The linear ranking of dice rolls

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.

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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

I can't make head or tail of this and almost suspect that there is no system to be discovered, that rolls with the same total are listed in arbitrary order. Looking at the first line above, 11134 outranks 11116; eliminating the three aces that the two rolls have in common, we can infer that 3-4 outranks 1-6. This suggests the "Earth" system, where rolls with the same total are ranked according to the Low. (See this post for an explanation of the terminology.) However, in the next two lines we can see that 1-4-4 outranks 3-3-3, and 1-3-6 outranks 3-3-4, which is inconsistent with that system.

I'll spend a little more time looking at the list in Graf's paper to try and find some pattern to the rankings. At any rate, the focus on the Sum first seems to support either the Air or the Earth Hexactys, as opposed to the systems proposed by Opsopaus.

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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.