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.
