Dice and the Tarot trumps

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.