From the Narrow Desert

In a recent post, Kevin McCall wrote,

We might compare [a particular hypothetical distribution of IQs] to rolling all 5 Platonic solids: one 4 sided die, one 6 sided, one 8 sided, one 12 sided, and one 20 sided, which would have a completely different distribution from rolling five 6-sided dice.

And I commented,

Would rolling a mix of polyhedral dice really result in anything significantly different from a normal bell curve. I haven’t done the calculations, but my assumption is that it would not.

Then I almost immediately retracted this statement ("Never mind. I've checked it, and my assumption was totally wrong!") because I'd put the possible rolls of three dissimilar dice (a d4, a d6, and a d8) into a spreadsheet and it had given me a histogram that looked nothing like a normal distribution.

But now I have to retract that retraction and reaffirm my original assumption. The weird-looking histogram is an artifact. Rolling the three dice mentioned yields one of 16 possible values, from 3 to 18, but the spreadsheet software (Google Sheets) for some reason made a histogram with only 13 bars. Most of the bars represent a single value, but 3-4, 9-10, and 15-16 are grouped together, which is why those three bars are abnormally tall. Making a 16-bar histogram by hand, I find that rolling mixed dice does after all yield a normal bell curve.

The moral of the story: If it comes down to trusting either your own instincts or the basic competence of Google programmers, go with your own instincts every time!

But you already knew that.

One of the cups on the Magician's table in the Tarot de Marseille -- the one marked A below -- was originally a dice shaker, probably a leather one.

This emerged suddenly as an unarguable intuition after many hours spent picking over the details of the Magician's table in early Tarot decks. I don't expect my own intuitions to carry much weight with anyone else, though, so here is a bit of circumstantial evidence to back me up:

The cup has a strange square shape -- vertical sides, horizontal bottom -- unlike a normal cup. In the Viéville deck it is unambiguously rectangular in shape, including the mouth. Compare the strangely shaped "cup" in Viéville to the backgammon dice cups and the medieval British dice shaker below.

Leather is brown, a color not included in the standard 8-color palette of the Tarot de Marseille (an exception is the François Héri deck of 1718, which uses it only for the Hermit's habit). What color would be used for leather, then? Well, belts and shoes are normally made of leather, so that should give us a clue. Looking at the 12 decks in Historic Tarots gallery at the Tarot of Marseilles Heritage website, 8 out of 12 use the same color (yellow) for the Magician's belt, shoes, and cup; and each of these individual elements is yellow in 10 out of 12 decks.

Incidentally, this syncs up with the Pythagorean Tarot of John Opsopaus, which, without any pretense of restoring the original TdM, patterns the Magician after the December illustration in the Chronograph of 354 and puts a rectangular purgos, or ancient Greek dice-shaker, on the table. (The picture below is from the original card drawn by Opsopaus himself; the published version, done by another artist, has a round purgos.)

In his article "Tarot Divination Without Tarot Cards" (qv), John Opsopaus proposes a system of correspondences between the 56 cards of the Minor Arcana and the 56 possible rolls of three dice. This is analogous to his systems for mapping the 21 trumps to the 21 possible throws of two dice -- but is necessarily more complicated because the Minor Arcana are structured in a way that the trumps are not. While the trumps are numbered in a linear fashion, the Minor Arcana are grouped into four suits, each of which has 10 numbered (or "pip") cards and four face (or "court") cards.

Just as 21 is a triangular number, 56 is a tetrahedral one. Since 10 is also a tetrahedral number, the pips of each suit can be assigned to the smaller tetrahedron at one of the corners of the larger one. Once this is done, there remain 16 points at the center, arranged in the shape of a truncated tetrahedron, and these can be assigned to the courts.

The diagram below (which is my own work but is based closely on Opsopaus) shows how the 6th tetrahedral number can be divided into four smaller tetrahedra (red, yellow, green, blue) and a truncated tetrahedron (purple). Purple points represent court cards, and the other four colors represent the four suits of pips.


This is a great way of dealing with the pips, but the problem is that the courts are also divided into four suits, and there seems to be no natural way of quartering our central truncated tetrahedron.

One way of dealing with the courts is to associate each of the court ranks with one of the suits -- which has traditionally been done by way of mapping both the court ranks and the suits to the four classical elements. One popular system is Kings/Clubs/Fire, Queens/Cups/Water, Knights/Swords/Air, Knaves/Coins/Earth.

Notice that our truncated tetrahedron is made up of four hexagons, each of which faces one of the four pip-tetrahedra. For example, in the diagram above, the top surface of the truncated tetrahedron is a hexagon, comprising the rolls {334, 344, 335, 345, 355, 346, 356}, and facing the red tetrahedron. If we assign the red tetrahedron to the suit of Clubs, say, which is associated with the court rank of Kings, then the seven rolls on the red-facing hexagon will correspond to the seven court cards which are Clubs and/or Kings. The roll 345, which is in the center and which thus belongs exclusively to the red-facing hexagon, corresponds to the King of Clubs. Each of the remaining six rolls is shared with one of the other hexagons. For example, the rolls 355 and 356 belong to both the red-facing and the blue-facing hexagons. If we assign blue to Cups, corresponding to Queens, then one of these rolls will be the King of Cups, and the other will be the Queen of Clubs; we can perhaps assign the higher toll, 356, to the former on the grounds that a King should outrank a Queen.

These examples are just examples. I have not yet thought out which tetrahedron should correspond to which suit or any of the other details. Nevertheless, Opsopaus's basic schema seems to have a lot going for it.

In past posts, I have discussed possible ways of pairing the 21 Tarot trumps with the 21 possible rolls of two dice, concluding in the end that the "Air Hexactys" system was the best. In this system, dice rolls are ranked first by total (e.g., any roll that totals 7 outranks any roll that totals 6); and then, among rolls with the same total, by the higher of the two numbers rolled (e.g., among rolls totaling 7, 1-6 outranks 2-5, which in turn outranks 3-4). The dice rolls, thus ranked from lowest to highest, were then paired with the trumps from 1 to 21. This system seems to "work" and to make symbolic sense, as briefly discussed in my post "The root trumps of the Air Hexactys."

Now, the main reason for assuming a link between dice rolls and Tarot cards is the fact -- unlikely to be a coincidence -- that there are 21 possible rolls of two dice (2d6) and 56 possible rolls of three (3d6), corresponding precisely to the 21 trumps (the practice of numbering the Fool and counting it as a trump is a relatively recent development) and the 56 suit cards (Minor Arcana). The next step, therefore, is to try to find a dice-to-cards mapping that works for the Minor Arcana as well as the Air Hexactys works for the trumps.

Here are some of the challenges facing anyone who would attempt such a mapping:


Four suits: The trumps have a simple linear structure, being numbered from 1 to 21; all that is required is to establish a linear ranking of the dice rolls, which is not difficult. The Major Arcana, on the other hand, are organized in four suits. There is no obvious way to divide up the 56 possible rolls of 3d6 into four equal categories.


Rank within each suit: In most traditional games played with Tarot cards, the long/black suits (swords and clubs) are ranked, beginning with the lowest, A 2 3 4 5 6 7 8 9 10 J C Q K -- the same ranking used in most modern card games, except that aces are low. However, the round/red suits (cups and coins) rank the pips (but not the court cards) in reverse order: 10 9 8 7 6 5 4 3 2 A J C Q K. Although there are a few exceptions (most notably the French jeu de tarot), I think we can take this idiosyncratic ranking system as a very old and probably original feature of the Tarot pack.


Relative rank of suits: Due to the quirk just mentioned, it seems that if we want to rank all 56 Minor Arcana linearly, we have to group them by suit first and then by rank within each suit. It wouldn't make sense, for example, to begin with the four aces, then the four deuces, and so on, because long aces are low but round aces are high. I suppose we could begin with the long aces and round tens, then the long deuces and round nines, etc., but this seems very unnatural.

The problem with ranking by suit first is that Tarot games do not rank the suits. Players must always either follow suit, play a trump, or discard a card. In no case can cards from two different suits be played in the same trick, so the question of whether, say, the King of Swords outranks the King of Clubs never arises.

Modern "esoteric" or divinatory tarot is not a trick-taking game, so "rank" as such is not an issue. Nevertheless, it has become customary to think of coins/pentacles as the "lowest" suit, followed by swords, then cups, and finally clubs/wands as the highest. Waite's chapters on the Minor Arcana in The Pictorial Key to the Tarot, for example, begin with the King of Wands as the highest card and proceed down the ranks to the Ace of Wands; then the Cups, from King to Ace; then the Swords, and finally the Pentacles. (Waite makes tens high and aces low for all suits, contrary to the Continental tradition of which he was perhaps ignorant; Tarot has never been a game in England.)

Historically, the four suits do have a natural rank, since they can be traced back to Chinese suits representing different denominations of money. Coins, which have preserved their character as single coins, are the lowest denomination. Next come Clubs, which were originally strings of 100 or 1000 coins. (Ancient Chinese coins had a hole in the center and were strung together.) Cups derive from the Chinese character for 10,000; and Swords, from the character for ten -- meaning, in context, ten myriads, or 100,000. This ancient ranking has been preserved in many modern Anglo-French-suited games, where Spades (Swords; cf. Italian spade) are the highest-ranking suit, followed by Hearts (Cups), Clubs, and finally Diamonds (Coins). (Bridge, in which Diamonds outrank Clubs, is an exception.)


Relation to trumps, especially root trumps: Because each of the trumps has been associated with a roll of 2d6, each roll of 3d6 will be associated with between one and three of the trumps. For example, the roll 1-4-6 would be associated with 1-4 (Lover), 1-6 (Hanged Man), and 4-6 (Sun) -- or, alternatively, with the three root trumps associated with 1, 4,  and 6 -- namely, Magician, Death, and World. For rolls where the same number occurs three times, only one of the trumps will be linked; for example, 4-4-4 should be related to Death. (The Four of Swords would seem a natural choice for this roll.)


Sparse symbolism: Compared to the trumps, the Minor Arcana contain relatively little imagery. Prior to Waite's innovative "scenic pips," a card like the Seven of Swords portrayed nothing more nor less than seven swords (stylized as arcs in the Marseille tradition). Such cards have no immediately obvious meaning, and the traditional meanings ascribed to them vary widely. This makes it hard to judge whether or not a particular dice-card pairing is a "good" one. This can be viewed either as a bug (because it offers little guidance) or as a feature (because it allows the cards to accommodate a wide variety of systems).

Chinese-style dice; the ace and the four always have red pips, with an extra large one for the ace

Rolling five six-sided dice (5d6) yields one of 26 possible numbers, from 5 to 30, which means it can be used as a means of randomly selecting a letter of the alphabet. Of course, the possible results of a 5d6 roll vary widely in frequency -- with, for example, 17 being 780 times as frequent as 5 -- approximating a bell curve. The letters of the alphabet also vary in frequency, though not quite so widely as 5d6 results, and not in so close an approximation of a normal distribution. Still, if the most frequent letters are mapped to the most probable 5d6 results, the results seem to be serviceable enough. Here's the key for converting numbers to letters.


And here's how the letter frequencies associated with this key match up against the actual empirical frequency of each letter in English.


Using this system, you can play (a reasonably close facsimile of) Scrabble without having a board or tiles. You just need five dice, a sheet of graph paper (with squares large enough to write in), some scrap paper, pencils, and a copy of the key. Instead of drawing tiles, roll 5d6 and jot down the the letter that corresponds to the number rolled. When you "play" a letter by writing it in a square on the graph paper, scratch it off and then roll the dice to "draw" a replacement. In a classroom setting, use a whiteboard instead of paper.

Having read my earlier posts on the subject (here and here), Kevin McCall has the following thoughts on mapping the Tarot trumps to rolls of the dice.

As far as the association of cards of the Major Arcana with dice rolls, it seems like another way into the system rather than an ordering of die rolls could be by considering the symbolism of the numbers 1 – 6 and then trying to associate each number and each pair with a card.  After reading about Pythagorean number symbolism and tarot symbolism, here are some of my thoughts (highly speculative):

I think the Air Hexactys is the best of the four, but perhaps with some modification:

We know the Magician is (1,1), Priestess is (1,2), World is (6,6) and Judgment is (5,6).  I think that 1 must mean magic or beginning, 2 female or passivity, 3 male or activity, 4 terrestrial but in a negative sense, 5 combining 2 and 3 as representing balance, and I think Opsopaus is right that 6 represents finality and the celestial.

In that case, the Magician (1,1) would be pure beginning
The Empress (2,2) would be pure feminine, and the Emperor (3,3) pure masculine.
Then, it makes sense that the Priestess is (1,2) for magic, feminine and then the Hierophant “should be” (1,3) for magic, masculine.  Then, the Lovers “should be” (2,3) for masculine and feminine coming together.

Justice “should be” (5,5) as complete balance.  The Chariot (3,5) combining activity, victory with balance and self-control.  The Hanged Man (1,6) at the apex of the pyramid makes sense because I think this card represents the midpoint and beginning (1) and ending (6) coming together.   Temperance should be (1,5) for wisdom combined with balance, i.e., good judgement.  Fortitude should be (2,5) for passive balance, resistance rather than the activity represented by the Chariot.

The Devil, Death, and the Tower should all have 4, since these are all cards with a negative connotation, i.e., “terrestrial” cards, with terrestrial in a pejorative sense.  Death as (4,4) as given in the Air Hexactys makes sense because if 4 represents change, time, corruption, then Death is of all the cards, most representative of this archetype.  I’m not as sure about the Devil and the Tower, but these could be (2,4) and (3,4) respectively, with the Devil being passive corruption, sin (in a “Luciferian” sense, as Steiner would put it) and the Tower combining a dramatic change (with 3 as activity) and 4 as change but in a destructive, harmful sense.  The Hermit as (1,4) can also make sense, in particular if Opsopaus’s version of the Hermit as Time is correct, in either case representing either beginning in an earthly sense, caused by time or a lonely search for wisdom in the world.

I think Opsopaus is right that the World, Sun, Moon, Star, Judgement, and Hanged Man are all celestial cards, so they should all have a six somewhere.  (3,6) for Sun makes sense because this would be masculine/celestial.  (2,6) for Star gives contemplative/celestial.  (4,6) for Moon would mean the moon is bridge between earthly and celestial.  And, the idea of the sublunary realm, with the sphere of the moon being the lowest heaven, the boundary between nature and the Heavens makes sense for this correspondence. 

We can view Judgement and Wheel of Fortune as a pair.  Judgement representing is perfectly fair and final judgement, while the Wheel of Fortune is arbitrary and fickle.  Fortune metes out rewards and punishments on earth, while Judgement does in Heaven.  Then, Fortune should be (4,5) corresponding to Judgement as (5,6).

*

My comments:

McCall's approach here is similar to what I did in my post on the Tarot-astrology correspondences worked out by the Golden Dawn, which resulted in reordering the Justice and Strength cards. Before considering the accepted (Marseille) ordering of the Arcana and how to align it with the accepted (Sepher Yetzirah) ordering of the planets, signs, and elements, I looked at each card and considered what astrological correspondences, if any, seemed to make sense for it, irrespective of traditional ordering schemata. I could then evaluate the Golden Dawn mappings against these "natural" correspondences. Dice-Tarot mappings like those given by McCall here, based on qualitative considerations without taking into account the order of the trumps or the ranking of rolls, could play a similar role vis-à-vis the various systems I have been examining, helping to choose among the four Hexactyses (and, for that matter, among the various historical orderings of the trumps, of which the now-standard Marseille sequence is just one; Opsopaus's original Fire and Water Hexactyses were in fact applied to a variant of the Ferrara sequence).

Regarding the specific mappings proposed by McCall, I am in broad agreement with him as to the basic symbolic meaning of 1, 2, 3, and 6. However, the traditional meaning of 4 is rest, stability, stasis -- quite far from McCall's "change, time, corruption." (Interestingly, despite our near-opposite understandings of 4, we are agreed that it is appropriately mapped to Death -- which can be seen either as the ultimate change or as final rest.) I also have trouble accepting 5 as "complete balance"; For an odd number to represent balance is, well, odd. Five more often represents disruption, breakdown, crisis -- but also creativity, novelty, transcendence. Basically, it's something completely different being added to the stable arrangement represented by 4 and shaking things up.

By the way, while McCall thinks the symbolism of 2 and 3 as feminine and masculine means that 3-3 "should be" the Emperor, but I find the Triumphal Chariot even more appropriate. The charioteer is just as much a man as the emperor is, and I find the achieved status of the conquering hero to be far more archetypically masculine than the ascribed status of a passive throne-sitter.

I'll probably revisit this idea later and consider what numerical associations seem, irrespective of "ordering" considerations, most natural and appropriate to me.

Trying to judge the relative merits of the four possible systems (described here) for mapping dice rolls to Tarot trumps, I tried to find historical examples of the rolls of two or more dice being mapped to a linear series.

Remembering that John Opsopaus had mentioned (here) that "San Bernadino's sermon of 1243 draws an analogy between the 21 rolls of two dice and the 21 letters of the (medieval) Roman alphabet," I tried to track down the sermon in question (which turns out to be Contra Alearum Ludos, actually delivered in 1423 by St. Bernardino of Siena) to see in what order the rolls had been assigned to the letters of the alphabet. Opsopaus cites an article by M. G. Kendall (qv), which quotes Bernardino as follows: "Missale vero taxillum, esse volo: [. . .] in eius missali solum alphabetum, hoc est viginti una literae comprehendantur, ac totidem puncta in decio concludantur." Kendall translates the last bit as "just as that missal is composed of a single alphabet of twenty-one letters, so in the [game of] dice there are twenty-one throws," and comments, "The twenty-one possible throws are undoubtedly those with two dice." I find this interpretation unconvincing. Bernardino is comparing the missal to a single die (taxillum), and puncta obviously refers to the points on the die rather than to the number of possible throws of two dice. (Each face of a die is marked with a different number of points, from one to six, and 1 + 2 + 3 + 4 + 5 + 6 = 21). So, not only does Bernardino not list specific mappings from rolls to letters, but it seems unlikely to me that he had possible rolls of the dice in mind at all.

*

I had a bit more luck with an article by Fritz Graf (qv) on Greek oracular texts used in astragalomancy (the rolling of four-sided "dice" -- actually tali, the knucklebones of animals, with sides numbered 1, 3, 4, and 6 --  as a form of divination). The standard method was to roll five tali (5d4, to use D&D terminology), for a total of 56 possible rolls, and the texts list these possible rolls as a numbered list. While we're more interested in the 21 possible rolls of 2d6, these astragalomantic texts are still useful as an indication of how the ancients put dice rolls in linear order.

The first thing to notice is that the rolls are ordered according to their total value. Any roll that totals 15, for example, "outranks" -- i.e., corresponds to a higher number on the list than -- any roll that totals 14. This is a point against the "Fire" and "Water" systems of Opsopaus, discussed here, which rank rolls according to the value of the highest or lowest die rather than the total.

Among rolls with the same total, the ranking system is not so clear. Here are the relevant data (the asterisk marks a lacuna in the text, incorrectly restored by Graf and corrected by myself):

• 11134 > 11116
• 11144 > 11333
• 11136 > 11334
• 11344* > 13333 > 11164
• 11444 > 11336 > 13334
• 11346 > 33333 > 11166 > 13344
• 13444 > 33334 > 11446 > 13336
• 14444 > 33344 > 13346 > 11366
• 33336 > 33444 > 13446 > 11466
• 33346 > 34444 > 14446 > 13366
• 11666 > 33446 > 13466 > 44444
• 33366 > 34446 > 14466
• 33466 > 44446 > 13666
• 34466 > 14666
• 44466 > 33666
• 16666 > 34666

Update: I went back through the astragalomantic oracle text and examined all the pairs of rolls that have the same total and three tali with the same value -- in other words, pairs of rolls that are identical except that one has 1-6 where the other has 3-4. There are 20 such pairs. For 15 of the pairs, 3-4 outranks 1-6.

• 111-34 > 111-16
• 114-34 > 114-16
• 116-34 > 116-16
• 144-34 > 144-16
• 333-34 > 333-16
• 334-34 > 334-16
• 136-34 > 136-16
• 344-34 > 344-16
• 146-34 > 146-16
• 336-34 > 336-16
• 444-34 > 444-16
• 346-34 > 346-16
• 446-34 > 446-16
• 366-34 > 366-16
• 466-34 > 466-16

For the remaining five, 1-6 outranks 3-4.

• 113-16 > 113-34
• 133-16 > 133-34
• 134-16 > 134-34
• 166-16 > 166-34
• 666-16 > 666-34

I can't for the life of me figure out what makes these five different. I've tried everything I can think of, including poker-style rankings (seeing if, for example, three of a kind always outranks two pair or vice versa), but there just doesn't seem to be any pattern. My tentative conclusion is that my initial impression was right, and that rolls with the same total are listed in arbitrary order.

In the previous post, I discuss the idea that each of the 21 Tarot trumps originally corresponded to a particular roll of two dice and look at four possible systems of trump-dice mappings. Since John Opsopaus has already discovered two of them and dubbed them the Fire Hexactys and the Water Hexactys, I have used the other two classical elements to give corresponding names to the remaining two systems. Here I want to look in more detail at the system that seems to me to be both the most natural one: the Air Hexactys -- illustrated below using the Jodorowsky-Camoin version of the classical Tarot de Marseille trumps.


In the diagram above, the cards are laid out in 11 columns corresponding to the 11 possible values of a roll of two dice (from 2 to 12). Where two or more rolls have the same value, they are ranked according to the higher of the two numbers rolled. (Thus, for example, in the third column, the roll 1-3 outranks 2-2 and is placed above it in the diagram.)

The cards in the bottom row of the diagram correspond to the six doubles, from "snake eyes" on the left to double sixes on the right. These six trumps, then, indicate the basic meaning or character to be associated with each of the six faces of the dice; and the 15 remaining trumps represent combinations of these six basic elements.

For any trump in the diagram, following the two diagonal paths down to the bottom row will lead us to the two "root trumps" whose meanings it combines. Take, for example, the 8th trump, called Justice, corresponding to the roll 2-4. Following the diagonal path down and to the left leads us to 2-2, the Empress; following the other diagonal down and to the right leads us to 4-4, the trump with no name ("Death"). Justice, corresponding to a roll that combines 2 and 4, should represent some combination of the symbols and ideas found on the Empress and Death cards -- and such proves to be the case. Like the Empress, Justice depicts a woman seated on a high-backed throne, the shape of which is suggestive of a pair of wings; unlike the Empress, though, this woman is armed with a sword -- a deadly weapon corresponding to the Grim Reaper's scythe. In fact, it turns out that all three of the trumps that depict deadly weapons (the other two being the Lover and the Wheel of Fortune) are arranged in a diagonal line leading to Death. Likewise, all the trumps featuring crowned males are connected to the Chariot.

Many other such connections are evident.

The Hermit card depicts an old man carrying a lantern, and the dice connect it to the Magician and the Moon. The hermit is a wizardly figure -- another type of "magician" -- and his lantern indicates that he is traveling by night. The only person who ever carried a lantern by day was Diogenes the Cynic, known as "the Dog." Either way, the Moon card, with its night scene featuring dogs, is indicated.

The Wheel of Fortune is connected to the Chariot and Death. A chariot of course has wheels, and the charioteer wears a crown like the sphinx on the wheel. The sphinx's sword matches the Reaper's scythe, and in a broader sense both the Wheel and Death represent the ultimate futility of everything, and how people rise only to fall in the end.

The Tower -- which depicts a tower being destroyed and people falling to their deaths -- connects to Death and the Moon. The Moon card features towers.

Strength (a woman controlling a wild animal) connects to the Empress (a woman in control) and the Moon (wild animals).

The Pope has a crown and scepter (ferula) like the charioteer, and the two monks in front of him are in the same positions as the charioteer's horses. Like the empress (but unlike the charioteer and the emperor), he holds his scepter in his left hand.

The Hanged Man occupies a special position, at the apex of the triangle, and is linked to its two other corners, the Magician and the World. He is dressed in motley, as is the magician. His legs are in the same position as those of the dancer of the World, and like her he is surrounded by a stylized representation of the zodiac.

The ease with which these and other connections jump out at me from the diagram, suggests that the Air Hexactys constitutes a meaningful arrangement of the trumps and will repay further contemplation.

In my post on the Magician, I mentioned that the number of cards in the Tarot deck (56 suit cards, 21 trumps, and the Fool) likely has something to do with dice, there being 56 possible rolls of three dice and 21 possible rolls of two. (This was drawn to my attention by John Opsopaus, who got it from Gertrude Moakley, according to whom it was first discovered by Maurice G. Kendall.) Although we can probably assume from the numbers that each card originally corresponded to a particular roll of the dice, no record of those correspondences exists, leaving them a matter of inference and guesswork.

Matching the Minor Arcana to dice rolls is complicated by the latter's being divided into four suits, with no obvious way of so dividing the 56 possible rolls of three six-sided dice. The Trumps are somewhat more straightforward, though, since they represent a simple sequence from 1 to 21. All that is required is to rank the 21 possible rolls of two dice from lowest to highest. These possible rolls can be represented schematically in a triangular arrangement, as below, and our task is to convert that triangle into a one-dimensional sequence. (Opsopaus calls this representation of the sixth triangular number a hexactys, by analogy with the Pythagorean tetractys.)


Obviously, "snake eyes" (two aces, 1-1) is the lowest possible roll, followed by 1-2, and these will correspond to trumps 1 and 2 (the Magician and Papess in the Marseille ordering), respectively; at the other end, the highest two trumps (the Judgment and the World) answer to 5-6 and 6-6.

Beyond that, though, several systems are possible. When two dice are rolled, there are three numbers to consider: the number of the higher die, the number of the lower die, and the total. For convenience, I will call these the High, the Low, and the Sum. The ranking of the rolls will depend on which of these three numbers takes precedence over which. Take the third trump, for instance. Should it be 2-2 or 1-3? Both rolls have a Sum of 4, but 1-3 has a higher High, and 2-2 has a higher Low. Logically, there are four possible rules we could follow:

1. Rank rolls according to the High. Among rolls with the same High, rank them according to the Low or the Sum. (The ranking will be the same either way.)
2. Rank rolls according to the Low. Among rolls with the same Low, rank them according to the High or the Sum. (The ranking will be the same either way.)
3. Rank rolls according to the Sum. Among rolls with the same Sum, rank them according to the High.
4. Rank rolls according to the Sum. Among rolls with the same Sum, rank them according to the Low.

Opsopaus considers only the first two systems, saying that they both have their merits and that he is unable to choose between them. He calls the first the Fire Hexactys because he represents it as an upright triangle (like the alchemical symbol for fire) with 1-1 at the top and the sixes at the bottom. The second is the Water Hexactys because he represents it as an inverted triangle (like the alchemical sign for water) with the aces at the top and 6-6 at the bottom. In keeping with the elemental theme, I have dubbed the third system the Air Hexactys (because, as I have diagrammed it, vertically higher rolls outrank lower ones) and the fourth, the Earth Hexactys (for the converse reason).

The diagram below gives the mappings for all four hexactyses. The numerals are the numbers of the Tarot trumps, while the dice rolls are represented by the color scheme introduced above (where red is 1, orange is 2, etc.)


How to decide which of the four systems is the best? I consider the Air Hexactys the most intuitively natural mapping. When rolling dice, it's natural to focus on the Sum first. We think "I rolled seven" first; distinguishing the various "seven" rolls (1-6, 2-5, 3-4) is secondary. I also think it's most natural to consider the High before the Low -- just as in poker, if two players both have Two Pair, we look at the value of each player's higher pair first to determine who wins.

Another thing to consider is that the bottom row of each diagram consists of "doubles," which will be rolled only half as frequently as the other rolls. It makes sense that these less-frequent trumps should be "special" in some way -- as the Magician (1-1) and World (6-6) obviously are. The unnamed 13th Arcanum ("Death") is another obviously special trump, and the Air Hexactys (uniquely) assigns it to a double, 4-4. Appropriately, 4 is the number of death in East Asian cultures, just as 13 is in the West.

Other possible considerations include the trump at the apex of the triangle (another "special" position), and which trumps are adjacent to which others in the diagram. We might also consider whether the six rolls including a given number have anything in common. I will look at these points in detail later, to see whether or not my initial preference for the Air Hexactys is confirmed.