CHAPTER TWO: SIXES
5. DELPHIC DICE
COMBINATIONS WITH TWO DICE
CHINESE DICE AND ARROWS
THE CUBE OF SPACE
II. 5. DELPHIC DICE
Hermes the messenger and culture-carrier brought the art of writing and the science of number, the mason's craft of piling one stone upon another (and much else) to archaic Greece, including a magical cube: the die. The complex and frequently contradictory nature of Hermes suggests that he is a very old god. Some elements (among them dice and gaming) reveal him to be a cultural adaptation of the god Thoth, the Egyptian lord of magical lore. In some Greek myths, it is Hermes who gets credit for inventing dice, which--as god of shepherds--he is said to have fashioned from the astragaloi, or the knucklebones from either sheep or goats. In another, even earlier tradition, a Mediterranean precursor of Hermes, perhaps closer to the Egyptian Thoth, used the more nearly cubic heel bones of the boibalis, or Libyan antelope. In The White Goddess, Robert Graves relates tali, or dice, to divination and the alphabet. The figure of Talus, one manifestation of the lame-king type, like the biblical Jacob or the Fisher King from the Grail legend, was associated with smithcraft:
Talus in one account was the son, or maternal nephew, to Smith Daedalus, in another was forged in the furnace of Smith Hephaestus. Dionysus, because of his titles pyrigenes and ignigena ("engendered by fire")--a reference to the autumnal Toadstool-Dionysus engendered by lightning--may have been equated with Talus in this sense....Mercury was not only patron of dice-players but prophesied from dice. He used five dice with four markings on each, in honor of his Mother, precisely like those given an Indian King at his coronation in honor of the Mother; and if, as I suppose, he used them for alphabetic divination he had his own alphabet of fifteen consonants and five vowels. The game of hucklebones is still played in Great Britain with the traditional set of five. In the case of six-sided dice, however, three made a set in ancient times; these would provide the diviner with eighteen letters of the alphabet, as in the thirteen-consonant Beth-Luis-Nion [alphabet].
[Robert Graves, The White Goddess, p. 330-332. ]
In a different study, Graves says the Proverbs of Zenobius hold that:
Athene was first credited with the inventions of divinatory dice made from knuckle-bones, and these came into popular use; but the art of augury remained an aristocratic prerogative both in Greece and at Rome....The [later] Apollonian priesthood constantly trespassed on the territory of Hermes, an earlier patron of soothsaying, literature and the arts; as did the Hermetic priesthood on that of Pan, the Muses, and Athene....The barbarous Hellenes took over and exploited, in the name of their adopted god Apollo, the Creto-Helladic civilization which they found in Central and Southern Greece--boxing, gymnastics, WEIGHTS AND MEASURES [our emphasis], music, astronomy, and olive culture were all pre-Hellenic--and learned polite manners.
[ Robert Graves, The Greek Myths, Section 17, Volume I, p. 66 f.]
The patriarchal subjugation of olive culture and its related institutions toward the end of matriarchal, pre-Hellenic times may also help explain certain logical and poetic anomalies in the choice of weapons by Odysseus (as an avatar of Hermes) for blinding the Cyclops Polyphemus. For, as pointed out by Professor (and Master) Denys Page,
the Odyssey, alone among all versions of this folk-tale, substitutes a log of olive wood for the spit....Why was the nature of the weapon changed? We can only guess.
[ Denys Page, The Homeric Odyssey (The Mary Flexner Lectures delivered at Bryn Mawr College, Pennsylvania), Clarendon Press, Oxford (1955, and William Clowes, London, 1966) pp. 9 ff.]
We will guess that the one eye of the giant is a "blind," symbolizing the vulva of Athena (or Athene), the archaic goddess of the Olive, in whose honor a sacred olive tree continued to be cultivated in a courtyard of the Erechtheum temple on the Acropolis down through classical times. The most appropriate instrument to express the violent conquest of matriarchal religious institutions by the usurping male priesthood (as an emblematic rape of this particular aspect of the goddess) was therefore likely to have been an olive wood stake.
After a period in which six matriarchal Greek states under the rule of Hera, and six patriarchal States under the rule of Zeus, agreed to an amphictyony or federation, civil war broke out.... This seems to have been the occasion on which Athene--who had not taken part in the war--was forced to disavow her control of handicrafts, such as pottery and weaving, and to claim that she had been reborn from Zeus's head as the Goddess of Wisdom.
[ Robert Graves, "What has Gone Wrong?" Difficult Questions, Easy Answers, p. 115. ]
Athena was also well-established at an early date in Lybia. From the Libyan Lake Tritonis, the hero Perseus (as one of the heroic avatars, or embodiments in human form of the divine Hermes, the messenger and culture-bringer) passing conveniently by the Phoenecians along the way and landing at Mycaenae ("the place of the mushrooms"), brought to the Greeks the magic of the alphabet, the letters for which he carried in the wallet adorned with the aegis of Athena: the image of a Gorgon's mask, which was "to emphasize the secrecy of his lesson." Perseus/Hermes wrested from the Three Graeae--or was given by them--a single magic eye:
This eye symbolizes the gift of perception: Hermes is enabled to master the tree-alphabet, which they (the three Graeae, Athene as Triple-Goddess, or the Three Fates) have invented. They also give him a divinatory tooth, like the one used by Fionn in the Irish legend; a sickle, to cut alphabetic twigs from the grove; a crane-skin bag or wallet in which to stow these safely; and a Gorgon mask to scare away the curious.
[ Graves, Greek Myths, Section 73.9, Volume I, p. 245. ]
In his compendious studies of the Delphic myth and its origin, the late Professor Joseph Fontenrose drew attention to the account, in the Homeric Hymn to Hermes, of divination by psephoi, mantic pebbles or lots:
the evidence for a lot Oracle indicates the use of pebbles (psephoi) in the tripod basin.
[ Joseph Fontenrose, The Delphic Oracle: Its Responses and Operations With a Catalogue of Responses, University of California Press, Berkeley (1978), p. 222. ]
The poet Diane Ackerman, with her charming incantation on the sense of hearing, in a section with a title ("Jaguar of Sweet Laughter") borrowed from the Popol Vuh, a book sacred to the Maya, and immediately following her observation that "the Saxon word for lovemaking was 'fuck' (from Old English fokken..."), develops the sonic, self-referential association, supplying a delightful etymology:
Sounds so captivate us that we love hearing words rhyme, we like their sounds to ricochet off one another. Sometimes we prefer words to sound like what they mean, in the aural equivalent of a pun: hiss, whisper, chirp, slither, babble, thump. The word murmur makes us murmur just to say it, which is why these lines by Alfred, Lord Tennyson sound so perfectly full of a summer glade:
The moan of doves in immemorial elms,
And murmuring of innumerable bees.
The Greeks called this phenomenon "onomatopoeia," but there are forms of it so subtle that their origin has disappeared into etymological history. For example, the word "poet" comes from an Aramaic word that denotes the sound of water flowing over pebbles.
[Diane Ackerman, A Natural History of the Senses. Random House, New York (1990), p. 183-184.
Professor Fontenrose supposed the sortilege with psephoi to be the only pre-Apollonian (that is, around B.C. 750) oracular activity to have taken place at Delphi. According to myth, Hermes learned his oracular technique from three honey-eating Parnassian nymphs, called the Thriai, most likely identical with the Muses--first three, then later nine--whose attentions were captured by Apollo. Hermes received his initiation in the divinatory art of casting lots with dice (probably three of them) in the Corycian cave on the slopes of Mount Parnassus, a site that Fontenrose once thought was the primordial location of the Delphic oracle itself. So Hermes' astragaloi may have been one and the same as the original "pebbles" of the Corycian cave, the bones of the demon/god called Typhon, Python, or Dionysos. The name of the cave--probably because of its internal shape--was derived from corycus a large leather bag filled with flour, fig grains or sand and hung up in the gymnasium for boxing practice: a punching bag. The theory of an archaic oracle at the cave was given up by Professor Fontenrose,
since recent excavations show the cave unused between Mycenaean times and the sixth century B. C.
[ Fontenrose, Delphic Oracle, p. 4, n. 4. There was another, even older Corycian cave in Asia Minor, one chasm of which was considered an approach to the underworld, p. 406-409. See also, by Joseph Fontenrose, Python: A Study of the Delphic Myth and Its Origins, University of California Press, Berkeley (1959), Chapter XIV, "The Corycian Cave," and "The Three Sisters of Parnassos," pp. 426 ff. ]
In their typical early form, Greek astragaloi (dice) were approximately rectangular in shape. They were not likely to land on their ends, since one of these was more or less pointed, and called keraia, while the other was unnamed. Only four sides were marked, of which two were broad and two narrow. Of the broad sides one was convex (pranes), the other concave (uptia); these were marked to represent the values three and four. The narrow sides could be differentiated because one was flat (chion) marked "1." The other was indented and as the rarest was considered the luckiest (koon) and marked "6." This helps to explain another difference between modern dice and those of ancient Greece in that it was the underside of the die which counted, not the side facing up. As dice came to assume a regularized cubic shape, all six sides would come to be marked.
[ Harry Thurston Peck, Harper's Dictionary of Classical Literature and Antiquities, Cooper Square Publishers, New York (1963), p. 1521 f.]
COMBINATIONS WITH TWO DICE
The most familiar expression of the number six in the province of solid geometry is the 3-dimensional cube with six faces. A cubic die on which all six faces are distinguished--as commonly used in games--obviously has just six probabilities for each cast. In the teeth of Einstein's merely wistful assertion that "God doesn't throw dice," an entire discipline studying probability and statistics has been erected (without which much of modern physics could not have been done) based instead upon the assumption that He sometimes does...or at least that He might. Starting with the simplest case, casting lots by using a single cubic die, the analogical mind could associate with each of these possible values all sorts of fanciful correspondences with either "the world of things six in number," or "the world in its sixness."
Dice supplied the instrumentality for generating both the oracular wheel of fortune and all forms of playing cards. In one form or another, dice are also among the oldest known implements used for sortilege, or the casting of lots. Not surprisingly, therefore, dice were regarded as religious in nature long before they were used as secularized equipment for vulgar gambling and games of chance. The person who performed the sortilege and read the lots in Greece was known as the Hierophant; his name means just that: "one who shows or reveals the secret or the mystery." This title originally had been given exclusively to the High Priest presiding at the Eleusinian Mysteries, where the Great Gods were honored.
Since the early Renaissance in Europe, "The Hierophant" has been the name given to the Tarot card of the Major Arcana (also called trumps or "Triumphs"). To this particular card is allotted the cardinal number five--usually indicated by the Roman numeral "V." In the lore of Qabala (variously spelled, as Cabala or Kabbalah, etc.) this number and card both came to be associated with the letter vau (or vav) as written in Hebrew. Nevertheless, a connection with sixness is borne out because in the sequence of cards in the Major Arcana, the Hierophant is the ordinal sixth, since the unnumbered or "zero" card (the Essence, conventionally but incorrectly called the Fool) is counted as the first, that is, in order. The archetypal image of the Hierophant, as one who shows the secrets and reveals the mysteries, in Western European iconography was drawn to resemble the Pope. This card is further interpreted as symbolizing the distractions of externally adoring sacred knowledge, which block the Essence from assimilating and embodying it as genuine wisdom, or internally realizing the Unity. At another level, this card represents the teacher herself/himself, the proverbial Gypsy Master who knows how to roll the dice and deal the cards, or the "mediumistic being," in the words with which Marcel Duchamp, himself, chose to define the essential function an artist.
The object symbolically associated with the Semitic form of the letter vau is a tent-peg, hook or nail, also recalling for cross-cultural Renaissance Qabalists, the Crucifixion. But when read in Hebrew, the letter is equivalent to "and," the English conjunction. The Greek word kleis KEY, and the Latin clavis "nail," are related lexically; and through the Latin we get CLEF, CLOVE, CLAUSE, CLOSE, CLOISTER, CLOISONNÉ, CLAVICLE, CLAVIER, CEMBALO, CHORD and other words, including (through clava) CLUB. The Indo-European root is klu, which (probably through the Germanic, Old English and Frankish forms) yields LOT, LOTTO, LOTTERY, ALLOT. The connections can be appreciated in Greek, where klerousis means "election by lots." In vulgar Latin the sortiarius was the "caster of lots," from the Latin sors, lot or fortune, as with the diviner of lots in China, the "reader," or sortilegus, casting the I Ching.
The primary function of apportioning lots, for the Greeks, belonged to Lachesis among the three Moirai or Fates. Apportioning was understood as the primary action, in that the unspun wool would be apportioned by weight before being given to a slave; then it could be checked again by weight when the work was finished. It was this executive function later usurped by Zeus, preserving the archaic metaphor of a length of thread with the length of one's lifetime.
[ Richard Broxton Onians, The Origins of European Thought: About the Body, the Mind, the Soul, the World, Time, and Fate (New Interpretations of Greek, Roman, and kindred evidence also of some basic Jewish and Christian beliefs). Cambridge (1954). Part III, "Fate and Time," & Chapter II, Peirata, p. 335).]
The word for a die--a cube, like the gaming piece--comes from the proto-Indo-European zero-grade root do, through the Latin datum, "that which is given." The derived French form of the word appears in Marcel Duchamp's posthumous work, a twenty-year secret, Etant donnés. Closely related to datum is the Latin verb dare which can mean both "to give" and "to play." The English language received the word "die" in this sense through the Old French dé, "a playing piece." If nothing else, With Hidden Noise, like so many of the other Readymades, is a playing piece: a curious puzzle, a perplexing intellectual and aesthetic enigma, an involvement with chance and fun. Although the poet and critic Octavio Paz addresses it to Duchamp's magnum opus, we mention here his opinion that
The direct antecedent of Duchamp is not to be found in painting but in poetry: Mallarmé. The work that most closely resembles the Large Glass is Un coup de dés.
[ Octavio Paz, Marcel Duchamp: Appearance Stripped Bare, translated by Rachel Phillips & Donald Gardner, Viking, New York (1978), p. 77.]
It should be perfectly obvious that with one cubic die, in the casting of lots there are six probabilities. Not quite so obviously, with two cubic dice, twenty-one different numerical combinations can be obtained, assuming the dice are so similar as to be indistinguishable; this comprises what mathematicians call "the unordered set." However, with dice of different colors, say, it is easily demonstrated that there are, in fact, thirty-six different combinations, "the ordered set" (since 6 x 6 = 36).
The number 36, the ordered set of combinations possible with two dice, is regarded as a solar number in mythology and traditional lore. One illustration of this is the famous glyph of the Minoan Labyrinth associated with Daedalus and the Minotaur at the ancient capital Knossos on the island of Crete. Drawn in the characteristic way--as it has been shown for years on the cover of Daedalus, the quarterly Journal of the American Academy of Arts and Sciences--the archetypal maze requires precisely thirty-six turns to penetrate the convoluted space from the outside and to arrive (like Theseus) at its heart, presumably the lair of the Minotaur. Theseus then had to retrace his steps and in another thirty-six turns, following the clew given to him by the most holy Ariadne, to emerge once again into the light of the sun from the magical prison, an emblem of sleeping consciousness.
Theseus, the Attic solar hero, thus completes his journey to the Underworld--or the voyage within, to the depths of the psyche--in seventy-two steps, which is no mere accident. If the average number of heartbeats per minute is taken to be seventy-two, we may observe that the ratio of 72:60 also has a cosmic correspondence with the mean rate of the precession of the equinoxes: since the rate of precession is reckoned to be fifty seconds of arc per year, the sun will precess sixty minutes (one degree) of arc in seventy-two years, the length of a full lifespan, known as our human eon, also spelled "Aeon," or in the transliterated Greek form we use here to specify this sense, aion. Relating all this in a cultural frame of reference, the late, distinguished scholar of folklore, mythology and cultural history, Joseph Campbell points to:
the observable fact that at the moment of the Spring equinox (March 21) the heavens are never quite in the position they were the year before, since there is a very slight annual lag of about 50 seconds, which in the course of 72 years amounts to 1 degree (50" x 72 = 3600" = 60' = 1 degree) and in 2160 years amounts to 30 degrees, which is one sign of the zodiac.
[ Joseph Campbell, The Masks of God: Oriental Mythology (Volume II), p. 117.]
Recent scholarship has begun explore the intriguing and long-overlooked subject of the precession of the equinoxes: an ancient and recurrent, if at times obscure, theme both in world literature and in the history of science. For example, another illustrative legend concerns the Jews of Alexandria who celebrated an annual festival,
the excuse for which was that the Five Books of Moses had been miraculously translated there into Greek by seventy-two doctors of the Law (`the Septuagint') who had worked for seventy-two days on them, each apart from the rest, and agreed exactly in their renderings at the conclusion of their task.
[ Graves, The White Goddess, p. 276. See also, Giorgio de Santillana and Hertha von Deschend, Hamlet's Mill: an essay on myth and the frame of time, David R. Godine, Publisher, Boston (1977). ]
Dice are the original means for generating the odd and even, black and red (and even the green) slots of the modern roulette wheel, such as that appearing on Duchamp's Monte Carlo Bonds (1924). The thirty-six combinations of the ordered set are represented on the roulette wheel in a traditional array, each of the numbers corresponding to a physical slot distinguished by color as either red or black. It is not clear to the present author how the colors have come to be distributed among the numbers, although the principle may be analogous to the distribution of numbers spatially among the pie-slice sections of a dart board, as the attempt to achieve an approximately mean sum for the values of adjacent sections. Taking the example of a simple 36-sectioned or 36-slotted wheel, the odds of the spun ball landing in any given numbered slot are 1:36. If a payoff is offered at 35:1, then the wagering between the house and the bettor would almost fairly break even, a push. Marcel Duchamp's betting strategy--a sort of martingale (doubling the stakes after each loss) was to have been financed with the proceeds raised from the sales (rather slow) of the Monte Carlo Bonds (1924). The entire scheme was designed to just break even, possibly as an exercise in statistical inframince ("infra-thin"), Duchamp's outré aesthetic.
On certain roulette wheels another slot may be added; it is often numbered zero (0) and traditionally colored green. This new slot changes the odds against a bettor to 1:37, so that if the payoff remains at only 35:1 there is a thicker slice of statistical pie for the house. Clearly then if the same payoff rate is maintained, every new slot added to the wheel, such as a double zero (00) or triple zero (000), has the effect of increasing the house margin. This may provide a modern, venal reason for adding zeros to the wheel, but there could also be some deeper rationalization as well, if we were to take the zeros as emblematic of dice originally used to generate the roulette wheel values: two dice (for 36 probabilities) represented by the 0 and the 00 slots. But what about the triple zero? Interpreting the zeros as transformations of dice, we may read them not as the values of particular throws, but self-referentially, as symbols of the system-generating dice, themselves descended from the mantic psephoi in the oracular basin at Delphi. But why three? Three is the magic number par excellence, that of the Thriai on Mount Parnassus and other emanations of the great female Triple Goddess: the Three Graces and the Three Muses or, in their more shadowy aspect, the Three Fates. But what is the system generated by the casting of three dice?
Contemporary scholars are beginning to understand better the old relationship between dice and cards. We know that the casting of cubic dice generated all forms of playing cards, and also that the modern bridge deck evolved from a tradition of the Tarot pack. The European history of the cards can be traced back to the early Renaissance, or possibly as far back as the Middle Ages. However, both the individual representations on Tarot cards and the idea of a system made to symbolize the structure and function of the psyche may be very much older, and must be sought for perhaps "as far as China." For despite being surrounded by a great deal of ignorance, superstition, and wishful or willful misunderstanding, the tradition of the Tarot provides generally useful representations of the psyche that seem to ring true at the archetypal level. This probably accounts for the continuing respect the Tarot cards receive from many people, who regard the full deck as one version of the Book of God.
What survives today of the Tarot tradition may indeed represent a reconstitution of Cumaea's windblown, scattered leaves of prophesy, renowned in the classical account of Virgil's Aeneid, leaves that are cited in the last Canto, at the highest level, of Dante's Paradiso, the loose leaves that reappear in the later visions of William Blake and Ezra Pound, and in Gerard Manley Hopkins's poem, "Spelt From Sibyl's Leaves." We might also discover this image in the leaves of the geometry textbook of Duchamp's Unhappy Ready-made (1919), on the Crotti's balcony in Paris (like Bob Dylan's answer, my friend) "blowin' in the wind." Recall that Dürer's engraving Melancholia I, the very saturnine "pondering" figure of Geometry-personified--created when that German artist had grown weary of painting--was printed on single, loose-leaf sheets.
The traditional pack of Tarot cards contains seventy-eight cards, which are technically of three different kinds: related to one die, two dice and three dice. The first of the cards corresponds to the single die: it is usually unnumbered, or numbered "zero" (and sometimes incorrectly as XXII). In the genuine esoteric teachings of the Tarot this card, properly called the Essence, always comes first; the so-called "zero" card is accorded primacy as the first of eighty cards, symbolizing the Divine Essence within every human being. Rather surprisingly--perhaps as a "blind"--it may be known in vulgar convention by many misleading and frequently derogatory names.
In the early Italian sources, it is invariably called il matto (the Madman), its usual name today. In Piedmont it is called il Folle (the Fool) and in Sicily, il Fuggitivo (the Fugitive). In France it was originally called le mat, a term without any other meaning (save the irrelevant [sic!] one of checkmate in chess). Subsequently, it was called l'excuse, from its role in the game. In German-speaking lands this was corrupted to der Sküs, an otherwise [to Dummett, anyway, apparently] meaningless term.
[ Michael Dummett, The Visconti-Sforza Tarot Cards, Braziller, New York, 1986, p. 100. Madness could be considered psychic checkmate. See also by Michael Dummett, with Sylvia Mann, The Game of Tarot: from Ferrara to Salt Lake City, Duckworth, London (1980). This volume records yeoman efforts in gathering information about playing cards and variations of what the authors take to have been a "particular type of card game." With respect and gratitude for this rich store of information, we have come to a different understanding about the nature and origins of Tarot. At the present writing there appears to be no single book in print that adequately addresses this important and largely misunderstood topic; it would be a privilege of high order to essay. ]
If the Tarot is a "game" as supposed by Mr. Dummett then it must be in the sense of Johan Huizinga"s Homo Ludens or Hermann Hesse's Magister Ludi; still, the connection between cards and chess cannot be "irrelevant" for us, given Duchamp's lifelong addiction to the latter. It is worth noting that Marcel Duchamp's principal written contribution to the literature of chess deals with a technical situation infrequently arising in the end game, but explicitly concerned with the issue of checkmate.
[ See L'Opposition et les cases conjugues sont reconcilies by Duchamp and V. Halberstadt, limited edition, Edition de l'Echiquier, Paris and Brussels (1932). Michel Sanouillet and Elmer Peterson, in "A Disclaimer" say their collection does not include Duchamp's "writings on chess, which are too technical for most readers," (Salt Seller, p. viii). ]
The group of cards called the Major Arcana or "trump" cards, which should be counted as 24, is usually composed of 21 cards, plus the "zero." Often these are distinguished by Roman numerals and bear some additional name or title. Obviously, the unordered set of twenty-one combinations that can be thrown with two dice was used to generate these trump cards. The unordered set may also figure in the history and logic of other card games like "21," or blackjack. Robert Graves describes the number 21:
being a number sacred to the Sun since the time of the Pharaoh Akhenaton who introduced into Egypt about the year 1415 B.C. the monotheistic cult of the sun's disc.
[ Graves, The White Goddess, p. 274.]
Twenty-one (21) is one of the triangular numbers, so-called because the can be displayed as the sum of successive integers. Similarly, the fifteen balls arranged in a triangle in the standard rack on a pool table: 1 + 2 + 3 + 4 + 5 = 15. This is another way of saying that 21 represents the sixth term in the integer sequence associated with the most basic combinatorial number, called the binomial coefficient. In the domain of number theory, they are also referred to as "the coefficient of binomial expansion," which figured in Sir Isaac Newton's "first important invention in mathematics," an expression of the binomial theorem for negative and fractional exponents (leading to infinite series), which he called the Rule, and initially communicated to Leibnitz.
[ N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York and London (1973), p. 17 and Sequence 1002. Isaac Newton, "On the Binomial Theorem for Fractional and Negative Exponents, (Letter of June 13, 1676), Commercium Epistolicum (1735 ed., p. 131-132); reprinted in James R. Newman, editor, The World of Mathematics, Simon and Schuster, New York (1956), Volume I, with Commentary, p. 519 ff. More recently, see Sloane's successor to the remarkable Handbook: N.J.A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego (1995), especially sequence M2535, with accompanying figures illustrating some of the polygonal numbers. Triangular numbers have a general form that may be represented as: P(r,1) = 1/2r(r+1), or n(n+1)/2.]
With three dice, 216 is the total number of combinations that can be thrown: "the ordered set," which is easier to appreciate when each die is of a different color. We may recall this number (times ten, or with the simple addition of a zero in place value notation, i.e. in the form of 2160) as the number of years for the sun to precess through one sign of the zodiac. Such a period, as approximately one-twelfth of the whole Precession of the Equinoxes, is taken to represent a zodiacal "age," as in the "Age of Aquarius," or the "Age of Pisces," and so forth. But in "the unordered set," the number of different combinations cast with three indistinguishable dice is 56. So we see that the unordered set was used to generate the non-trump card of the Tarot deck: the Minor Arcana is composed of 56 cards--each of them originally corresponding to a specific combination determined by casting three dice. In the full deck, all three of these "generative" dice are each represented by a "wild" card instead of only one wild Joker as in the pack of the Fool.
The group of 56 cards--collectively called either the Minor, or sometimes the Lesser Arcana--is composed of four suits of 14 cards each. It is from these, and not from the Major Arcana, that our modern bridge deck derives, along the way having subtracted one court card from each suit. The suits are called wands or batons (clubs), pentacles or discs (diamonds), cups (hearts), and swords (spades). In the hierarchical sequence of the Tarot tradition, the suits are ranked in the same ascending order as in modern contract bridge. The 14 suit cards are made up by: 10 numbered cards and 4 court cards (king, queen, knight and page) rather than the royal 3 (king, queen and jack) of today's common deck that now contains only 52 cards. For the Minor Arcana, a traditional order of cards within each respective suit first considers the interrelationship of the four court cards, then the ten numbered cards counting downward from the ten to the deuce, then the ace which in each respective suit is accorded extraordinary value as a token of the Divine Grace that requires human receptivity.
CHINESE DICE AND ARROWS
The many different forms of Chinese playing cards derive their designs from several distinct sources, among which dice is paramount, while others include chess, the counting of cash by coin and string, and lexical or literary associations. To the 21 combinations with two dice, "the Chinese, with affectionate solicitude," have (like Western gamblers) given vernacular names; the names of the first two, we note, reiterating names of the first two hexagrams (kua) from the I Ching:
Double sixes = heaven
Double aces = earth
Double fours = man
Ace and three = harmony, or the geese
Double fives = plums Double threes = long threes or door chain
Double twos = the bench
Five and six = tiger's head of the chopper
Four and six = the screen
Ace and six = the water bucket
Ace and five = the hammer
These first eleven values listed above are known as civilians, and the last ten which follow are termed the military:
Four and five, three and six = nines
Two and six, three and five = eights
Three and four, two and five = sevens
Ace and four, two and three = fives
Two and four = bighead six
Ace and two = three chickens.
The order in which they are here given is the order of their value, which it is of great importance to remember. It is, in fact, one of the earliest pieces of useful information acquired by the young...of either sex, in China. This order...is the same for both dice and dominoes: indeed it seems open to considerable doubt whether we should not, from the peculiar standpoint of the Chinese, regard dice, equally with dominoes, as playing cards. As this seems to verge on paradox it shall not be pressed, but it may be conveniently noted here that Chinese dice are sold not in pairs but in sets of six, and that the ace and four spot are coloured red, the other spots black. Both peculiarities are copied into the dominoes or cards derived from them.
[ The British Museum, Catalogue of the Collection of Playing Cards Bequeathed to the Trustees of the British Museum by the late Lady Charlotte Schreiber, Longmans, London (1901), p. 185 f. ]
In his compendious and marvelous work, Joseph Needham includes extensive analyses of divinatory techniques and their historical impact upon scientific thought in China, underscoring the importance of dice with unique authority.
One of the oldest of such procedures was the "pitch-pot" game, the throwing of arrows into a pot....It would only need markings or numbers on the arrows to have an object which by compression would become a cubical dice, and this again by extension or unfolding would give rise to dominoes on the one hand and playing-cards on the other. Cubical dice are ancient, examples having been found in Egypt and India, and from Graeco-Roman times; it is generally supposed that they reached China from India, and this we may provisionally accept. (The transmission must have occurred quite early. Waley has suggested that the prominence of the number six in the Book of Changes was derived from the six sides of a cubical dice.) But it is also now rather well established that dominoes and playing-cards were originally Chinese developments from dice. There are indications, says Carter, that the transition from dice to cards...took place about the same time as the transition from manuscript rolls to paged books. These cards, at their first appearance towards the close of the Thang [or, T'ang Dynasty], must have been among the earliest examples of block printing. After the beginning of the Sung, their evolution forked into two directions, one leading to playing-cards as we know them, the other to dominoes, from which again in its turn the famous game of "Mah Jongg" derived. But while the story of playing-cards in China includes firm dates as early as [A.D.] 969, when one of the Liao emperors had card games at court, the earliest reference in Europe is 1377 in Germany. ...The whole question is important with regard to the origins of block printing, for the Chinese cards had long been printed when the Europeans came first in contact with them, and it seems that from 1400 some of the European cards were also printed. The earliest European dated religious prints, the Virgin and Child of 1418 and the St. Christopher of 1423, coincide closely with St. Bernardino's famous sermon against card-playing. Carter concludes that playing-cards have a very important place in the transmission of the art of printing to Europe.
[ Needham, Science and Civilisation in China, Volume IV, "Physics and Physical Technology," p. 328 f. Needham's references are to Arthur Waley, "The Book of Changes," Bulletin of the Museum of Far Eastern Antiquities (Stockholm, 1934), 5, p. 121; and to T. F. Carter, The Invention of Printing in China and its Spread Westward, Columbia University Press, New York (1925, 2nd edition 1955). ]
Professor Needham ends this particular volume of his brilliant series with a summation including the following points (here selected) that are relevant to our interests:
(1) The Game of chess (as we know it) has been associated throughout its development with astronomical symbolism, and this was even more overt in related games now long obsolete.
(4) Numbered dice, anciently widespread, were on a related line of development [to chess] which gave rise to dominoes and playing-cards (9th century China).
(19) Magnetical science, unlike Euclidean geometry and Ptolemaic planetary astronomy, was an essential component of nascent modern science the antecedents of which were not primarily Greek. All the preparation for Pierre de Maricourt, and hence for the ideas of Gilbert and Kepler on the cosmic role of magnetism, had been Chinese; and they in their turn, with their belief that gravity must be something like magnetic influence, were an important part of the preparation for Newton. The field physics of still later times, firmly established in Clerk Maxwell's classical equations and more congruent with organic thought than Greek atomic materialism, can again be traced back to the same root. Much is owing to the faithful and magnificent experimenters of medieval China.
[ Needham, Science and Civilisation in China, Volume IV, p. 332, 334. ]
THE CUBE OF SPACE
One student of the Tarot, after referring to the revival of interest in the Renaissance tradition during the 19th century, provides a general reference for the traditional associations for the cards of the well-known Waite deck:
In the main, I am in agreement with Arthur Edward Waite, who discusses the various theories as to its [the Tarot's] origin in his Key to the Tarot, published in London by Wm. Rider and Sons. Dr. Waite concludes that the Tarot has no exoteric history before the fourteenth century. The oldest examples of Tarot designs now preserved in European museums were probably made about 1390. According to an occult tradition, in which I am inclined to place confidence, the actual date of its invention was about the year 1200 A.D. The inventors, this tradition avers, were a group of adepts who met at stated intervals in the city of Fez, in Morocco. After the destruction of Alexandria, Fez became the literary and scientific capital of the world. Thither, from all parts of the globe, came wise men of all nations, speaking all tongues. Their conferences were made difficult by differences in language and philosophical terminology. So they hit upon the device of embodying the most important of their doctrines in a book of pictures, whose combinations should depend on the occult harmonies of numbers.
[ Paul Foster Case, The Tarot: A Key to the Wisdom of the Ages, Macoy, Richmond, Virginia (1947), p. 2-3; for the Hierophant, see p. 74 f. A new edition of this book's interesting approach contains color illustrations, but unfortunately they seem to be based on relatively modern European models. It would have been far more informative had the author been inclined to publish additional information relating to the early paper-making and printing activities that can be associated with medieval Fez, or the archaic instances of cubic temples, such as those atop ziggurats of the 3rd millennium BCE.]
Although relying on some problematical references, Mr. Case's representation of the "Cube of Space," in the context of his description and analysis of the Major Arcanum IV (the Emperor), does present information on the directions and orientation of the cube, proposing a system of associations extended so as to include the 22 Hebrew letters of the traditional Qabala.
These directions are given in The Book of Formation, and have to do with a very important esoteric teaching, in which the manifested universe is presented as a cube, shown in occult diagrams with its western and southern faces visible to the observer....The six faces of this cube and its interior center are assigned to the seven double letters of the [Hebrew] alphabet. The three interior coordinates correspond to the three mother letters [A = aleph, M = mim and SH = shin]. The twelve boundary lines represent the twelve simple letters.
[ Case, The Tarot, p. 64 f., especially p. 67-68. ]
Salvador Dali is easily the best known of twentieth-century artists who have applied themselves to designing a Tarot deck with 78 cards; and although 78 is the most familiar conventional number, for perhaps inappropriate to discuss in the present context, we believe that Dali, like many before him, was not playing with a full deck of cards. In graphic terms there have been some very sumptuous packs of cards designed or painted by major Renaissance artists, including the famous series by Andrea Mantegna. Several of the Major Arcana also were copied by Albrecht Dürer from a set of Italian Tarocchi. But perhaps the real treasures among earlier Tarot cards were those painted on thick cardboard, with gold embossing, by Bonifacio Bembo, the favorite painter of Bianca Visconti-Sforza, for whom they were painted some time around 1450.
[ Rachel Pollack, Salvador Dali's Tarot, Salem House, Salem, NH (1985). For Mantegna, see Erica Tietze-Conrat, Mantegna: Paintings, Drawings, Engravings, Phaidon, New York (1955). For Dürer, see Panofsky, Dürer, p. 31. For Bembo, see Gertrude Moakley, The Tarot Cards Painted by Bonifacio Bembo for the Visconti-Sforza Family: An Iconographic and Historical Study, The New York Public Library (1966); and Michael Dummett, The Visconti-Sforza Tarot Cards, George Braziller, New York (1986). Also published with reproductions of this deck was Italo Calvino's text, The Castle of Crossed Destinies, in a gorgeous edition by Franco Maria Ricci (1969); and more modestly, as translated by William Weaver, Harvest/HBJ Book, San Diego (1977). Professor Dummett provides a useful bibliography in The Game of Tarot, p. 70, n. 24.]
Other writers have sought to relate Duchamp to the Tarot while apparently under the considerable handicap of not having done any real work within a living, authentic Tarot tradition and--so it would seem-- without guidance from an authoritative teacher. These efforts are not necessarily frivolous; there is always the chance that one may stumble across the secret teachings, and discover "by accident" some valid clues to the mysteries of the ancients. Or, looking at it the other way, the teachings of the Tarot tradition (precisely because they address the archetypal level of consciousness) might well serve as an effective instrument of elucidation for any system--whether organic, mechanical, electronic, or theoretical--only if it be sufficiently complex and cybernetically structured.
With Marcel Duchamp's elusive work known as The Large Glass, or The Bride Stripped Bare by Her Bachelors, Even, (La Mariée mise à nu par ses célibataires, même), given the assemblage of weird and unorthodox materials it should come as no surprise that some aspects inevitably correspond with the archetypal imagery and symbolism of the Tarot. In view of that monumental piece's detailed, arcane iconography, its richly allusive frame of self-reference, and one the most complex and disingenuous titles in the history of art, we might find it much more surprising if none did. For example, there happens to be a card in the Major Arcana of the Tarot called "The Hanged Man," and there is, as we know, a Pendu femelle element in the Large Glass...but, with all due respect, we might ask "So what?" May we surmise, because Duchamp's "hanging thing" is explicitly female in gender, a correspondence with the sibylline reference of "The Waste Land's" dedicatory Latin quotation: "in ampulla pendere"? Eliot called that arcanum by name in his 1922 poem, for The Hanged Man was among the cards read by "Madame Sosostris, famous clairvoyante...known to be the wisest woman in Europe." But the gender does not correspond any better than the date (although Duchamp did not declare The Large Glass definitively incomplete until 1923). Anyway, the poet was not necessarily knowledgeable about the mysteries of the Tarot. Indeed, this is made clear in the notes Eliot himself provided, where we may read an honest, modest, and cautionary disclaimer:
I am not familiar with the exact constitution of the Tarot pack of cards, from which I have obviously departed to suit my own convenience. The Hanged Man, a member of the traditional pack, fits my purpose in two ways: because he is associated in my mind with the Hanged God of Frazer, and because I associate him with the hooded figure in the passage of the disciples to Emmaus....
[ T. S. Eliot, "Collected Poems 1909 - 1935," The Complete Poems and Plays 1909 - 1950, Harcourt, Brace, New York (1960), p. 51. ]
In sum, it appears likely that neither Eliot nor Duchamp knew beans, spilled or otherwise, about the Tarot. But Duchamp did display a continued interest concerning things in the domain of Hermes: all aleatory phenomena, chance, odds, probability, numbers, chess, gambling, games, play, humor, wit, puns, irony, paradox, and secrets. For our interest in With Hidden Noise, we may need only recall Eliot's line 117:
What is that noise?