Musical Theory and Ancient Cosmology
First published in The World and I, February 1994 (pp.371-391). ©
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[Précis]
In ancient Mesopotamia, music, mathematics, art, science, religion, and
poetic fantasy were fused. Around 3000 B.C., the Sumerians simultaneously
developed cuneiform writing, in which they recorded their pantheon, and
a base-60 number system. Their gods were assigned numbers that encoded the
primary ratios of music, with the gods' functions corresponding to their
numbers in acoustical theory. Thus the Sumerians created an extensive tonal/arithmetical
model for the cosmos. In this far-reaching allegory, the physical world
is known by analogy, and the gods give divinity not only to natural forces
but also to a "supernatural," intuitive understanding of mathematical
patterns and psychological forces.
The cuneiform mathematical notation, invented by Sumer, was fully exploited
by the virtuoso arithmetical calculations of Babylon, politically ascendant
in the second millennium. The notation employs few symbols, which are distributed
in patterns easily understood by the eye. Thus, few demands are made on
memory. In Mesopotamia, mythology took concrete form; for example, important
activities of the gods can be read as "events" in a multiplication
table notated as a matrix of Sumerian bricks. Classical Greece abstracted
all of the rational tonal concepts embedded in this Sumerian/Babylonian
allegory for two thousand years, simply waiting to be demythologized. Moreover,
because the religious mythologies of India, China, Babylon, Greece, Israel,
and Europe use Sumerian sources and numerology, theology needs to be studied
from a musicological perspective.
[Main Article]
If science is conceived of as knowledge and philosophy as love of wisdom,
then the invention of musical theory clearly is one of the greatest scientific
and philosophical achievements of the ancient world. When, where, and how
did it happen?
Assuming that Cro-Magnon man processed sound with the same biology we possess,
humans have shared some fifty thousand years of similar auditory experiences.
Musical theory as an acoustical science begins with the definition of intervals,
the distance between pitches, by ratios of integers, or counting numbers,
a discovery traditionally credited to Pythagoras in the sixth century B.C.
Not until the sixteenth century A.D., when Vincenzo Galilei (Galileo's father,
an accomplished musician) tried to repeat some of the experiments attributed
to Pythagoras, was it learned that they were apocryphal, giving either the
wrong answers or none at all. Today, as the gift of modem archaeological
and linguistic studies, our awareness of cultures much older than that of
Greece has been phenomenally increased; this permits us to set aside the
tired inventions about Pythagoras and tell a more likely story, involving
anonymous heroes in other lands.
My story is centered in Mesopotamia. It demonstrates how every element of
Pythagorean tuning theory was implicit in the mathematics and mythology
of that land for at least a thousand years, and perhaps two thousand, before
Greek rationalists finally abstracted what we are willing to recognize as
science from its long incubation within mythology.
What seems most astounding in ancient Mesopotamia is the total fusion of
what we separate into subjects: music, mathematics, art, science, religion,
and poetic fantasy. Such a fusion has never been equaled except by Plato,
who inherited its forms. Socrates' statement about the general principles
of scientific studies in book 7 of Plato's Republic, with the harmonical
allegories that follow directly in books 8 and 9, guides my exposition here.
The Mesopotamian prototypes to which they lead us fully justify Socrates'
treatment of his own tale as an "ancient Muses' jest," inherited
from a glorious, lost civilization. Scholars who have become too unmusical
to understand mankind's share in divinity, as Plato feared might happen,
still can lean on him for understanding, for all of his many writings about
harmonics and music have survived. (I must suppress here, for reasons of
space, the extensive harmonical allegories of the Jews, whose parallel forms
infuse the Bible with related musical implication from the first page of
Genesis to the last page of Revelation.)
Music was as important in ancient India, Egypt, and China as it was in Mesopotamia
and Greece. All these cultures had similar mythic imagery emphasizing the
same numbers, which are so important in defining musical intervals; this
raises doubts about whether any people ever "invented" acoustical
theory. For instance, in any culture that knows the harp as intimately as
it was known in Egypt and Mesopotamia, its visible variety of string lengths
and economy of materials (strings require careful and often onerous preparation)
encourage builders, as a sheer survival strategy, to notice the correlation
between a string's length and its intended pitch.
Similarly, in China, where by 5000 B.C. the leg bones of large birds, equipped
with tone holes appropriate for a scale, appear as paired flutes in ritual
burials, the importance of suitable materials conditioned pipemakers to
be alert to lengths. The basic ratios could have been discovered many times
in many places, more likely by loving craftsmen and practitioners than by
philosophers. Certainly, the discovery came no later than the fourth millennium
B.C., before even the first Egyptian dynasty was founded or the Greeks had
reached the Mediterranean shore.
A NEWLY EMERGING PERSPECTIVE.
In the fourth millennium B.C., the Sumerians, a non-Semitic people of
uncertain origin, developed a high civilization in Mesopotamia, now the
southern part of Iraq. For reasons that have been vigorously argued but
remain unclear, they developed a base-60 number system. Waiting to be recognized
within it--and in ways obvious to any scribal adept, although invisible
to the illiterate--were the main patterns of harmonical theory that appear
later in India, Babylon, and Greece. Sumerian tombs of this early period
yield a harvest of harps, lyres, and pipes, and the literature surviving
on clay tablets abounds in elaborate hymns.
In the cuneiform writing of the Sumerians, which was invented concurrently
with the base-60 number system, the pantheon of deities is rationalized
by assigning to the high gods the base-60 numbers that, as we shall see,
encode the primary ratios of music. The glyph, or symbol, for heaven
or star, followed by the appropriate number, functions as a "god
nickname." (See fig. 1. The numerical values of the deities are given
in Budge 1992.) The numbers reveal their significance in triangular arrays
of pebble counters.
Furthermore, in the mythology of their religion, the responsibilities and
behavior of the gods correspond with the functions of the god numbers in
base-60 acoustics. Sumerian cosmology is grounded in the metaphorical copulation
of the male A and female V numerical arrays, from which the Greek "holy
tetraktys" is abstracted.
For example, the head of the pantheon and father of the gods is the sky
god An (the than Anu), god 60, written in cuneiform as an oversize 1 sign
(see fig. 5). Because base-60 numbers enjoy potentially endless place value
meanings as multiples or submultiples of 60 (like the unit, 1, in decimal
arithmetic), An = 60 (written as 1) functions as the center of the whole
field of rational numbers. In mathematical language, An is its geometric
mean, being the mean between any number and its reciprocal.
Anu/An, therefore, is essentially a do-nothing deity, as he was later accused
of being-, a reference point, perfectly suited to represent simultaneously
the middle band of the sky, the center of the number field, and the middle,
reference tone (the Greek mese) in a tuning system. He was fated
to be deposed by more active leaders among his children, as harmonical logic
focused more clearly on structure and sheer virtuosity in computation became
subordinated to deeper mathematical insight.
Theology, from its birth as "rational discourse about the gods"
and in many later cultures influenced by Sumer, is mathematical allegory
with a deeply musical logic. Tuning theory today remains a fossil science
with no change at all in its basic parameters--structured by the gods themselves
in numerical guise--since it premiered in Sumer about 3300 B.C.
To glimpse this new vision requires that we lay aside our algebra, our computers,
and our pride in rational superiority and represent numbers to ourselves
as the ancients did: concretely. We must learn to do musical arithmetic
with a handful of pebbles in a triangular matrix, as the Pythagoreans teach
us, imitating the pattern of bricks in the Sumerian glyph for mountain.
Then, like Socrates, we must show ourselves the harmonical implications
of that arithmetic with a circle in the sand, for that circle is the cosmos,
viewed as endlessly cyclical, like the tones of the musical scale (fig.
2).
In what follows I am presenting Mesopotamian arithmetic as Plato still
practiced it in the fourth century B.C., studying his mathematical allegories
for clues to earlier examples. Plato is the last great harmonical mythographer
of the European world; never again did a major philosopher so thoroughly
ground his thinking in music.
In retrospect, decoding Sumerian-Platonic harmonics proves astonishingly
simple. Anyone, even a child, who can count to ten and sing or play the
scale can make self-evident the scale constructions that once modeled the
cosmos.
Because 60 is integrally divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and
30, base-60 arithmetic can correlate many subsystems, allowing fluent manipulation
of fractions. This very early mastery of fractions ensured adequate arithmetical
definition of pitch ratios--presumably as string-length ratios on early
harps, approximate length ratios on the flutelike panpipes, or tone-hole
ratios on the aulos--no matter how many tones are involved and whether pitch
patterns rise or fall.
About 1800 B.C., the Babylonians became politically ascendant and reorganized
the Sumerian pantheon, keeping its god numbers and related mathematical
terminology. They developed base-60 computation to a level of arithmetical
virtuosity not equaled in Europe until about A.D. 1600 and not understood
in modern times until the middle of our own century (see Neugebauer 1957).
Not until 1945, when Neugebauer and A. Sachs published the translation of
cuneiform tablet YBC 7289 from the Yale collection, did the world learn
that ancient Babylon (1800-1600 B.C.) possessed a base-60 formula for the
square root of 2 accurate to five decimal places (1.41421+), or the formula
for generating all Pythagorean triples (a triangle with sides of 3, 4, and
5 units is merely an example) a thousand years before Pythagoras explicated
the first one.
The Greeks, still thinking in terms of Egyptian unit fractions (so that
a descending whole tone of 8:9, for instance, was constructed by laboriously
adding to the reference length 1/8 of itself), would have been astonished
to learn that the Egyptians, whom they revered, had like themselves been
far surpassed in computational facility by an ancient neighbor.
The paucity of surviving Sumerian mathematical texts requires scholars to
make many inferences from later Babylonian survivals, and much Sumerian
literature remains untranslated or inaccessible. Thus, as further linguistic
evidence becomes available, the story I tell here will require revision,
becoming more certain in dating, clearer in meaning, and richer in detail.
To look ahead in history and see the persistence of Sumerian/Babylonian
methods, Ptolemy, in the second century A.D., in the Harmonica, recorded
all of the some twenty Greek tunings known to him with sexagesimal (base-60)
fractions. Between about 500 B.C. and A.D. 150, Babylonian and Greek astronomy
thrived on base-60 computation. It was still used by Copernicus in the fifteenth
century and endures in modern astronomy The Chinese calendar is still reckoned
by 60s. Astronomy, however, as the science of precise measurement that it
later became, "was practically unknown in ancient Sumer; at least as
of today we have only a list of about twenty-five stars and nothing more"
(Kramer 1963).
HOW BASE-60 SURVIVES IN TIME MEASUREMENT
Analog clocks and watches equipped with rotating hands for hours, minutes,
and seconds are living fossils of the Sumerian arithmetical mind-set (fig.
3).
a. Numbers have visible and tangible markers on the dial (representing the
fixity of the recurring temporal cycle), restricting burdens on memory and
permitting operations to be reduced to counting and adding.
b. Sixty can be conceived of, when we please, as a large unit (one rotation
of the second or minute hand), conversely giving the small unit the implication
of 1/60.
c. The large unit, alternately, can be conceived of as a higher power of
60 (correlating the simultaneous rotations of both second and minute hands),
for 602 = 3,600 seconds is also one hour, conversely giving our small unit
the implication of 1/3,600.
d. Twelve hours constitutes a still larger unit (one rotation of the hour
hand) of 12 x 60 = 720 minutes, and 12 x 3,600 = 43,200 seconds, conversely
giving the smallest unit the implication of 1/720 or 1/43,200.
e. We avoid confusion between these alternate arithmetical meanings the
same way the Sumerians did, namely, by remembering the context of the questions
we are trying to answer.
f. The existence of alternative ways of expressing a unit, as in the examples
above, indicates and emphasizes the importance of reciprocals.
Musicians, following Plato, still project their tones into a circle that
eliminates cyclic octave repetitions (Plato, in the Timaeus, insists
that God makes only one model of anything). Thus today, using our modern,
equal-tempered scale, we can identify any musical interval as some multiple
of a standard semitone, to the envy of calendarmakers, who, having to deal
with the irregularities of days, months, and years, are jealous of our perfect
twelve-tone symmetry. But the nearest approximation of our twelve-tone,
equal-tempered scale in small integers remains that provided by ancient
base-60 arithmetic.
SUMERIAN NUMBERS
Sumerian numbers were impressed on small clay tablets with a stylus,
at first round, later triangular, held slanted for some numbers and vertically
for others (fig. 4). Numbers from 2 to 9 were built up by repetitions of
the unit, made with the edge of the stylus. A 10 was imprinted with the
end; a 60 was made as a large 1 by pressing the stylus more firmly into
the clay. The equation 602 = 3,600 was scratched in as a circle (see van
der Waerden 1963). Only a few symbols were needed, and repetition made them
easy to decode, minimizing burdens on memory. The idea of a number was actually
embodied in the strokes required to notate it (fig. 5).
Computation was made easy by tables of "reciprocals, multiplications,
squares and square roots, cubes and cube roots, ...exponential functions,
coefficients giving numbers for practical computation,...and numerous metrological
calculations giving areas of rectangles, circles" (Kramer 1963). Many
copies of these tables have come down to us.
The standard multiplication tables pair each number with its reciprocal
and give special prominence to the favored subset of "regular"
numbers, whose prime factors are limited to 2, 3, and 5 (larger prime factors
necessarily lead to approximations in the reciprocals). "Regular numbers"
up to 60 are shown (fig. 6) with their reciprocals, transcribed, for instance,
so that the reciprocal of 40/60 = 2/3 reads 1,30, meaning 90/60 = 3/2. Notice
that only the most important fractions of 60 are deified (1/6, 1/5, 1/4,
1/3, 1/2, 2/3, and 5/6). The tone names are nearest equivalents in modern
notation. Several values require three sexagesimal "places" (indicated
by commas); auxiliary tables freely employ six, seven, and even more places.
SUMERIAN SYMMETRY OF OPPOSITES
A telling clue to the psyche--of Sumerians, of Plato, and of ourselves--is
affection for the symmetry of opposites. Inverse, or bilateral, symmetry
conditions base-60 computation, as it conditioned Platonic dialectics. ("Some
things are provocative of thought and some are not...Provocative things...impinge
upon the senses together with their opposites." Republic 524d)
When facing a mirror we exhibit to ourselves, with varying degrees of perfection,
this symmetry of left/right opposites across an imaginary "plane of
reflection."
The old-fashioned scale, or balance beam, epitomizes this notion (fig.
7). The balance owes its functioning to gravity, but its appeal to us, its
attractiveness, is due to our ear, which in addition to being the organ
of hearing is also the personal organ of balance. Our empathic human feelings
for the balance beam affect the inverse, or bilateral, physical symmetry
because of the experience of balancing our own bodies, an activity dependent
on the ear, not the eye. All of the computations presented later will be
aligned in this basic symmetry, with Anu/An = 60 (meaning 1) on the balance
point. Sumerian art greatly elaborates this symmetry of opposites (fig.
8).
THE DEIFICATION OF TONE NUMBERS
The deified Sumerian numbers, taken over by Babylon, are 10, 12, 15, 20,
30, 40, and 50, all fractional parts of "father" Anu/An = 60,
head of the pantheon. Their fractional values and god names are indicated
here with a brief description of their mythological functions.
Anu/An, 60, written as a large 1, "father of the gods" and earliest
head of the pantheon, is any reference unit. He is equivalent in our notation
to 60/60 = 1, where he functions, according to modern concepts, as "geometric
mean in the field of rational numbers."
Enlil, 50 (5/6), "god on the mountain" possessing fifty names,
is mankind's special guardian and was promoted to head the pantheon circa
2500 B.C. Enlil deities in base 60 what the Greeks knew as the human prime
number, 5, in their base-1O harmonics. By generating major thirds of 4:5
and minor thirds of 5:6, he saved Sumerians tremendous arithmetical labor,
as we shall note in due course.
Ea/Enki, 40 (2/3), "god of the sweet waters" and perhaps the busiest
deity in Sumer, "organizes the earth," including the musical scale.
He deities the divine prime number, 3, in the ratio of the musical fifth
2:3, the most powerful shaping force in music after the octave. (Notice
that the trio of highest gods (40, 50, 60) defines the basic musical triad
of 4:5:6 (do, mi, sol, rising, and mi, do, la, falling). The ratio 4:5 defines
a major third and the ratio 5:6 defines a minor third, taken either upward
or downward within the matrix of the musical octave.)
Sin, 30 (1/2), the Moon, establishes the basic Sumerian octave matrix as
1:2 30:60.
Shamash, 20 (1/3), the Sun, judges the gods.
Ishtar, 15 (1/4), is the epitome of the feminine as virgin, wife, and everybody's
mistress.
Nergal, 12 (1/5), is god of the underworld.
Bel/Marduk, 10 (1/6), the biblical Baal, originally was a minor deity but
eventually became head of the Babylonian pantheon in the second millennium
B.C. He inherited all the powers of the other gods, including Enlil's fifty
names, in a giant step toward a "Pythagorized" monotheism built
on the first ten numbers.
GREEK HARMONICAL PRINCIPLES IN SUMERIAN ARITHMETIC
Here are the principal arithmetical symmetries of base-60 Sumerian harmonics,
summarized in the inverse "heraldic" symmetry displayed above
but expressed as modern fractions. Every tone in the scale will be found
to participate in numerous god ratios, and all other ratios are their derivatives
via multiplication (which is what Plato means by "marriage" in
his elaborate metaphor in the Republic). All of the harmonical concepts
in my analysis, however, are Greek. Plato's formula for this particular
construction can be found in the Republic, book 8; his discussion
of general harmonical principles is in the Timaeus.
All pitch classes generated by the prime numbers 2, 3, and 5, up to the
index of 60, are represented here (fig. 9). Remember that all doubles are
equivalent, so that 3, 6, 12, and 24 define the same pitch as 48, for example.
a. Tones are defined by numbers.
b. The significance of a number lies only in its ratio with other numbers.
c. Numerosity is governed by strict arithmetic economy. Because Sumerian
double meanings were assumed, the numbers 30, 32, 36,... are in smallest
integers for this context. This economy is obscured somewhat by writing
ratios as fractions; mentally eliminate the superfluous reference 60s.
d. Every number is employed in two senses, as great and small, displayed
here as reciprocal fractions.
e. The double meanings of great and small require the basic model octave
to be extended across a double octave from 30/60 = 1/2 to 60/30 = 2.
f. Tones are grouped by tetrachords (that is, in groups of fours) whose
fixed boundaries always show the musical proportion 6:8 = 9:12, defining
the octave (6:12 = 1:2), the fifth (2:3, that is, 6:9 and 8:12), and the
fourth (3:4 or 6:8 and 9:12).
Notice how the arithmetic mean 9 and the harmonic mean 8 establish perfect
inverse symmetry (see fig. 10) and define the standard whole tone as 8:9.
These ratios define the only fixed tones in Pythagorean tuning theory, and
they are invariant. Pythagoras reputedly and plausibly brought this proportion
home from Babylon in the sixth century B.C. In base 60, these "framing"
numbers necessarily are multiplied by 5 into 30:40 = 45:60.
Notice that Ea/Enki, god 40, defines these frames (DA falling and G:D
rising) in his double role as 40:60 and 60:40 and thus literally "organizes
the earth" (as represented by the string) into do, fa, sol, do, harmonic
foundations of the modern scale.
g. The Enlil = 50 tones of pitch classes b and f always belong to the opposite
scale, for the god shares these tones with 36 (that is, 30:36 = 50:60 and
30 and 60, "beginning and end," coincide); thus, Enlil is free
to supervise the system by reminding us of the symmetry of opposites.
Enlil's promotion to head the pantheon possibly symbolizes this insight.
He plays a very active role, also generating several intervals that actually
reduce numerosity, whereas the primal procreator, Anu/An = 60, a do-nothing
deity of little account in Sumer and Babylon, remains purely passive.
Platonic dialectics, however, emphasize anew the importance of an invariant
t4 seat in the mean," thus turning Anu/An's passiveness as geometric
mean into the greatest possible Socratic virtue as "the One Itself."
h. The falling or descending version of this scale, as notated [in Figure
9], is in our own familiar major mode. It is more commonly notated one tone
lower, on the white keys of the C octave. The rising scale on the right,
its symmetric opposite, is the basic scale of ancient Greece, India, and
Babylon. It is more simply notated one tone higher, on the white keys in
the E octave.
My choice of D as reference pitch is dictated by the necessity of showing
opposites simultaneously, in the Sumerian normative arithmetical habit that
Plato later required of his students in dialectic. Future philosopher-guardians
in idealized cities needed to become expert in weighing the merits of contradictory
claims, requiring the ability to see opposites simultaneously. Music provided
the opportunity to do this, par excellence, and so childhood training began
with it.
AN OVERVIEW OF CALENDAR AND SCALE
To coalesce the musical opposites shown above into one Sumerian/Platonic
overview, eliminating all octave replication and laying bare the irreducible
structure ("God's only model"), we need only project these tones
into the same tone circle.
From Plato's mythology (in the Critias) come "Poseidon and his
five pairs of twin sons" (see fig. 11), aligned in perfect inverse
Sumerian symmetry across the central vertical plane of reflection. (Poseidon,
at twelve o'clock, Greek successor to the water god Ea/Enki, is self-symmetric,
being both beginning and end of the octave no matter whether we traverse
it upward or downward.) These eleven tones constitute the only pitch class
symmetries up to an index of 60.
But to coalesce opposite fractions so that the numbers--like the tones--show
the same ratios when read in either direction, we must expand the numerical
double 1:2 into 360:720 (see fig. 12). If we confine ourselves to three-digit
numbers, there is, in addition to Poseidon's ten sons, only one other pair
of symmetric numbers, namely, 405 and 640 (since 405:720 = 360:640). These
are notated here as C and E to indicate their very slight and melodically
insignificant difference from c and e. This microtonal "comma"
difference of 80:81, barely perceptible in the laboratory and then only
by a good ear, was taken by the Greeks as the smallest theoretically useful
unit of pitch measure and is approximately 1/9 of their standard whole tone
of 8:9. The whole-tone interval between A and G (in figs. 11 and 12) invites
similar subdivision, and symmetry requires a point directly opposite our
reference, D. This locus is defined by the square root of 2, lying beyond
the ancient concept of number, and so we must search for an approximation.
A musically acceptable candidate (its error is actually less than a comma)
now appears at a-flat = 512, or, alternately, g-sharp = 512, only slightly
askew our ideal value and with the "god ratio" of 4:5 with C or
E.
Plato's Poseidon and his ten sons are shown again (in fig. 12), together
with the new symmetry pair C/E and the alternate a-flat/g-sharp pair (one
of which is always missing in the 360:720 octave). My vertical pendulum
now swings gently back and forth to either side of six o'clock as the numbers
are read alternately in rising and falling scale order (that is, as great
and small).
At 512, where a-flat is not quite equivalent to g-sharp, the ancients had
little choice but to accept this arithmetical compromise with perfect inverse
symmetry.
How did they rationalize such a complicated, inverse symmetry, one ultimately
defeated because of the compromise? Remembering the quite ancient correlations
of scale and calendar, let us apply imagination to their problem.
This base-60 model can be imagined as an appropriate correlate to the lunar
calendar of Sumer and Babylon, as it later became the map of an idealized
circular city in Plato's Laws, calendar and musical scale being assumed
to have a similar cosmogony. Notice the following correspondences:
a. The basic seven-tone scale requires the thirty digits in the 30:60 octave,
and 30 is deified as Sin, the Moon, and the basic octave limit.
b. The two opposite seven-tone scales and the symmetrically divided tone
circle correspond with Sumer's two agricultural seasons, in which irrigation
during the dry summer complemented the rainy winter harvest.
c. In the octave double between 360 and 720, which coalesces opposites,
there are 360 units to correspond with the schematic calendar count of 12
x 30 = 360 days. (Eventually, astronomers in India and Babylon defined these
units as tithis, meaning 1/360 of a mean lunar year of 354 days,
hence slightly less than a solar day. Greek astronomers eventually defined
the same 360 units geometrically as degrees. Neither development is relevant
to ancient Sumer.)
d. Tonally acceptable but acoustically inaccurate semitones, alternately
small (24:25) and large (15:16), correspond with the lunar months embodied
in ritual, alternating between 29 and 30 days.
e. Between a-flat = 512 and g-sharp 512 (in the opposite sense), a gap corresponds
with the excess of a solar year over 360 and the defect of a lunar year
of 354 days from 360. (Five and a quarter extra solar days are about a 1/69
of 360, while the gap in the reduced comma is actually about 1/60 of an
octave, a remarkable near-correspondence.)
Because any successful agricultural society must find some way to accommodate
lunar, solar, ritual, and schematic cycles with the growing cycle, we need
not suppose that Sumerians or anyone else ever really believed the year
contained 360 days. Only a musicology dedicated to numerical precision and
economy finds 720 days and nights (that is, 360 days and 360 nights) cosmogonically
correct.
MATRIX ARITHMETIC
All of the tonal, arithmetical, and calendrical relations discussed above
are coincidences. They exist among base-60 numbers whether or not anyone
is aware of them, mainly because 60 is divisible by three prime numbers,
2, 3, and 5, and no others, and 60 is being used in the way we use a floating-point
decimal system.
If Sumerian mythology did not offer persuasive evidence that Sumerians were
conscious of tonal implications, then their establishment of a base-60 system,
which included such perfect models for a lunar-oriented culture and for
Pythagorean harmonics two thousand years later, would be pure serendipity,
meaning that it resulted from "the gift of finding valuable or agreeable
things not sought for." But the most interesting evidence for Sumerian
harmonical self-consciousness is yet to be shown via Plato's kind of triangular
matrices, functioning as "mothers" in harmonical arithmetic.
In Plato's Greece, the harmonical wisdom of Babylon and India was transformed
into political theory. Men now acted out the roles once assigned to gods.
Plato's four model cities--Callipolis (in the Republic), Ancient
Athens and Atlantis (both in the Critias), and Magnesia (in the
Laws)--were each associated with a specific musical-mathematical model,
all generated from the first ten integers. All are reducible to a study
of four primes: 2, 3, 5, and 7.
In the Republic and Laws, idealized citizens--represented
as number--generate only in the prime of life. For Plato, this means that
2 never really generates anything beyond the model octave 1:2, for this
"virgin, female" even number--with all of its higher powers--designates
the same pitch class as any reference 1. (Multiplication by 4, 8, 16,...
generates only cyclic identities, different octaves of tones we already
possess. They are Plato's "nursemaids," carrying tone children
until they are old enough to "walk" as integers; hence, as he
says, his "nurses" require exceptional physical strength.)
The multiplication table for the 3 x 5 male odd numbers, however, generates
endless spirals of musical fifths (or fourths) and thirds; within the female
octave 1:2, new pitches are generated at the same invariant ratios. The
Greek meaning of symmetry is to be in the same proportion. Thus, a "continued
geometric proportion" (like 1, 3, 9, 27,...or 1, 5, 25,...) constitutes
"the world's best bonds," maximizing symmetry, which is obscured
by mere appearances when these values are doubled to put them into some
preferred scale order. The multiplication table for 3 x 5 graphs multiple
sets of geometric tonal symmetries (Plato's only reality) as far as imagination
pleases.
Greece inherited its arithmetical habits from Egypt, including an affection
for unit fractions in defining tunings (the ratio 9:8 was thought of as
"eight plus one- eighth of itself," and so on). It awoke to number
theory only when it became acquainted with Mesopotamian methods. Thus, the
travels of Pythagoras, whether legendary or not, played an important role.
Those methods apparently were new enough in Plato's fourth century B.C.
to invite his extensive commentary, yet old enough so any novelty on Plato's
part was absolutely denied by Aristoxenus (fl. circa 330 B.C.) within fifty
years.
Plato is responsible for an astonishing musical generalization of the base-60
tuning formula as 4:3 mated with the 5. His 3, 4, and 5 correspond with
Sin = 30, Ea = 40, and Enlil = 50 and remind us that all tones are linked
by perfect fourths, 4:3, which define possible tetrachord frames, or by
perfect thirds, 4:5. The last Pythagorean who really understood Platonic
"marriages" may have been Nicomachus in the second century A.D.;
he promised an exposition but none survives.
BABYLONIAN REORGANIZATION OF THE PANTHEON
In the second millennium B.C., the Babylonians reorganized the inherited
Sumerian pantheon in a way that very strongly points toward its Pythagorean
future. To avoid destruction by Enlil, who is disturbed by their confusion
and noise, the gods reorganize under the leadership of Marduk, god 10, the
biblical Baal, to whom all the other gods cede their powers.
Herein lies a beautiful reduction of Sumerian expertise with reciprocal
fractions to a more philosophical overview of harmonics as being generated
exclusively by the first ten integers (Socrates' "children up to ten,"
in the Republic, beyond which age he doubted citizens were really
fitted for ideal communities).
To celebrate their survival after Marduk defeats the female serpent Tiamat,
sent to destroy them, the gods decree him a temple; the bricks require two
years (2 x 360 = 720) to fabricate. This mythologizes 720, the Sumerian
unit of brick measure, and the smallest tonal index able to correlate seven-tone
opposites into a twelve-tone calendrical octave. When Marduk's tonal/arithmetical
bricks are aligned in matrix order, we see that the general shape of his
temple (with an index of 720) is an enlarged form of Enlil's temple (with
an index of 60); Enlil now confers his fifty names on Marduk. This temple
makes Marduk's face shine with pleasure, we are told.
Let me conclude our discussion of Marduk's victory over the dragon, Tiamat.
'GREAT DRAGON' TUNING
It is now a normal part of a child's musical education to learn to view
the scale as a spiral of musical fifths and fourths, as they are actually
tuned--for the convenience of the ear--and to be shown those tones in a
tone circle. That up-and-down, alternating cycle of pitches inspires, I
propose, the dragon and great serpent lore of ancient mythology (fig. 13).
Serpentine undulations are visible to any harpist in the lengths of successive
strings when taken in tuning order (as they still necessarily are), and
the undulations can be seen in any set of pitch pipes when similarly aligned,
as in China. Because the same tone numbers function reciprocally as multiples
of frequency and of wavelength, they have the same double meanings today
that they enjoyed in Sumerian times. It is entirely appropriate, therefore,
to represent this spiral both forward and backward, simultaneously, with
intertwined serpents.
In the mythological account, Marduk slays the dragon (which is presumably
the continuum of possible pitches represented by the undivided string) by
first cutting it in half to establish the octave 1:2. Further cutting presumably
"sections" the other pitches. No numbers larger than Marduk's--meaning
10--play any role in geometrical sectioning of the string.
This "serpentine" double meaning--rising and falling musical fifths
and fourths--lies at the very heart of our consciousness of musical structure.
Sumer did not hesitate to make the double serpent the center of symmetry,
as on this steatite vase of Gudea (fig. 14), priest-king of Lagash circa
2450 B.C., where they are flanked symmetrically by gryphons.
Large and unwieldy numbers can be avoided if the 4:5 and 5:6 ratios introduced
by Enlil are used to define the seven-tone scale (in which case all the
numbers are of two digits). Used for the twelve-tone scale, his numbers
need only three digits. Thus, in Sumer, Enlil = 50, base-60 deification
of the human, male prime number 5, grossly reduces our computational labors
from six-digit Pythagorean numerosity (in which a twelfth tone requires
311 = 177,147) to no more than three, and without noticeably diminishing
melodic usefulness (fig. 15). Only the five central tones (CGDAE) from the
Great Serpent appear in figure 12, where they are indicated by solid radial
lines. All other tones are owed to Enlil.
Historically, European music reintroduced this Just tuning system in the fifteenth
century A.D. to secure perfect 4:5:6 triads for its new harmonies without exceeding
twelve tones. The ancients probably loved it more for its arithmetical economy
than for its triadic purity. Microtonalists today, equipped with a powerful
new technology, are again searching for an effective employment of these ancient
Sumerian god ratios.
SOME PERSONAL CONCLUSIONS
The ultimate origins of music theory, as opposed to the Sumerian codification
that I deduce here, remain lost in the far more distant past, like the origin
of our sense for number. They are grounded in a common aural biological heritage,
some of which we share with other animals, and are by no means dependent, as
Aristotle noted, on precise numerical definition. As eminent contemporary musicologist
William Thompson explained in our correspondence,
In adapting to our complex environment, our sensory ingestive systems have become...forgiving
filters, enabling us to generalize....This, I'm convinced, is a product of very
early adaptive behavior, a part of our survival good fortune...in that our neural
system has developed myriads of networks which are overachievers when it comes
to doing some simple jobs.
Socrates never believed in the possibility of perfect justice. The great aim
of Plato's Republic was to help readers become more "forgiving filters"
for alternative cultural norms. There remains a certain fuzziness about a scientific
definition of musical intervals, as there is about the Republic's days
and nights and months and years, and art has turned that into something for
which we all can be grateful. Sumerian "overachievers"--and these
"black-headed people," as they called themselves, proved historically
to be as aggressive as the great heroes they knew or invented--achieved a tremendous
synthesis of cultural values. They challenge us to do as well.
__________________
Additional Reading
E.A. Wallis Budge, Amulets and Talismans, reprint, Carol Publishing Group,
New York, 1992.
Samuel Noah Kramer, The Sumerians, University of Chicago Press, Chicago,
1963.
Otto Neugebauer, The Exact Sciences in Antiquity, University Press of
New England, Hanover, N.H., 1957.
William Thomson, Schoenberg's Error, University of Pennsylvania Press,
Philadelphia, 1991.
B.L. van der Waerden, Science Awakening, Scholar's Bookshelf, Princeton,
N.J., 1963
Hermann Weyl, Symmetry, Princeton University Press, Princeton, N.J.,
1952.