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On Number as Artifact (Part 3: Conclusion)
Fred Fisher

There is a longstanding cultural connection between the spiral and the labyrinth

In a pithy paragraph in his Prelude to Science, Richard Furnald Smith writes,

The first day of the lunar month may vary from place to place according to the visibility of the first crescent, but in general the system is simple and uniform. Anyone can tell the date in any given month simply by noting the phase of the moon. There is one maddening defect, however. The lunar months and the solar year do not come out even. Twelve lunar months are 11 days too short, while thirteen lunar months are 18 days too long.[1]

The numbers 11 and 12 play a prominent role in Homeric, Pythagorean, and Socratic thought - in fact, in Western culture generally. But try to find the numbers 13 and l& This corroborates what is already known of Western calendar-making, which since Mesopotamia and Egypt has always favored the 12month year. Interestingly, the pre-Columbian cultures in Mexico and Central America have shown a no less demonstrable affinity for 13 and 18, suggesting quite a different calendar orientation. This indicates the role of number as artifact and underscores the need for prehistorians to become aware of its significance.

Literally all pre-Columbian lore celebrates the number 13 - a number clearly not thought to be unlucky in early Central America. It was combined with the vigesimal 20 (the Mayans counted toes as well as fingers) to make the unusual 260-day ritual calendar. The 360-day "round yea? was reached, on the other hand, by multiplying 20 and 18. Thus both of the superfluity numbers cited by Smith, above, come into play in the Mayan system.

There is always an aspect of strangeness when encountering unfamiliar cultures for the first time. But there is nothing stranger about the pre-Columbians, from a European standpoint, than this infatuation with 13 and 18! If one can draw any conclusions initially at all, certainly one must feel that the Mayans were determined that the lunar-solar gap must be closed: the year must under no circumstances by left incomplete. The fact that what resulted was a gross amount of overlap was not to be denied, but metaphorically the concept of closure had to be honored.

We have already seen in Peruvian culture a way in which there can be 13 lunar months in a year. It occurs if one observes, not the synodic month, but rather the sidereal month of between 27 and 28 days. The Chinese established a system of 28 1unar mansions" or hsiu in accordance with the sidereal month idea, although there is nowhere near the evidence of Chinese interest in 13 that one finds in the Western hemisphere. But the Chinese had a saying that may shed some light: "The pitchpipes and the calendar give each other a mutual order, so closely that one could not insert a hair between them."[2]

To a musician there is no more important clue to ancient number speculation than this. When nothing else can be called up to explain a knotty problem in early science, this statement often stands forth with the needed key. The writer first came upon it in the book Hamlet's Mill - a work which may very well grow in reputation in coming decades as having been a seminal influence upon man's understanding of his cultural roots.[3] The sentence may have appeared first in the writings of Tai Te, first century A.D., but it was eventually incorporated into lore attributed to Confucius (born 6th Century B.C.). The evidence suggests that it is appropriate to Chinese scientific thought of no matter what era or dynasty.

In the previous installment we mentioned the calendar and the musical scale as being in a very real sense the two horns of a mathematical dilemma that involved the concept of irrational numbers. By and large early scientists found irrationals abhorrent. In the effort to circumvent them set-theory calculations were often pushed to stratospheric heights. In astronomy these involved the notion of the Great Year. In acoustics the meganumbers were the result of computations like the 30 circlings of the scale 12 X 30 = 360) that piqued Edith Borroffs curiosity in 1976. Meganumbers were invariably preferred to fractional numbers. But the fractions, often highly complex ones, were there nonetheless. They could not be escaped.

What lay behind the persistence? The answer is: closure was expected. In an orderly universe, how could the answer possibly be something so ragged and unconvincing as a part of a number! By thinking this way the ancient arithmeticians perhaps delayed by many centuries some much-needed progress in the direction of expertise - with logariths, for example. The scale problem is mathematically more difficult than the calendar because the only real solution to it is logarithmic (Equal Temperament). Always the spiral was extended upward, hoping thereby to find the desired closure between fifths (the tuning interval) and octaves. Meanwhile astronomers persisted in the supposition that, after a certain length of time, all of the heavenly bodies would return to a position of synchronization. Once again, the idea of closure.

Early thinking about both space and time was cyclic. Needham comments on this with regard to Chinese science:

The Chinese physical universe in ancient and medieval times was a perfectly continuous whole. Chhi condensed in palpable matter was not particulate in any important sense, but individual objects acted and reacted with all other objects in the world. . . . No justification is necessary for the statement that Chinese natural philosophers tended to think in terms of cyclical recurrences. The fact has been noted by many observers. In its simplest and oldest form it had to do with little more than the rhythm of the seasons and the rise and fall of the individual lives of men, yet Chuang Chou in the -4th century welded it with poetic fire into his Taoist philosophy of the ataraxic acceptance of the Tao of Nature.... In the words of the Tao Te Ching, 'returning is the characteristic movement of the Tao'.[4]

Needham goes on to observe that "the cyclical formulation was imposed from the start by the very subject-matter of certain sciences," citing calendar-making and the circulation of the blood, along with the water-cycle of meteorology and (one must presume) the behavior of ocean tides.[5] But it would not be reaching to suggest that music, which the Chinese took as a basis for their system of measurements generally, underlay and informed this view in large part. To quote Tai Te once again, "And all this is what is meant by saying that the root of all living creatures was also the origin of rites and music, and the maker of good and evil, as well as of social order and disorder."[6]

[*!* Image: The scale spiral. The calendar spiral]

A circle that does not close but rather redounds upward indefinitely is, of course, a spiral. In terms of integer mathematics, both the calendar and the scale are spirals. The fascination of Neolithic and Bronze Act. cultures with spiral motifs suggests that there was indeed some understanding of the integer problem, and perhaps a recognition that closure could never really be found. Spirals occur in the Mediterranean, in Celtic culture, in Africa and Central America. They are positively ubiquitous in China. Unfortunately, there tends to be a lack of appreciation for this and other forms of ornamentation in early art, the supposition being perhaps that the design carries no scientific signification. Closer to the truth might be the intimation that only recently has the scientific significance become recognized. Jill Purce sees in spiral motifs the "journey of the soul" - not exactly an objective view but nonetheless a typical assessment up till now.[7] J. E. Cirlot sees two types of spiral: the inward-winding and the outward-winding.[8] This is more helpful, since the lunar-solar problem corresponds clearly to the former, the scale problem to the latter. (See figure above)

Perhaps all analogies break down at some point. But the evidence is there for all to see, and it must mean something. The Greeks for example enshrined the double spiral in the capitals of their Ionic pillars.

[*!* Image: The Ionic double spiral]

The Chinese and the Mayans were lavish with them in their decorative patterns. There is in fact hardly a more pervasive symbol in early thought, and even so it is one about which few convincing explications have been advanced.

In both of the figures (page 3) the amount of miss has been exaggerated. The gap in the scale amounts to about 2%. The calendar miss is only slightly larger. Granted that musical tuning is not an easy subject concerning which to verbalize. The reader may wonder w familiar pitches such as F and Bb are spelled "E#" and "A#" respectively. This is done for acoustico-theoretical reasons. If the circle were joined at C the new tone would not be C but actually B#: too sharp, by an amount known as the Pythagorean comma. One can just as easily go around the circle in the opposite direction, i.e. toward flats. But the result will be the same, with a comma of exactly the same size occurring at the join.

Admittedly this is difficult and unfamiliar content for most readers, but the problem is not only technical. Acoustics - or what the Greeks call harmonike - is simply not the direction in which Western thought is moving these days. As one recedes deeper and deeper into the past, a common ground is found in which such matters can more comfortably be talked about. De Santillana linked music to myth and other ancient interests and deftly labeled it the "shop talk" of an earlier time. He was absolutely right.[9] The main point that needs to be made is that the two types of spirals connote two related conundrums. Today one concerns oneself with other pressing matters, but at one time both of these problems were of concern to thinkers, and neither was capable of being resolved within the then-existing parameters of mathematics.

This series of articles began by considering two ways of measuring the length of a month: synodic and sidereal. In the second article we adduced the problematic 13, a number not much liked by modem society by nevertheless one which seems to have figured rather prominently in megalithic thought. We raised the question, as we have just done with the double spiral, of what this obviously symbolic number could have meant.

It seems to have interchanged rather freely with the number 12. Today's scientist does not deal with number in this way, but there is what must be called an unsettling lack of specificity in dealing with numbers if one delves into the deeper cultural past. Again, Borroff is enlightening:

Perhaps the most puzzling to modem scholars is the use of different numbers for parallel symbolic and practical considerations - not the same numbers put to different uses, but different numbers, most significantly 3 and 3.1605, 12 and 13, 360 and 365. The Egyptian symbolic [pi] was 3, their practical [pi] was 3.1605: their symbolic circle of the year was of 360 days, but their practical year was the same as ours (and in Mesoamerica's ancient calendar, more accurate still; ancient Chinese texts held that the body contained 360 veins but proposed 365 acupuncture points; and open circle was twelve, the closed circle was thirteen (a double concept that we have inherited in the 'baker's dozen').[10]

Two astronomically respectable ways of achieving the lunar year of 354 days are 29 X 12 and 27-1/4 X 13. The first uses the standard synodic month, the latter the less familiar sidereal month. It was pointed out earlier that 28 X 13 is the "solar" year of 364 days and it was shown that this, too, appears ton have been a legitimate set of "sky numbers" among several early cultures including the builders of Stonehenge.

But it is difficult, even for a musician, to think of 13 as a "musical" number. It shouldn't be. Music happens to be one field in which both ends of a dimensional measurement are traditionally included in the measurement. It's undeniably a perverse situation, but undeniably it exists (no doubt it's one of the things that go toward making music an art today, and not a science). Distances in music are called "intervals". In the interval of a fifth, both ends are counted:


the same with the octave, a word that means "eighth;"


and the same with any other interval. But the ancients counted chromatically and would indeed have counted to thirteen in measuring the octave:

C C# D D# E F F# G G# A Bb B C

Ten years ago, in his epochal work The Myth of Invariance, Earnest McClain proposed a new line of inquiry - a realm of number theory "in which music sets the problems."[11] He recognized that "musical patterns elevate certain numbers to a prominence pure number theory would not accord them."[12] Even before this engineers, mathematicians, and astronomers were finding an unexpected source of fascination in the numbers and patterns inherent in the calendar circles of North America, England, and the European mainland, as well as in the temples and pyramids of Central America, Egypt, and the Orient. The numbers and patterns in the two cases are surprisingly the same.

Man may not have created numbers. Certainly many cultures believed them to be God-given. Ideas about number have changed. But the numbers themselves have stayed the same, and in many cases so have the problems to which they refer. No cataclysm of any imaginable sort could possibly act to alter the nature of acoustic science and the scale comma. they are there for all time: one of the great intellectual challenges in man's mental evolution.

Thus to track the history of thought it is now incumbent upon prehistorians, as Nicolas Rashevsky advocated, to turn to number. But not as a pure mathematician - rather as early man himself did. Two man's earliest sciences were harmonike and astronomy. There seems little doubt that modem-day musicians and archaeoastronomers have something to tell us.


1. Richard F. Smith, Prelude to Science (NY: Scribners, 1975), pp. 4142.

2. Joseph Needham, Science and Civilization in China vol. II (Cambridge University Press, 1965), p. 270.

3. Giorgio de Santillana and Hertha von Dechend, Hamlet's Mill (Boston: Gambit, 1969).

4. Needham, Science and Civilization Vol. IV part 1, section 26 (Cambridge University Press, 1962), pp. 8, 10.

5. Ibid., p. 11.

6. Needham vol. II, p. 270.

7. Jill Purce, The Mystic Spiral (NY: Avon Books, 1974).

8. J.E. Cirlot, tr. Sage, A Dictionary of Symbols (NY: Philosophical Library, 1962), pp. 290-292.

9. Santillana, p. 312.

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

11. Ernest McClain, The Myth of Invariance (NY: Nicolas Hays, 1976), p. 4.

12. Ibid.

Fred Fisher, who now resides in Denton, Texas, earned a Bachelor's Degree in music at Northwestern University, and both a Master's Degree in music and a Doctorate in Musical Arts at Eastman School of Music in Rochester, New York, in the field of piano. He taught at the University of Oklahoma for nine years and at North Texas University for 14. He has published in a variety of journals and magazines including the Cimmaron Review, Clavier, Connecticut Review, and Piano Quarterly. After retirement from university teaching in 1982, Dr. Fisher has continued his scholarship with a particular interest in the relationship between mathematics, acoustics and astronomy and in the understanding and application of these relationships in diverse cultures since ancient times.


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