Quote:
Originally Posted by Michael Sternbach
Am I getting this right that one of the things we are looking for is a scheme that combines two of the Major Arcana to form a Minor?

Ah, well yes, there does end up being something like this; essentially there's what amounts to a multiplication table of majors with majors. If you look down a column in the table, you would find eight places where a minor arcana card occurs.
Eight times 21 would be 168, not 56, so for each minor, there are three different ways of forming it as a pair of majors. So actually for each minor, there is a set of three majors, call them A, B, and C. A*B = the minor, as does B*C, as does C*A; they go in a circle. (B*A or C*B would get us a different minor.)
Quote:
Originally Posted by Michael Sternbach
If so, we can derive the 36 numbered Pips from the Trumps by using their astrological correspondences according to the Golden Dawn system (I will be referring to their Thoth names in the following as these are the ones that come naturally to you and me). For instance, the Three of Wands equals the Sun in Aries, so that would be the Trumps XIX and IV combined.
The Aces belong to the Fifth Element or Spirit, which as such is represented by The Magician. The other elemental cards are The Fool = Air, The Hanged Man = Water, The Aeon = Fire, The Universe = Earth, so you would simply combine each of those with The Magician. That The Magician is also representing Mercury doesn't need to concern us, since we won't combine Mercury with any of the elements (only with signs) to derive any of the other Pips, so there will be no crossover. The Universe also has a second attribution, which is Saturn, but this doesn't need to bother us either here, as we combine The Magician as Spirit and The Universe as Earth with each other to get to the Ace of Disks.

Hmm. I think I can make the Magician do that, except the part with the Fool. Since I'm using that card as the identity element, it simply becomes whatever it's combined with.
Ah no, actually I just played with some examples and it doesn't work. The result was interesting though. Right now I have five or six potential ways of splitting the 56 minor arcana into four suits. It looks like no matter which method I use, the Magician combines with other majors to make a set of minors which are of at most two suits! Four from each of those suits. (EDIT: Wait, no, I'm not sure now. May be possible to get four different suits for some or all majors... I need to redo the calculations.)
So unless I find a different way of splitting them up, each numbered major will be in a sense associated with exactly two suits.
Since looking at things that way picks out eight minors for each major, maybe I should say the cards governed by the emperor and empress give us the 16 court cards!
Quote:
Originally Posted by Michael Sternbach
Things get a little trickier when we consider the Courts. For instance, the Prince of Cups is Air of Water, so we would think of the combination The Fool/The Hanged Man, but the Queen of Swords is Water of Air or The Hanged Man/The Fool, that's the same combination, even though in reversed order. I don't think that we can simply swap the two constituents of a mathematical element in this scheme (as far as I understand it).

Not so! For the most part, combining elements in opposite order will always get something different. But it will always stay the same *type* of card, major or minor (or root or outer). The only time when changing order won't matter is when combining two majors gets another major. The higher symmetry of the majors prevents the result from changing.
In the case of the minor arcana, if two majors A*B make a minor, then the reverse order, B*A, make the inverse of that minor. (Each minor has a different minor as its inverse; the two combine to make the fool.)
Quote:
Originally Posted by Michael Sternbach
Moreover, in four cases we would have combinations of an element with itself, such as Air of Air (The Fool/The Fool) for the Prince of Swords.
We could work around most of these difficulties by assuming the planets to be equivalents to the elements though, as is done both in traditional astrology and Tarot. Thus we consider Mars = Fire, Venus = Water, Jupiter = Air, Saturn = Earth, and these are at once the elemental attributions of The Tower, The Empress, Wheel of Fortune, The Universe.
So we now understand the Prince of Cups to be Fortune (Air/Jupiter) combined with The Hanged Man (Water), whereas the Queen of Swords is The Empress (Water/Venus) combined with The Fool (Air).
The only problem we are still left with seems to be that Earth of Earth (Princess of Disks) is the Universe combined with itself.

Quite interesting.
Though this doesn't quite add up in terms of the card combinations the group defines, what I'm finding is that there are a lot more ways of combining two elements of a group. For example the whole problem with the fool not changing anything is solved if we choose some 'mediator' card for these specific operations. Suppose for example the chariot were the mediator. Then there may be a way to have Magician combine with Chariot and *then* Fool, Hanged Man, Aeon, or Universe, and have these be the four aces.
Or any number of other tricks. I'll have to try a few.
I should be able to finish my more detailed explanation post (with animations!) on Thursday. In the mean time here are some more random notes on the structures I've been finding.
Arrange the 21 numbered trumps in a circle, in order. The sets of three "related" cards I've been talking about occur in the following manner. Choose a "template", which is a set of three numbers such as {1, 2, 3}. The only requirement in your choice is that one of the numbers must be a multiple of 3; one must be a (possibly different) multiple of 3 minus 1; and one must be a multiple of three minus 2. For example {4, 8, 12} would be a valid template.
The template represents how many cards you are going to go forward (clockwise) from a starting point. But the starting point cannot be anywhere on the circle: you start on a card whose number is a multiple of 3.
The three cards you land on by using the template are related; to use my earlier terminology, they are part of the same Aset. There are 7 sets of 3 you can get from a single template.
The "Bsets" are found using a different template, but it's related to the first. Add either {0, 3, 9} or {0, 6, 9} to the first template in any order, for example {4, 8, 12} could become {13, 11, 12} or {4, 14, 21}, by adding either {9, 3, 0} or {0, 6, 9}. Whatever you choose, this new template will define a valid choice of Bsets.
Choosing two such templates determines which group elements the major arcana stand for. So really, choosing which majors are related to any one major determines which ones are related to all the others.
Also, these sets of three majors have influence over the minor arcana. For each set of three majors, there are eight minors which belong in the set too. I've become pretty sure these 8 cards consist of two minors from each suit.
Most of the tricks I try to make suits give me 8 suits of 7 cards each. So I need to find an elegant way of matching them up with each other. But suggestions as to how the Tarot deck's suits elegantly split in half would be appreciated too.