Musical Math: 7 and 12 in Ancient Science and Cosmology
In his book The Myth of Invariance Ernest McClain uses a musical analysis of the imagery in Indias oldest sacred text, the Rig Veda. Mr. McClain shows clearly the very ancient origin and mutual dependence “of science, of our calendar, of musical theory, and of our civilization.” From the earliest origins of science in prehistory, the relationships between number and tone have been fundamental in mans attempt to understand his world.
To quote Mr. McClain, “What seemed most certain to our ancestors was that physically nothing endured. In this sea of restless change man discovered an island he could trust, the octave of ratio 1:2–the basic miracle of music functioning as a matrix for all smaller intervals and providing a metric basis for a tonal algebra. From what we know at the present time it seems likely that the octave invariance was recognized in India, Sumer, Babylon, Egypt, and Palestine well before the variant cycles of sun, moon, and planets were coordinated with even modest accuracy. Calendrical periods of 30, 60, 360, and 720 units and their multiples belong to the essential arithmetic of a systematic mathematical harmonics. Their source was not astronomy though they found a ready application in early astronomy, which knew them to be unsuitable for its own cycles.”
In other words, the observations of the fundamental environmental cycles that regulated mans life, the daily cycle of light and dark, the monthly lunar cycle of approximately 30 days, and the yearly seasonal cycle found a ready harmonious correspondence with the “musical arithmetic” that was developed by ancient man. It is not presumptuous to assume that these early observations of these cycles were a contributing factor to the beginnings of mathematics. For as Plato observed: “The sight of day and night, of months and the revolving years, of equinox and solstice, has caused the invention of number.whence we have derived all philosophy”.
Early man saw in the rules and mathematics of the division of the octave into scales a universal applicability of the division of any whole into its parts. This “Musical arithmetic” according to Mr. McClain, “fueled a radiant vision of universal harmony while providing both a model and a motive for the development of a rigorously abstract number theory and a related geometrical algebra.” Plato expressed this vision of a universal harmony thus: “To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves.by the revelation of a single bond of natural interconnection.”
In mans early observations of the primary cycles of time that governed his life, the day, the month, the year, he saw a metaphor for all change and incorporated it with a view of time as cyclic. This view was reinforced when he began to explore the transient, dynamic rhythms of music. To quote Marius Schneider: “In view of the inconstancy of the world of form, primitive man questions the reality of static (spatial) phenomena and believes that transient (temporal) dynamic rhythms are a better guide to the substance of things.” As we shall see later, this perspective is remarkably similar to the perspective being forced upon us by the new sciences of chaos and complexity.
So how was the connection between astronomical cycles and the musical scales made? The mathematics of Rg Veda man was limited to rational numbers, defined as: “A number capable of being expressed as an integer or a quotient of integers, excluding zero as a denominator.” Thus he was limited to expressing his mathematical relationships as integers. But even though this early science and math may have been primitive from our technological perspective, it does not follow that the philosophy was. As Mr. McClain points out, in studying these source texts “we are dealing with a primitive science of music and number, and a mature philosophy.” It is the egoism of our times to presume that the holistic concepts of early man were in fact as primitive as their technology.
As it turns out, and Mr. McClain so clearly demonstrates, “the smallest integers which can define a diatonic scale with two similar tetrachords—a fundamental concept in both Hindu and Greek tunings—occupy a space of thirty units in the octave double 30:60”. This “cycle of thirty units harmonizes the month with the diatonic scale”. In addition, the smallest integers which can define the tones of the reciprocal diatonic scale “in chromatic order lie within the octave double 720:360”, thus harmonizing the idealized year of 360 days with the chromatic scale. The two fundamental cycles of time, the month and the year, were thus harmonized with the diatonic 7 tone and chromatic 12 tone musical scales. And the harmonization was not forced. It was a harmonization that was required by the mathematics of whole integers.
Another factor that contributed to our early mans observation that the science of scaling in music harmonized with the basic astronomical cycles has to do with what I referred to earlier as “the comma of Pythagoras”. As we noted previously, the cycle of fifths brings us to a musical frequency of 2076 which is 7 octaves from a starting frequency of 16. The exact frequency of the 7th octave is 2048. This produced a discrepancy of 28 units which represents overshooting the 7th octave by .0137. As we referred to above, the octave double of 360:720 is the smallest integer set that can produce the chromatic ordering of notes. Thus this scale of 12 notes was harmonized with the number 360, which is the number of the idealized year.
Now we also know that the actual year is 365.25 days long, not 360. The 5.25 days represents overshooting the idealized year by a factor of .0146, very close to our musical discrepancy of .0137. This seemed even more verification to our early scientists that there was a fundamental relationship that existed between musical math and calendrical reality. This was not all. The second most significant tuning of our ancient musical math produced a scale of 12 notes with a shortfall discrepancy between two notes that represented the 12th note on an idealized 360 unit mandala.
This discrepancy translated into a shortfall of .0333 in our musical math. Remarkably, this correlates with the calendrical shortfall of 11 days that exists between the number of days that span a lunar year of 12 lunations (354 days) and the idealized 360 unit calendrical cycle. This discrepancy of .0305 (11 days divided by 360) is very close to the discrepancy of .0333. In a way, we could say that this very possibly clinched it for our early scientist. He has now found significant correlations between musical math and the primary cycles governing his life, and a rational for assuming that this musical math, this scaling by 7 and 12, somehow pointed to a fundamental patterning that existed in nature.
Was this musical math unique to Rg Veda man, ancient Indian science. Far from it. It was extensively developed and formalized in ancient Greece through the work of Pythagoras and in the writings of Plato. Nicolas Campion, author of The Great Year also made note of this though referring to it more as “number mysticism”. He states: “In Greek number-mysticism the numbers 3 and 4 were closely related: added together they produce seven and multiplied, twelve, both of which numbers possessed deep cosmological significance.” I feel Mr. McClain does a much more exemplary job of showing that this was not to early man what could be referred to as just number-mysticism, but represented a mature philosophy based on empirically observed relationships between the mathematics available in that day and significant events in mans world.
Additionally, in ancient Sumer and Babylon this musical math and musical science was highly developed. The art of calculation in third millennium Babylon was comparable in many respects to the mathematics of the early Renaissance some thirty odd centuries later. The Pythagorean Theorem was known in Babylon more than a thousand years before Pythagoras. And Pythagorean string length ratios, considered traditionally to be the beginnings of science, were “recognized by both Mesopotamia and Egypt at least two millennia before the dawn of Greek civilization”. James Jeans points to the finding of two Egyptian flutes the dates of which is calculated to be about 2000 BC, both of which are based on a 7 tone diatonic scale. One can find in this ancient culture of Babylon very clear correlations between “the tones of the scale, the Babylonian-Sumerian deities, and the basic geometry of the square and the circle.” We find in the Epic of Gilgamesh repeated references to the number seven as ” the central measure of time and space”.
In ancient Hebrew culture we also find an absolute reverence for these numbers of the musical scales. As Nicolas Campion points out: “Certain numbers, such as 7 and 12, were thought to possess special cosmological significance, embodying the universal structure of time and space. Therefore, it was considered absolutely vital that political and cultic ritual place the highest importance on these numbers.” And again, “any number that was tied to temporal cycle, as were seven, twelve, and four, was representative not of a fixed state of affairs but of a dynamic process.”
To ancient man the transient temporal rhythms of music were a better guide to the true substance of things, to the underlying pattern of reality and of what could be perceived and experienced. And the fact of this was only reinforced for him through harmonization of the musical scales with the basic temporal cycles of his experience. This perspective remained true within mans science till the nineteenth century. In classical Greek culture, Pythagoras is generally credited with discovering the “deep connection between mathematics, music, and sound”(13), although it would probably be more accurate to state that Pythagoras was very influential in the propagation of this understanding rather than the originator, in light of Mr. McClains work. As outlined above, Mr. McClain shows the much more ancient origin of this understanding or perception. However ancient, it seems clear that “within the confines of the musical scale, the ancients constructed a theory of everything”.