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On Number As Artifact (Part 1: Introduction
Fred Fisher

[*!* Image: Music was still a science when a medieval monk drew this diagram of the relationship between the seven Liberal Arts. Musica is at lower right]

"Culture," an archaeologist once observed, "is highly perishable and therefore cannot be excavated. "(1) This is not an attitude most today would find encouraging, especially perhaps those who consider themselves students of man's cultural evolution. If culture is what makes us human, then (according to this view) archaeology is not going to be able, in the final analysis, to tell us very much about ourselves.

This dark cloud, however, just may have a bright lining. In recent years it has been shown that answers to many questions about the past can be learned through a new avenue of inquiry drawing upon philology and archaeology, but essentially mathematical in nature. Nicolas Rashevsky suggested in 1968 that mathematical methods might serve prehistory in ways heretofore unanticipated. He cautioned, further, that the notion was bound to be regarded abrasively by many:

There are undoubtedly many skeptics who will feel that the use of any kind of mathematics in historical research is foolish and doomed to failure. They may be right. There are certain things that have been rigorously proved to be impossible in science. The scientist bows before a rigorous proof of such an impossibility. Nobody, however, has yet given a similarly rigorous proof of the impossibility of explaining historical phenomena by mathematical means, or even of occasionally using mathematics to help explain the mechanism of history.(2)

Perhaps it depends on the cultural area being investigated. Today one tends to think of music as an art. But an important branch of musical thought acoustics- is definitely a science, and was so for many millennia of cultural history during which certain philosophical premises became engrained in human thinking. Plato's friend Archytas called acoustics and astronomy "sister sciences," and the Chinese saw music and its laws as a source and basis for all knowledge.

The bond between astronomy and acoustics was mathematical. In the liberal arts curriculum of the medieval academy, the four subjects comprising the quadrivium (arithmetic, geometry, music, and astronomy) were linked together by a common concern for number. Thus an epistemology taking number as its basis is a perfectly legitimate approach to the early history of both musical and astronomical thought. Why not try Rashevsky's idea!

Of course, the close bond sensed by the ancients between music and astronomy has little meaning for modern scientists. It isn't enough simply to investigate the two fields separately. To get the whole picture of man's past, they must once again be joined. The kind of specialized compartmentalization that serves modem science so well will not help the prehistorian. We need generalists!

Once again there is some good news to report. There are a few generalists out there: mavericks, by and large. Like Dr. Emest McClain, a retired Brooklyn College music professor, or Hugh Harleston Jr., a freelance engineer out of Mexico City currently engaged in publishing a series of guide books to the ruins at Teotihuacan. What drives both of these men, as well as a few other men and women around the world, is the realization that the only knowledge that will really open up the secrets of man's intellectual past is an eclectic, ecumenical approach to experience - a syncretistic view like that of early man himself.

The results that have already derived from this change in attitude speak for themselves. McClain, who is presently studying the acoustic properties of ancient Chinese bell collections, began by deciphering the acoustico- astronomical imagery of the Hindu Rg Veda and the writings of Plato.(3) The gleanings are surprising to say the least. If McClain is right about what he has found (much of it appears to be almost irrefutable), early mathematical thought has been sold short. While it is true that these thinkers lacked an understanding of logarithms and some other aspects of irrational numbers, the mathematics of the Rg Veda, man's oldest extant literary treatise, suggests remarkable sophistication on all three levels of mathematics, astronomy, and acoustics.

Commentators have assumed that the extravagant powers of 10 attributed to Indra's forces were simply generous tributes to the god, but numbers -factors of the form 10n are actually a part of the essential arithmetic. Again and again Indra is addressed as "Lord of a Hundred Powers." (Hymn 1.30.6 is one of many examples.) "Help us, 0 Indra, in the frays, yea, frays, where thousand spoils are gained." We hear that Indra slew "ten thousand Vrtras. " Or Indra won "ten thousand head of kine," and ten thousand other "gifts". It is by such extravagant multiples of 432 and 864 - the bounding numbers of our basic "Pythagorean" scale - that we have arrived at the Yuga numbers themselves, and a similar multiplication by 10 to the third power carries us to the still larger Kalpa and Brahma numbers to be studied next.(4)

What makes all of this easier is that the problems themselves are unchanging. The sky of the astronomers, with the exception of certain measurable factors such as the precession of the equinoxes, is to a considerable extent the same now as it was millennia ago. Even more resistant to change is the musical scale of the acousticians. There have been different solutions to the scale problem in various parts of the world at different times, but the problem remains one and the same.

To make this point clearer - and acoustical problems are abstruse by any reckoning - take a simple astronomical analogy. There are several ways of measuring the length of a month. The synodic month of approximately 29 days represents the period during which the moon returns to the same position with respect to the sun. The sidereal month, about two days shorter, refers to the time in which the moon returns to the same position with regard to the stars. There are also the draconitic, month, the anomalistic month, and so on, each measuring a somewhat different aspect of the lunar cycle. One society or culture might base its principal lore and behavior upon sidereal measurement of the moon, whereas others might choose the synodic period. These might cause remarkable divergence in cultural norms, but the phenomena themselves remain unaffected.

The "Pythagorean scale" to which McClain refers is really much older than Pythagoras and underlies scale systems from cultures as far afield as ancient China. It is also fundamentally the scale in use by Western culture today. The fact that Chinese music sounds completely different from Western music suggests how easy it is for concepts of music as an art to intrude upon discussions of acoustic theory.

A writer who has pointed the way to a better understanding of the numerical nature of Chinese thinking with a heavy emphasis upon Chinese musical science - is the Frenchman Marcel Granet. In La pensee chinoise ("The Chinese Way of Thinking") he suggests the extent to which a fetishism for simple number influences Chinese attitudes toward tuning the traditional five-tone scale.(5) Bounded by the 10:5 octave ratio (same 2:1 relationship as McClain's 864:43 2 octave), the intervening tones are represented by 7, 9, 6, and 8. This is not a "usable" musical pattern but it satisfies the Chinese demand for simplicity and order in all things, and it has some interesting cosmic ramifications, as we shall see presently.

The Westerner digging deeply into Chinese musical thought finds himself constantly on common ground, sensing at the same time the agelessness of the ideas with which he is involved. And McClain appears to have had precisely this same feeling with regard to Hindu thought: "Throughout the Rg Veda allusions to music and to the power of the musician are explicit and ubiquitous, and so is the insistence that the imagery be read as allegory."(6)

Almost certainly the single most significant finding to date, and the one most exemplary in linking the fields of acoustics and astronomy, was the suggestion in 1978 by Dr. Edith Borroff of the State University of New York at Binghamton regarding the number 360. Borroff, a musicologist who - unlike many of her profession - does not shy away from the implications of number and musical mathematics, established by means of an elaborate computer calculation that this familiar "artifact" number already long associated with geometry and navigation also has acoustical meaning.7

The number 360 had musical meanings for the Chinese since, as Granet shows, the tones 5, 8, 6, 9, 7, 10 of the "idealized" five-tone scale, if each is multiplied by a factor of 8 (a number much beloved by the Chinese), add up to 360:

40 + 64 + 48 + 72 + 56 + 80 = 360

This is not at all the content of the Borroff program but rather confirms the universal appeal of a number used for thousands of years, yet a number about which little is known. Except as a kind of luni-solar "round" number (a compromise between a 354-day lunar and 365-day solar year), the number 360 has no direct astronomical significance. Yet it has been pervasive in human thought from Sumer onward, turning up in Egypt, China, India, the Greco-Roman world, and even Central America. While the information contained in the Borroff calculation is too technical for a general- interest publication, suffice it to say that its implications could be far-reaching indeed. It feels extraordinarily right.

There are two contrasting pictures painted here: on the one hand the great sophistication of the Borroff findings, a calculation which - prior to computers might well have consumed one or more lifetimes, and on the other hand the naivete of the Chinese rationalization. Both lead to the important "cosmic" 360 though in different ways. The writer submits that both pictures of early mathematical man are valid, since number symbolism continued into Western culture well into the Middle Ages. And there is plenty of counter-evidence by means of which one can prove that Chinese mathematics, by and large, was not naive.

[*!* Image: The Chinese Lo-Shu magic square]

Arithmetic now appears to have arisen, not as an independent line of inquiry in its own right, but rather as an adjunct to those disciplines which required its services, such as astronomy and acoustics. Though in some respects (such as the veneration of integers) it remained rather primitive, it included such seemingly modem concepts as set theory. This is what the McClain studies demonstrate, and it seems clear that mathematicians, if they are to learn the roots of their own profession, need to have also some instincts for astronomy and acoustics. To a considerable extent it is the astronomers who have led the way. Archaeoastronomy is now a recognized discipline. There simply is no recognized equivalent so far in musicology, which continues to falter on a linguistic base.

The groundwork for change is being laid. Number is a valid cognitive tool, in historical pursuits as in engineering, architecture, chemistry, and economics. The Borroff and McClain findings may well serve to encourage scholars of early literature and other records to take numerical references seriously. As McClain notes at the beginning of The Myth of Invariance, "Historians of science have barely begun to cope with certain kinds of material available to them. "(8)

The kinds of ideas cherished by early man concerning the universe - as well as his place in it - may not be the truth as modem man sees it. Historically speaking, this cannot be regarded as a test of their relevance. Students of culture must take what they find, without standing in judgment. MIT's Giorgio de Santillana, one of the few true generalists of our time, saw the need clearly:

Our forebears built up their world view from the idea which today would be called geocentric; they concluded with speculations about the fate of man's soul in a cosmos in which present geography and the science of heaven are still woven together. Worse, maybe, they built them up on a conception of time which is utterly different from the modem metric, linear, monotone conception of time. Their universe could have nothing to do with ours, derived as it was from the apparent revolutions of the stars, from pure kinematics. It has taken a great intellectual effort on the part of many scholars to transfer themselves back to that perspective. The results have been astonishingly fruitful.(9)

[see next issue for Part II]

[*!* Image: The emperor Huang Ti, reputed inventor of the compass, the wheel, and ceremonial garments such as this outfit. As always, China's fetishism for number is evident. [from Christie, Chinese Mythology, 1968]


1. James Deetz, Invitation to Archaeology (Garden City, NY: Natural History Press, 1967), p. 7.

2. Nicolas Rashevsky, Looking at History through Mathematics (MIT Press, 1968), p. 4.

3. Ernest McClain, The Myth of Invariance (NY: Nicolas Hays, 1976); The Pythagorean Plato (Stony Brook, NY: Nicolas Hays, 1978).

4. McClain, Myth, p. 75.

5. Marcel Granet, La pensee chinoise (Paris, A. Michel, 1968), pp. 202-203.

6. McClaine, Myth, p. 11.

7. Edith Borroff, "Ancient Acoustical Theory and a Pre-Pythagorean Comma," College Music Symposium XVIII, 2 (Fall 1978), pp. 22-23. Also see the following article, in which the essential algebraic material is shown:

(3/2) 360 ) (3)360
X =   ---------   =   --------- + 360
(2/1)210 (2) 210

3360 approx. 1.5016,
or very close to
the value of the
3:2 perfect fifth.
  =   -------   =  

8. McClain, Myth, p. 4.

9. Giorgio de Santillana and Hertha von Dechend, Hamlet's Mill (Boston. Gambit, 1969), p. 322.

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