) repeated an anecdote that shows the power of music, so much praised by the Pythagoreans.
I have earlier described the great extent to which ancient history was composed on the basis of models. Burkert[115] (1972: p. 116) suspected that this was probably the case with Pythagoras in Croton, with the plot being borrowed from contemporary plays on The Suppliants by Aeschylus and Euripides. With the development of commercial wealth, Sybaris experienced the same kind of democratizing revolution that had swept the Aegean, and exiled the leading families who had monopolized the lands and established interest-bearing debt claims on much of the population. The exiles sought refuge in Croton—and then, curiously, the tyrant Telys of Sybaris demanded their return. “The exiles have taken refuge at the altars in the marketplace of Croton, emissaries of Telys demand their surrender and threaten war, the popular assembly is in doubt what to do until Pythagoras addresses them and reminds them of the obligation to protect suppliants, and the war and the famous victory follow.”[116]
Burkert observed that “The destruction of Sybaris was the worst atrocity wrought by Greeks against a Greek city in that era; the attempt to make the unheard-of comprehensible was bound to give rise to legends, and the contrast of ‘Sybaritic’ luxury with Pythagorean sobriety was a strong stimulus to the creation of moralistic and edifying fiction.”[117] The reduction of history to educational—and propagandistic—ends was underway.
Minar[118] (1942: p. 10) described how, after the Crotonite aristocracy defeated Sybaris in 510 BC, they divided the lands among themselves, “and when the Pythagoreans were overthrown [in 450 BC] it was no accident that abolition of debts and redivision of the land were first on the rebels’ agenda.” Burkert[119] (1972: p. 115), surveying the sources, stated that “the house of Milo, which was the meeting place of the Pythagoreans in Croton, was burnt down by their opponents, and only a few of those present escaped.”
Theopompus and Posidonius wrote that Pythagorean oligarchs such as the tyrant Athenion of Athens “at the first opportunity cast aside the mask of philosophy and became a tyrant.”[120] Appian, describing the wars of the great anti-Roman leader Mithridates, said that the south Italian Pythagoreans, “and in other parts of the Grecian world some of those known as the Seven Wise Men, who undertook to manage public affairs, governed more cruelly, and made themselves greater tyrants than ordinary despots.” And Diogenes Laërtius depicted the anti-Pythagorean revolt “as a blow for freedom from tyranny” (Burkert[121] 1972: pp. 118f.).
Pythagoreans and Music
What Pythagoras gave the ruling class was a propagandistic philosophy—a kind of numerological patter-talk—to justify their favored economic position. No ruling class ever before had needed this kind of rationale, for the simple reason that ideological divisions between oligarchs and democrats had not yet emerged. But in the fifth century BC and henceforth they did so throughout the classical Mediterranean world.
They made musical harmonics their rallying cry. After all, if one is going to speak of harmony in society—of “cooperation”—this is natural. And the fact is that the invention of the very word “kosmos” was attributed to Pythagoras.
How they built up their position as ideological and propagandistic advisers is instructive. Minar[122] (1942: pp. 23ff.) described how the Pythagoreans functioned much like political cult organizations (Greek “hetaireia”), a word which also meant conspiracy largely because of the Pythagorean use of such civic cults. They were secret organizations, with their own common meals and other cult practices. An allied type of association, the thiasos, was a mystical and secret cult “which quite soon falls into political activity and functions quite efficiently as the ruler of a large domain.” During the rule of the 400 in Athens (and again under the even more dictatorial 30 tyrants), these clubs joined together to seize power.
Most of these cultish clubs had about 20 members, but the Croton branch was as large as 300. They operated in ways which, in modern times, are more usually associated with underground left-wing politics. Minar[123] (1942: p. 27) wrote (alluding to Calhoun[124] [1913: pp. 37f.]) that “when the Pythagoreans were extending the rule of Croton over the neighboring cities, a common method was to organize a Pythagorean Society in each city, which then aided them in seizing power.” Their political program was to restore the so-called “ancestral constitution” (patrios politeia) and restore aristocratic rule as “in ancient times.”
According to Minar[125] (1942: pp. 42f.), Pausanias[126] (6.13.1) indicated that the Crotonite Pythagoreans were early supporters of the Sicilian aristocrat Gelon, who seized power over Gela in 491 BC and then over Syracuse, turning power over to their hereditary landed nobility, the Gamoroi. (Gelon was followed by Hiero, after whose death in 467 BC aristocratic tyrants were overthrown throughout the region, and democratic regimes set up. See Diodorus[127] 11.67–11.72.) These quasireligious cults formed the closed brotherhoods from which autocratic political leaders were chosen.
In these brotherhoods we find one of the earliest examples of political sectarianism. One report (cited in Iamblichus’s Life of Pythagoras, conveniently collected along with related documents in Guthrie[128] 1987: p. 119) was a verse, lightly paraphrased as: “Like the blessed gods, he revered his friends, but reckoned others as of no account.”
This kind of conspiratorial oligarchism resulted in the Pythagoreans being expelled from Magna Graecia. Croton’s aristocracy was overthrown in 454 BC and the surviving Pythagoreans expelled, along with their children. And even the hagiographer Iamblichus wrote (Guthrie[129] 1987: p. 120) that the Crotonians expelled the Pythagoreans, “banishing along with them all their families, on the two-fold pretext that impiety was unbearable and that the children should not be separated from their parents. They then repudiated the debts, and redistributed the lands.”
In sum, the picture of Pythagorean secret societies as drawn by Minar[130] (1942: p. 33) was not of scientific or even “mathematical” societies, much less innocuous musical groups. “[T]here is very little even in the most abstruse philosophical doctrine… [that] can be called scientific. The Pythagoreans made some genuine contributions in mathematics and perhaps in music, but it may fairly be said that one of their chief concerns was to discourage anything like experimentation. Their theorizing was directed very largely toward the crystallization of the same attitudes as are inculcated in the acusmata and the popular teachings. … The most important element of Pythagorean morality is armonia, which means in practice that subjects must be ‘persuaded’ and accept willingly the dominance of their natural superiors.” But only a few well-born individuals could enter the small company of the “elect” who determined public policy under Pythagorean-oligarchic regimes.
How much of this should be blamed on Pythagoras? Actually, it seems that he had much less to do with the principles of musical harmony—or Greek mathematics—than was long believed.
We are talking here not only about the origins of sectarian political cults and their conspiratorial approach (and successes) but also about western civilization’s earliest and most vaunted educational institution, Plato’s Academy and its curriculum based on “music.”
Suppose you were a democrat, and opposed the aristocratic program. You would be told that you had to go to school to “understand” the oligarchic mathematics of “harmonic proportions,” that is, the code phrase for “unequal equality.”
Plato’s Laws would ban certain kinds of music. Here too we find the censorial attitude toward culture associated with authoritarian cults through the ages, right down to the attempts to ban jazz (with its “blue notes,” that is, flat sevenths) and associated music of the people deemed unworthy of higher culture.
So: the oligarchs controlled education, and the kosmos. In Rome they succeeded in gimmicking the constitution. (Pythagoras is claimed to have helped Numa. This is cosmo-history. Numa was the “spiritual” element, Romulus the worldly. So Pythagoras—or more accurately, one of his followers—was assigned to Numa. But the real influence came with Rome’s sixth king, Servius. His constitution gave the preponderant balance to the wealthiest orders, who voted first. The votes of nearly 90 percent of the population were weighed so low as to barely count, except in a near-tie [such as never occurred during the entire history of the republic].)
They controlled the cavalry. And most of all, they were devious. They were willing to play dirty; the democrats hesitated to.
The harmonics of equality had about as much to do with fairness as gerrymandering has to do with democratic geographic representation.
So we have (1) religion as oligarchic, (2) cults as oligarchic, (3) leisure as oligarchic, and (4) sports as oligarchic (the gymnasia), hence “public figures” such as sports heroes (the modern equivalent would be movie actors). Viz. the figure of Alcibiades.
This begins to illustrate the antinationalist, cosmopolitan treachery of wealth. The wealthy go wherever their fortunes take them. They cultivate well-placed friends in other cities, in case they are exiled (a frequent occurrence from the seventh to sixth centuries BC as populations expelled their aristocracies, redivided the lands, and canceled the debts).
In this system favorable to the wealthy, there also needs to be (5) the educational system as oligarchic, (6) the electoral process weighted, and of course (7) the courts—i.e., winning by tricks. (The ultimate cheating is legitimizing the theft. This is done by propaganda.)
The oligarchy would also benefit from a distraction of those they lead—such as the Pythagorean Platonic musical theory of harmonies.
For propagandistic history, an oligarch would assign personal motives to their enemies (and imply that the oligarch’s own motives are of course only objective and pure). Thus the opponents of Croton’s Pythagorean oligarchs were accused by Iamblichus of being resentful at having been rejected as students of Pythagoras, corrupted by whatever success they may have won.
Key Concepts
This glossary of key concepts will help readers who are new to the subject of archaic human history.
Keywords: “Arthron,” “joint” (art, harmony), “canon,” “temperament,” and “chorus” (viz. “choreia” and “choregos”).
Key image: Calliope, the ninth muse, integrating the other eight muses by virtue of the aid to memory provided by music.
Lunar symbols: The untempered natural scale, the seven-stringed lyre, and the Sumerian temple harp “roaring like a bull.”
Solar symbol: The tempered scale and Apollo’s lyre.
Principle of regularity: In the visual arts, proportions were standardized in a cosmological way, and public figures were given traditional monumental qualities.
In music, tempering made similar intervals equal rather than sharp or flat as in the untempered “natural” scale.
Principle of periodicity: Each octave renewed itself from A to A’, making the number eight (for A’) the signifier of rebirth.
Public Renewal Ceremony: Temples were dedicated at New Year rituals by burying cosmo-historical proclamations in their foundations. Musical, lyric, and dramatic competitions accompanied early sports contests such as the Olympics and Delphi’s Pythian Games.
Integration with other dimensions of the archaic kosmos: Painting, sculpture, and historical narrative regressed individual characteristics to traditional “round numbers.” And like the calendar, the musical scale had to be tempered to produce equal intervals. This provided a metaphor for dealing with one’s appetites so as to live in society in cooperation with one’s fellows.
Public character: Poetry, epics, and religious prayers were sung and danced, above all on major public occasions. A herald sounded his trumpet to convene popular assemblies. Military forces were rallied by drums and trumpets beating out rhythms for maneuvers and sounding a code to indicate tactics.
Religious sanctification: Musical proportions governed the Creation and subsequent human history as rulers periodically restored harmony and quieted the “noise” of popular discontent.
Ultimate dissolution: The Pythagoreans made music into an oligarchic cult discipline by using the analogy of “geometric” as opposed to “arithmetic” harmony and justice, rationalizing general social inequality.
The “choregos” shifted from being “master of dance” to its wealthy patron, while “choreia” became a convulsive disease. “Calliope” descended from being the highest muse to the raucous merry-go-round organ.
Bibliography
Aristotle, Politics, Book 8 (the Internet Classics Archive, MIT Classics and Tufts University’s Perseus Digital Library Project).
Athenaeus, The Deipnosophists; or, Banquet of the Learned, C.D. Yonge (tr.), Vol. 3 (London: 1854), via the Internet Archive.
Athenaeus, The Deipnosophists: Or Banquet of the Learned of Athenaeus, C.D. Yonge (tr.), Vol. 3 (London: 1854), via Tufts University’s Perseus Digital Library Project.
Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950).
Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), pp. 93–103.
Guitty Azarpay, W.G. Lambert, W. Heimpel, and Anne Draffkom Kilmer, “Proportional Guidelines in Ancient Near Eastern Art,” Journal of Near Eastern Studies, Vol. 46, No. 3 (July 1987), pp. 183–213.
Donald Freeman Brown, “In Search of Sybaris,” Expedition Magazine, Vol. 5, No. 2 (January 1963).
Norman O. Brown, Hermes the Thief: The Evolution of a Myth (Great Barrington, Massachusetts: 1990 [1947]).
Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972).
George Miller Calhoun, Athenian Clubs in Politics and Litigation (Austin: 1913).
Francis Macdonald Cornford (tr.), The Republic of Plato (Oxford: 1941).
The Book of Daniel, New International Version.
Diodorus Siculus, Diodorus of Sicily in Twelve Volumes, C.H. Oldfather (tr.), Vol. 4–8 (Cambridge, Massachusetts: 1989), via Tufts University’s Perseus Digital Library Project.
Euripides, Bacchae, from The Tragedies of Euripides, T.A. Buckley (tr.), Bacchae (London: 1850).
Benjamin Farrington, Science and Politics in the Ancient World (London: 1939 [1946]).
Alexander Heidel, The Babylonian Genesis (Chicago: 1942 [1951]).
Isobel Henderson, “Ancient Greek Music,” in Egon Wellesz (ed.), Ancient and Oriental Music, (London: 1957), p. 341.
Herodotus, The History of Herodotus, G.C. Macaulay (tr.), Vol. 1. (New York: 1890), via Project Gutenberg.
Hermann Hilprecht, Explorations in Bible Lands During the 19th Century (Philadelphia, 1903).
Homer, The Iliad, A.T. Murray (tr.) (Cambridge, Massachusetts: 1924), via Tufts University’s Perseus Digital Library Project.
Homer, The Odyssey, A.T. Murray (tr.) (Cambridge, Massachusetts: 1919), via Tufts University’s Perseus Digital Library Project.
Charles Burton Gulick (tr.), Athenaeus: The Deipnosophists: In Seven Volumes, Vol. VI (London: 1937).
Kenneth Sylvan Guthrie (ed.), The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy (Grand Rapids: 1987).
Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976).
Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978).
James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986).
Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942).
W.A. Oldfather, “Pythagoras on Individual Differences and the Authoritarian Principle,” Classical Journal, Vol. 33 (1938), pp. 537–539.
Pausanias, Description of Greece, W.H.S. Jones (tr.) and H.A. Ormerod (tr.), in 4 Vols. (Cambridge, Massachusetts: 1918), via Tufts University’s Perseus Digital Library Project.
Philo of Alexandria, “On the Change of Names,” in Philo: In Ten Volumes (And Two Supplementary Volumes), F.H. Colson (tr.), Vol. 5 (Cambridge, Massachusetts: 1929).
Philo of Alexandria, “On the Creation,” in Philo: In Ten Volumes (And Two Supplementary Volumes), F.H. Colson (tr.), Vol. 1 (Cambridge, Massachusetts: 1929).
Philo of Alexandria, “On the Virtues,” in Philo: In Ten Volumes (And Two Supplementary Volumes), F.H. Colson (tr.), Vol. 8 (Cambridge, Massachusetts: 1929).
Plato, Epinomis. From Plato in Twelve Volumes, W.R.M. Lamb (tr.), Vol. 9 (Cambridge, Massachusetts: 1925), via Tufts University’s Perseus Digital Library Project.
Plato, Laws. From Plato in Twelve Volumes, R.G. Bury (tr.), Vols. 10 and 11 (Cambridge, Massachusetts: 1967 and 1968), via Tufts University’s Perseus Digital Library Project.
Plato, Republic, via Tufts University’s Perseus Digital Library Project.
Plato, Timaeus. From Plato in Twelve Volumes, W.R.M. Lamb (tr.), Vol. 9 (Cambridge, Massachusetts: 1925), via Tufts University’s Perseus Digital Library Project.
K.R. Popper, The Open Society and Its Enemies: The Spell of Plato (London: 1945).
Curt Sachs, The Rise of Music in the Ancient World: East and West (New York: 1943).
Marius Schneider, “Primitive Music,” pp. 1–82, in Egon Wellesz (ed.), Ancient and Oriental Music, (London: 1957).
Anthony Snodgrass, Archaic Greece: The Age of Experiment (London: 1980), p. 177.
- ↑ Curt Sachs, The Rise of Music in the Ancient World: East and West (New York: 1943), p. 31.
- ↑ St. Augustine, De doctrina christiana, II.xvi.25.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. 98.
- ↑ Aristotle, Politics, Book 8.
- ↑ Aristotle, Politics, Book 8.
- ↑ Aristotle, Politics, Book 8.
- ↑ Euripides, Bacchae, line 381, from The Tragedies of Euripides, translated by T.A. Buckley, Bacchae (London: 1850).
- ↑ Aristotle, Politics, Book 8, section 1339b.
- ↑ Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950), p. 53.
- ↑ Homer, The Iliad, A.T. Murray (tr.) (Cambridge, Massachusetts: 1924), via Tufts University’s Perseus Digital Library Project, Book I, lines 493–594.
- ↑ Homer, The Iliad, A.T. Murray (tr.) (Cambridge, Massachusetts: 1924), via Tufts University’s Perseus Digital Library Project, Book I, lines 603–604.
- ↑ Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950), pp. 52–53.
- ↑ Homer, The Odyssey, A.T. Murray (tr.) (Cambridge, Massachusetts: 1919), via Tufts University’s Perseus Digital Library Project, Book 8, line 99.
- ↑ Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950), p. 52.
- ↑ The Book of Daniel, New International Version, Chapter 3.
- ↑ The Book of Daniel, New International Version, Chapter 3.
- ↑ 17.0 17.1 The blind musician is a familiar character from the Bronze Age through classical antiquity. See Samuel Noah Kramer and John Maier’s Myths of Enki, The Crafty God, for Sumerian practice. This spread to classical Greece.
- ↑ Norman O. Brown, Hermes the Thief: The Evolution of a Myth (Great Barrington, Massachusetts: 1990 [1947]), pp. 29–30.
- ↑ Norman O. Brown, Hermes the Thief: The Evolution of a Myth (Great Barrington, Massachusetts: 1990 [1947]), p. 29.
- ↑ Marius Schneider, “Primitive Music,” pp. 1–82, in Egon Wellesz (ed.), Ancient and Oriental Music, (London: 1957), p. 40.
- ↑ Athenaeus, The Deipnosophists; or, Banquet of the Learned, C.D. Yonge (tr.), Vol. 3 (London: 1854), via the Internet Archive, p. 1,001.
- ↑ Joseph T. Shipley, The Origins of English Words: A Discursive Dictionary of Indo-European Roots (Baltimore: 1984), pp. 16ff.
- ↑ Marius Schneider, “Primitive Music,” pp. 1–82, in Egon Wellesz (ed.), Ancient and Oriental Music, (London: 1957), pp. 38f., 4.
- ↑ For examples of this disparaging of practical music see the Principles of Music written by Boethius at the beginning of the sixth century AD. See also Curt Sachs, The Rise of Music in the Ancient World: East and West (New York: 1943).
- ↑ Curt Sachs, The Rise of Music in the Ancient World: East and West (New York: 1943).
- ↑ Anthony Snodgrass, Archaic Greece: The Age of Experiment (London: 1980), p. 177.
- ↑ Charles Burton Gulick (tr.), Athenaeus: The Deipnosophists: In Seven Volumes, Vol. VI (London: 1937), p. 383.
- ↑ Curt Sachs, The Rise of Music in the Ancient World: East and West (New York: 1943).
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), pp. 19f.
- ↑ Isobel Henderson, “Ancient Greek Music,” in Egon Wellesz (ed.), Ancient and Oriental Music, (London: 1957), p. 341.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976).
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. 7.
- ↑ Marius Schneider, “Primitive Music,” pp. 1–82, in Egon Wellesz (ed.), Ancient and Oriental Music, (London: 1957), p. 45.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 38.
- ↑ Plato, Republic, via Tufts University’s Perseus Digital Library Project, 587b–588a.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 33.
- ↑ Plato, Republic, via Tufts University’s Perseus Digital Library Project, 572b.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. 4.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), glossary.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), glossary.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), glossary.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. 119.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976) and The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978).
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 55.
- ↑ Plato, Republic, via Tufts University’s Perseus Digital Library Project, 587c.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. xi.
- ↑ Francis Macdonald Cornford (tr.), The Republic of Plato (Oxford: 1941).
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 38.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), pp. 102–103.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 3.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. 163.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 4.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 131.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), pp. 140–141.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), p. 19.
- ↑ Ernest G. McClain, The Pythagorean Plato: Prelude to the Song Itself (Maine: 1978), pp. 128–129.
- ↑ Athenaeus, The Deipnosophists: Or Banquet of the Learned of Athenaeus, C.D. Yonge (tr.), Vol. 3 (London: 1854), via Tufts University’s Perseus Digital Library Project, Book XII, Chapter 1.
- ↑ Donald Freeman Brown, “In Search of Sybaris,” Expedition Magazine, Vol. 5, No. 2 (January 1963).
- ↑ Athenaeus, The Deipnosophists: Or Banquet of the Learned of Athenaeus, C.D. Yonge (tr.), Vol. 3 (London: 1854), via Tufts University’s Perseus Digital Library Project, Book XII, Chapter 1.
- ↑ Athenaeus, The Deipnosophists: Or Banquet of the Learned of Athenaeus, C.D. Yonge (tr.), Vol. 3 (London: 1854), via Tufts University’s Perseus Digital Library Project, Book XII, Chapter 1.
- ↑ Benjamin Farrington, Science and Politics in the Ancient World (London: 1939 [1946]), p. 32.
- ↑ Benjamin Farrington, Science and Politics in the Ancient World (London: 1939 [1946]), p. 29.
- ↑ Benjamin Farrington, Science and Politics in the Ancient World (London: 1939 [1946]), p. 30.
- ↑ Benjamin Farrington, Science and Politics in the Ancient World (London: 1939 [1946]), p. 29.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 72.
- ↑ Philo of Alexandria, “On the Virtues,” in Philo: In Ten Volumes (And Two Supplementary Volumes), F.H. Colson (tr.), Vol. 8 (Cambridge, Massachusetts: 1929), pp. 207–209, 72–76.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 73.
- ↑ Philo of Alexandria, “On the Change of Names,” in Philo: In Ten Volumes (And Two Supplementary Volumes), F.H. Colson (tr.), Vol. 5 (Cambridge, Massachusetts: 1929), p. 179, 72–73.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 74.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 74.
- ↑ Philo of Alexandria, “On the Creation,” in Philo: In Ten Volumes (And Two Supplementary Volumes), F.H. Colson (tr.), Vol. 1 (Cambridge, Massachusetts: 1929), pp. 55–57, 69–71.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), pp. 56f.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 57.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 57.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 52.
- ↑ Plato, Laws, Book II, p. 672e. From Plato in Twelve Volumes, R.G. Bury (tr.), Vols. 10 and 11 (Cambridge, Massachusetts: 1967 and 1968), via Tufts University’s Perseus Digital Library Project.
- ↑ Plato, Epinomis, section 982c–e. From Plato in Twelve Volumes, W.R.M. Lamb (tr.), Vol. 9 (Cambridge, Massachusetts: 1925), via Tufts University’s Perseus Digital Library Project.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), pp. 41, 5.
- ↑ K.R. Popper, The Open Society and Its Enemies: The Spell of Plato (London: 1945).
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 5.
- ↑ Plato, Laws, Book II, pp. 656–657. From Plato in Twelve Volumes, R.G. Bury (tr.), Vols. 10 and 11 (Cambridge, Massachusetts: 1967 and 1968), via Tufts University’s Perseus Digital Library Project.
- ↑ Plato, Timaeus, Sections 40a–d. From Plato in Twelve Volumes, W.R.M. Lamb (tr.), Vol. 9 (Cambridge, Massachusetts: 1925), via Tufts University’s Perseus Digital Library Project.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 19.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 14.
- ↑ Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950), p. 55.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 14.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 14.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 14.
- ↑ Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), pp. 93–103.
- ↑ Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), pp. 95–96.
- ↑ Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), p. 94.
- ↑ Guitty Azarpay, W.G. Lambert, W. Heimpel, and Anne Draffkom Kilmer, “Proportional Guidelines in Ancient Near Eastern Art,” Journal of Near Eastern Studies, Vol. 46, No. 3 (Jul., 1987), pp. 183–213.
- ↑ Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), p. 96.
- ↑ Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), p. 101.
- ↑ Hermann Hilprecht, Explorations in Bible Lands During the 19th Century (Philadelphia, 1903), pp. 236f.
- ↑ Guitty Azarpay, “A Canon of Proportions in the Art of the Ancient Near East,” in Investigating Artistic Environments in the Ancient Near East, Ann C. Gunter (ed.) (Madison, Wisconsin: 1990), p. 99.
- ↑ Hermann Hilprecht, Explorations in Bible Lands During the 19th Century (Philadelphia, 1903), p. 385.
- ↑ Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950), p. 54.
- ↑ Athenaeus, “From the Sophists at Dinner,” in “The Greek View of Music” in Oliver Strunk (selected and annotated by), Source Readings in Music History: From Classical Antiquity through the Romantic Era (New York: 1950), p. 55.
- ↑ Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato (Maine: 1976), p. 112.
- ↑ James L. Miller, Measures of Wisdom: The Cosmic Dance in Classical and Christian Antiquity (Toronto: 1986), p. 17.
- ↑ When I refer to Italy I mean not only Rome and neighboring Etruria, but southern Italy’s Greek colonies—Magna Graecia—in Sicily (led by Syracuse under the tyrant Dion and his successors) and the heel of southern Italy, especially Croton, which became the homeland of Pythagorean immigrants late in the sixth century BC, and the neighboring commercial entrepôt of Sybaris which the Crotonites destroyed around 510 BC.
- ↑ George Miller Calhoun, Athenian Clubs in Politics and Litigation (Austin: 1913), p. 66.
- ↑ George Miller Calhoun, Athenian Clubs in Politics and Litigation (Austin: 1913).
- ↑ W.A. Oldfather, “Pythagoras on Individual Differences and the Authoritarian Principle,” Classical Journal, Vol. 33 (1938), pp. 537–539.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942).
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972).
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), p. 134.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972).
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), pp. 66ff.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), p. 127.
- ↑ Herodotus, The History of Herodotus, G.C. Macaulay (tr.), Vol. 1. (New York: 1890), via Project Gutenberg, 3.39ff.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), p. 113.
- ↑ Athenaeus, The Deipnosophists: Or Banquet of the Learned of Athenaeus, C.D. Yonge (tr.), Vol. 3 (London: 1854), via Tufts University’s Perseus Digital Library Project, Book XII, Chapter 1.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), p. 116.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), p. 116.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), pp. 116f.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), p. 10.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), p. 115.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), page unknown.
- ↑ Walter Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, Massachusetts: 1972), pp. 118f.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), pp. 23ff.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), p. 27.
- ↑ George Miller Calhoun, Athenian Clubs in Politics and Litigation (Austin: 1913), pp. 37f.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), pp. 42f.
- ↑ Pausanias, Description of Greece, W.H.S. Jones (tr.) and H.A. Ormerod (tr.), in 4 Vols. (Cambridge, Massachusetts: 1918), via Tufts University’s Perseus Digital Library Project, 6.13.1.
- ↑ Diodorus Siculus, Diodorus of Sicily in Twelve Volumes, C.H. Oldfather (tr.), Vol. 4–8 (Cambridge, Massachusetts: 1989), via Tufts University’s Perseus Digital Library Project, 11.67–11.72.
- ↑ Kenneth Sylvan Guthrie (ed.), The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy (Grand Rapids: 1987), p. 119.
- ↑ Kenneth Sylvan Guthrie (ed.), The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy (Grand Rapids: 1987), p. 120.
- ↑ Edwin L. Minar Jr., Early Pythagorean Politics in Practice and Theory (New York, 1979, and Connecticut, 1942), p. 33.
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