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HORUS VOL III. Issue 1

An Ear for Numbers
Fred Fisher

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While teaching music subjects at a university in the Southwest highly regarded for its music school, I became disturbed by the apparent lack of interest in, or understanding of, Chinese music theory. Since it was becoming clear to me that ancient Chinese theory impinges on Western music theory in innumerable places and ways, I wondered why no professor on either the undergraduate or graduate level had ever bothered to discuss it while I myself was a student. Certainly the situation was being perpetuated.

With these thoughts in mind, I began issuing my findings in the nature of an informal research newsletter which I sent out to colleagues across the country who I felt might be interested and, of course, handed out to any of my students who demonstrated the tiniest bit of curiosity. In January of 1974 my efforts bore fruit in an unexpected way. I received a letter from a Dr. Ernest McClain of Brooklyn College. With the letter was a paper that had just been published in a now-defunct but excellent magazine I had not previously known about. It is not an exaggeration to say that this article by McClain, titled "Plato's Musical Cosmology," proved to be one of the turning-points in my life.

In his letter McClain noted, "my analysis of Plato's mathematical allegories, which has occupied several years and enjoyed the cooperation of a number of authorities in various fields, teaches me that very much of your speculation about ancient music must be true." He added, "Plato, I suspect provides the proof for some of your theories." The article itself was dazzling. Men of letters have always found mathematical references in the Socratic dialogues labored and abstruse. For the most part such departures from seemingly normal literary style were dismissed as irrelevant, even expurgated from some student versions. And now here is this fellow McClain cutting right through to the essence of this material as if it had been just butter all along.

The paper, as it turns out, was a sample chapter from a book manuscript which McClain was having trouble getting published. Although other chapters appeared in print subsequently, eventually the book itself was published, titled The Pythagorean Plato. If there is a central theme in McClain's writings, it is that no modem-day approach to early scientific thought is valid if it does not take into account the acoustical impetus - that indeed music must rank, as the Chinese felt, pre-eminent and primordial among the various scientific pursuits of antiquity.

Why was music first? Early man was decidedly aural in inclination, looking to music and music derived intellectual premises for a groundwork toward understanding the universe in its totality. Today knowledge is fragmented. Your children go off to college, each to study his or her allotted portion of this vast sea of information, carefully categorized like pigeonholes in a secretary desk. This is not bad, necessarily, but it won't work in studying the past. Early thinkers really believed that the entirety of knowledge somehow fitted together into a simple, graspable, central truth, and that this truth, once beheld, would surely prove to be much more than merely the sum of its parts. Obviously, also, modem science is visually oriented and is perceived through microscopes, voltmeter readings, and complex formulas notated upon blackboards. If there were a central truth (and this is a skeptical age at best), how could it be aural?

Remember, we are not concerned at this point with whether early man was right about his idea of truth. We are simply trying -- against the white noise of modern civilization -- to recapture what it was that he was looking for and how he expected to find it. McClain states,

Plato illustrated his philosophy with a chain ofmathematical allegories extending through the Symposium, Republic, Timaeus, Critias, Statesman, and Laws. These allegories allude to mysterious arithmetical and geometrical models; in modem times they seem riddles, obscuring Plato's meaning rather than illustrating it .... So many of Plato's statements turn out ... to be mathematical and musical metaphor that this whole set of dialogues should be examined anew from this perspective.

McClain proceeds, in his writing, through a musical analysis of the dialogues mentioned, using as leitmotif the avowed Pythagorean identification of tone with number, "thus relating it with everything in the universe which 'participates' in number." To a mathematician the result appears in the form of analogy "between various political systems, musical tuning systems, and 'certain subgroups of a topological group.` It will come as a surprise to many to learn that Pythagorean arithmeticians were 'into'. group theory. But this is the only way in which early scale tuning problems could be approached, since the modem system of logarithms is based upon irrational fractions rather than whole-number relationships, and early science eschewed irrationals like poison. "Plato," says McClain, "implicitly recognizes the commutative, associative, closure, identity, and inverse properties characteristic of our modem theory of mathematical groups."

I had no knowledge of Ernest McClain and his work prior to receiving his first letter and the enclosed paper, but we proceeded to become great pen pals as the weeks and months passed. It was reassuring to find a kindred spirit halfway across the land, albeit intimidating to realize just how good he was. In 1976 his first book saw the light of day, and it wasn't The Pythagorean Plato, though this followed soon after. It was The Myth of Invariance: The Origin of the Gods, Mathematics, and Music from the Rg Veda to Plato. Music, as McClain shows in this work, was the ultimate testing-ground of arithmetical precepts and, like a mocking older sibling, a source of neverending frustration.

What the ancients wanted to find they could not find. Why? Because it wasn't true. They wanted to believe this, obviously, not for mathematical but for musical reasons. It had to do with problems they were encountering in tuning their scales. There was a paradigm for the recalcitrant tones of the scale in the equally intransigent months of the year: 12 months do not a solar year make, and 12 equal and properly tuned halfsteps do not ultimate in an octave. The problems are similar, but the musical problem is by far the more complex - and much of what mathematics is today it owes to the cruel lesson of music's essential inharmonicity.

The Socratics, Pythagoreans, and ancient Chinese presumably did not use blackboards. To a considerable extent they studied their mathematics aurally, with devices like gongs, monochords, and flutes. They knew that a properly tuned fifth (C up to G for example) expressed the mathematical ratio 3:2 in some way, whether string lengths, tube lengths, or material density and mass. This was a "Y' value, and projections of fifths amount to set-theory projections of the powers of three. Even more fundamental to early thought was the octave, of ratio 2:1. "Octave" means "eight" -a notion of much later provenance and no doubt calculated so as to cause endless confusion to persons with logical minds. But the point here is that the octave is a '7' value, and projections of octaves must therefore be ever inharmonic with projections of fifths. On this crucial issue both Pythagorean cant and Chinese civilization eventually foundered.

But the imparting of knowledge remained orallaural for a long time. Socratic dialogue was an oral/aural method of getting at essential truths through argumentation. The Pythagorean monochord remained a staple of clerical teaching methodology into the Renaissance. The Chinese pursued their elaborate acoustic experiments well past the Han dynasty, ever hopeful of achieving more favorable results.

But the world of science was moving in a different direction - away from the ears and toward the eyes. Music was less and less apt to be listed among the sciences despite its evident involvement with mathematical principles. In the English-speaking West the big change is easily chronicled and coincides with the growing preference for the vernacular eLver Latin in the world of book publishing. In 1597 Thomas Morley published his Plaine and Easie introduction to practical music. It was the first of hundreds of such books. The age of music as a speculative, philosophical, scientific concern was over.

Music has of course been an art form for thousands of years, and its reputed connection to deep philosophical, yea cosmological, truths has waxed and waned. Plato saw music as being essential to the proper upbringing of rulers and statesmen and even a reasonable avenue of speculation for determining the proper way to lay out cities. The Chinese would have been in total concurrence with this view.

The farther one moves back into the past the more homogeneous seems human thought. The biblical assertion that before Babel all men were "of one tongue" does not appear unreasonable. Most of what McClain finds in Hindu and Pythagorean lore jibes with Chinese dogma, and Chinese civilization must be the most ancient and consistent of all that we know of today. Westerners often ridicule Chinese science, perhaps not realizing that what they find ludicrous in it is its adherence to norms as musical as they are ancient. There was a time when the highest technology was Chinese -but times change.

Ernest and I don't correspond much any more, but last I heard he was becoming fascinated with the tonal implications of a set of experimental Chinese bells dating from the fifth century B.C. Moving around like this from culture to culture seems so easy to do if, for you, music becomes a lingua franca that facilitates such ecumenism. And keep in mind that we are talking here about the theory, not the art of music of the culture in question.

All societies had to tune their scales, and all societies encountered the same problems in the attempt to do so. Thus all of antiquity is, in some sense at least, a book lying open for the musician to read. llere is a flaw in the metaphor, however: the musician is one scholar who will choose to read the book of prehistory with - his ears.

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