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On Number As Artifact: Part 2: Development
Fred Fisher

[*!* Image]

Astronomers find evidence of 13 as a number base among the builders and theorists of Stonehenge

Edith Borroff, whose important work with the number 360 was discussed in the previous article, [HORUS II:1] believes the ancients tended to use number in both symbolic and functional ways. This is certainly true of the Chinese, with their hypothetical musical scales based on simple ratios, yet at the same time sophisticated acoustical computations based on principles of set theory.

Modern mathematicians, finding the symbolic numbers and assuming their use in practical arithmetic, jumped at false conclusions about the ancients. Further, modern writers on the history of science have been impatient with the delight of the ancients in symbolic speculation, a delight which they did not understand and did not want to understand.[1]

There are many examples of symbolic number speculation with regard to calendar systems, despite the obvious pragmatic purpose of the calendar respecting agriculture and other seasonal activities. In the first installment we mentioned the sidereal month

of between 27 and 28 days - not a 'month' in the usual sense of the complete cycle of lunation, but rather in terms of the moon's position in the starry firmament. Did this figure in ancient calendar lore? There is every evidence that it did.

R.T. Zuidema, an authority on Inca political and social systems, believes the Incas "had a complicated system of simultaneously using four different types of months."[2] In his published study of an early Peruvian calendar textile he finds convincing evidence of the sidereal month:

Seven regular rows are in the upper part and four in the lower, each consisting of 7 x 4 = 28 rectangles. These 11 rows clearly refer to sidereal lunar months of 27-1/3 days rounded off to 28 days. The 13 months of 28 days form a year of 364 days, which is 1 days short of a solar year. [3]

Why would any culture pay attention to a month so completely out of synchronization with the observable evidence (new moon, full moon, etc.)? Were the ancient Peruvians struggling to find something to explain other phenomena that concerned them? The Belgian mythographer Claude Levi-Strauss notes that a paradox becomes much more assimilable in the preliterate mind if it can be balanced by a second paradox. [4] Did the sidereal month serve such a purpose?

The answer may be simpler than that. The menstrual cycle of a woman comes much closer to the demarcations of the sidereal than the synodic month. And if this seems far-fetched by modern standards, it is well to keep in mind that the Incas' neighbors to the north (the Mayas and Aztecs) observed a ritual year of 260 days (!) that corresponds rather neatly to the nine-month period of human gestation. [5]

[*!* Image: The number 13 played a very significant role in Mayan culture]

These are not familiar numbers by present-day calendar-making standards. Where do they come from? How did the early scientists use them? An interesting study of Stonehenge by Alban Wall (HORUS, Summer 1985) suggests that Zuidema's 13 x 28 has echoes in prehistoric Britain:

Many investigators of Stonehenge have pondered over why the number 56 was chosen in constructing the Sun (Aubrey) Circle.... 28 days and 13 cycles are whole number factors of 364 days. The 56 evenly spaced holes is twice 28 and simply subdivided the base cycle of 28 days in two. To count days around the circle, the marker was moved twice a day, probably one hole at sunrise and another at sunset [6]

A year, of course, is not 364 days long. What happened when this became apparent?

If the true length of the solar year was exactly 364 days, both the Sun and the marker would have returned to exact alignment with each other, at hole 22, every 364 days. However, the marker stone, at each successive summer solstice sunrise, advanced further and further beyond hole 22, indicating thereby that the true solar year was slightly longer than the assumed 364-day base year. When it became evident that the year was consistently at least 365 days, this was taken into account by not advancing the marker on the day of the solstice. The magnitude of the remaining error was readily determined by maintaining the 365-day count over a period of many years to determine the average number of years for an error of one day. [7]

Perhaps this puts things into better perspective! It's about time the pendulum swung toward a middle ground. of objectively evaluating our ancestors, from awesome reverence preceded by veiled contempt. Stonehenge is neither a 'mere pile of rocks' nor a futuristic digital computer. It is eloquent testament of man's struggle to understand his environment - a struggle that goes onward today. The builders of Stonehenge may even have regarded their monument as a learning device. Modern scientific laboratories are indeed that. The Stonehengers were interested to find out just how long a year was. The 13 x 28 of the Aubrey holes was a good, educated guess. But is was wrong - by approximately 1-1/4 days.

Old knowledge dies slowly. Even in our time, hoary English academics have been known to refer to the length of the year as "364 days", as though the Year Day itself did not count. There is much engrained lore of this kind to be found the world over. The French still honor an ancient vigesimal system of counting in their quatre-vingt ("four twenties") for the number 80, and another form of pre-decimal thinking in not using teen-endings until after seize: 16. (Teens are not introduced in our own language until after 12, the traditional relic of dozenal thinking).

To the historiographer, how one counts may be just as important as what one counts---and in fact the 'how' may well provide some insight occasionally into the 'what'. How many ways are there to compute a 364-day year using integers only? Two times 182 is 364, so is 4 x 91, 14 x 26, 13 x 28. None of these sets looks very 'decimal'. None was intended to, of course. They are intended rather to suggest predecimal ways of approaching number, since early man was not particularly decimal in his orientation to the world. He was lunar among other things. The exemplary work of Alexander Marshack suggests that this was the case as long as 20,000 years ago, when man began notating lunar phases on pieces of bone and mammoth tusk. [8]

Early man was trying to find a set-theory way of counting the day of the year. Decimal methods (10 x 36) didn't even come close. Duodecimal counting (12 x 30) produced the same result and thus also had to be discarded. The smallest base that could produce 364 was 4. Prehispanic Americans indeed idolized this number, and now one reason is perhaps clear. But the Chinese had found a number which, when multiplied by 73, would produce the optimum year-number, 365. The number was 5. Chinese infatuation with this number has been well documented. It is the digit which the Chinese placed at the center of their famous Magic Square and, since Chinese counting remained largely decimal, the central digit of the first nine:

1 2 3 4 5 6 7 8 9

Five is also the sum of two important symbol numbed for the Chinese. Three is the number of Heaven, two the number of Earth. Five is the number of tones in the scale which the Chinese use for musicmaking (as distinguished from the 12-note theoretical scale). 1n the Chinese conception of Numbers," notes Granet, "there is an admirable reconciliation of the strictest conformism.... and fantasy." [9]

Primitive science is syncretistic and wastes nothing. We have already seen (in the first article) that the Chinese used 72 (5 x 72 = 360) as a central number in their hypothetical representation of the pentatonic musical scale (the pentatonic was the standard scale for music-making, but in this form dictated by the simplest integers- it would have sounded out of tune). Zuidema, in his study of Peruvian calendar systems, finds both 72 and 73, with the latter multiplied by 5 to make the 365-day solar year. [10]

There is reason for the ancient scientists to have felt that 72 and 73 were the 'right' calendar numbers, since there is a gap or surplus which occurs in musical tuning systems which equates almost exactly with the 73:72 discrepancy denoting the 365:360 year. Called the 'Pythagorean comma', it is reflected by the ratio 531441:524288, but a rough equivalent -in far simpler and more memorable terms- is the ratio 74:73.[11] It was this comma, and the ramifications of its projection mathematically, that incited Edith Borroffs computer findings concerning 360, and her conclusion that mathematicians had achieved these same computations - minus computers - by the Sumerian highpoint.

We mentioned the Levi-Strauss contention that paradoxes can be accepted in primitive thought if they can be balanced. The calendar problem and the tuning situation serve as perfect examples of two ancient paradoxes that beg to be united in thought, i.e., balanced. The one against the other. Most people comprehend the calendar problem without great effort: the cycles of sun and moon do not really coincide, despite efforts by mankind over the millennia to make them appear congruent. The tuning problem is more recondite. So much so that many practicing musicians confess to not really understand it.

One group that understands it well, is of course, piano tuners. Piano tuners are pragmatists in the sense that it is their job to make bad-sounding pianos sound better. But the nature of the tuning craft is such that, the better the acoustic problem is understood, the better it can be dealt with. The writer, while a graduate student in music, took up tuning not only because he hoped to make his own recently acquired piano sound better, but also in hopes of understanding more fully what could be called the "mechanics" of the musical scale. What he learned was that the various tones of the scale, if in proper acoustic proportion with one another, are going to be out of tune with the octave. In other words fifths, for example, of ratio 3:2 (this is how the scale is tuned) are not harmonious with the octave, of ratio 2:1. The simple mathematical truism which explains this discrepancy is that powers of 3 cannot coincide with powers of 2 (mathematicians would qualify this with the phrase "integral powers").

The proper ratio then, if one wants to come out harmoniously with the octave, could be expressed

  3 --


How much do fifths have to be shrunk to make the system work? The ancients, unused to irrational numbers, had no way of expressing this modification mathematically. Today, in the tuning method known as Equal Temperament, a logarithmic value is used. It was a Chinese prince who, in 1584 A.D., first worked out the mathematics of this nagging problem - a vindication of Chinese mathematical skills, if one were needed.[ 12]

[*!* Image: Cattle scapulae at an ancient Chinese burial site. Scapulimancy was a frequent Chinese divination method]

[*!* Image: Is ther e a connection between the musical scale and divination methods of this type? The Tonal Wheel. Arrangements of Divining Scapulae.]

No one wrestled with the problem more indefatigably than the Chinese, whose legends indicate concern with this tuning dilemma before the middle of the third millennium B.C. In a sense it was, among other things, an awareness of the scale problem that marked off the cultured, civilized society of antiquity from the barbaric and uncultured contingents. Not only the Chinese but others, such as Plato's followers, expected those in positions of political power to be aware of scale tuning and acoustic science, so that they might govern equably and properly. Such was the gravity of the situation and only an enhanced understanding of mathematic laws and principles was going to remedy it. And it is this dependence upon mathematics that makes the scale problem kin to that of the calendar in early scientific thought. Both were paradoxes, and the more one studies the two problems the more one is aware of how balanced they must have seemed in the prehistoric mind.

The tones of music are 12, and they bulge the octaves at the seams. The months of the year also are 12, but instead of crowding the year they fall short. Much of the efficacy of duodecimal mathematics hinged, in the archaic view, upon a resolution of this strange double paradox. And it was neo-Pythagorean elements, ultimately, who sounded the death-knell by championing decimal methods of reckoning throughout Europe. Ten-base and decimal metrication were on the threshold of sweeping that planet. The only drawback in this development has been that modern man is now almost totally oblivious of his cultural roots. If the veil is to be lifted, it will be up to the astronomer and the musician to lift it.


1. Edith Borroff, "Ancient Acoustical Theory and Pre-Pythagorean Comma," College Music Symposium XVIII, 2 (Fall 1978), p. 21.

2. R.T. Zuidema, "The Inca Calendar," in A. Aveni, ed., Native American Astronomy (Austin: University of Texas Press, 1977), p. 227 3. Ibid., p. 226

4. "The inability to connect two kinds of relationships is overcome (or rather replaced) by the positive statement that contradictory relationships are identical inasmuch as they are both self-contradictory in a similar way!' This statement occurs in his essay, "The Structural Study of Myth," which has appeared in several anthologies on structuralism and myth. See for example R. and F. de George, eds., The Structuralists (Garden City, NY: Doubleday Anchor, 1972), p. 180.

5. Fred Fisher and Gerald Edmundson, "Acoustics and the Mesoamerican Calendar," Interdisciplina I, 2 (Winter 1975-76), p. 52, note 5.

6. Alban Wall, "The Stonehenge: What Is It?" HORUS 1, 2 (Summer 1985), p. 7.

7. Ibid.

8. Alexander Marshack, The Roots of Civilization (NY: McGraw-Hill, 1972).

9. Marcel Granet, La pensee chinoise (Paris: A. Michel, 1968), p. 230-231. Translation by the author.

10. Zuidema, p. 224.

11. Ernest McClain, The Myth of Invariance (NY: Nicolas Hays, 1976), p. 76.

12. Joseph Needham, Science and Civilization in China vol. IV part 1, section 26 (Cambridge University Press, 1962), pp. 220-224.

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