In the Golden Dawn system, 7 of the Major Arcana are associated with astrological planets, and 12 of them with zodiac signs. For the logic behind these correspondences, see my old post "Why Waite switched Justice and Strength"; essentially, it's a slight modification of a tradition based on the Hebrew alphabet and laid out in an appendix to the Jewish mystical text Sepher Yetzirah.
Mapping the 22 Major Arcana to the 22 letters of the Hebrew alphabet and importing traditional Kabbalistic correspondences was a relatively straightforward operation. When it came to the Minor Arcana, though, the Golden Dawn had to create a new system more or less from whole cloth.
The aces and court cards were not given astrological correspondences per se but only elemental ones, as each suit and court rank was associated with one of the four classical elements. Pentacles and Pages mapped to Earth, Swords and Knights to Air, Cups and Queens to Water, and Wands and Kings to Fire. Thus, for example, the Ace of Pentacles was called "The Root of the Powers of Earth" and was mapped to the three Earth signs (Taurus, Virgo, and Capricorn). The Page of Pentacles was "Earth of Earth," the Knight was "Air of Earth," and so on.
There still remained the 36 non-Ace pip cards, 9 in each suit. Each of these was assigned to a particular Planet/Sign combination -- Jupiter in Capricorn, Mars in Taurus, etc. There are 84 such combinations, of which 36 were (seemingly arbitrarily) selected for representation in the Tarot.
This makes for a rather spotty system. For example, if I want to look at the cards that correspond to my birth chart, I find cards for Moon in Libra (Two of Swords), Venus in Aquarius (Five of Swords), and Mars in Pisces (Ten of Cups), but that's all. My other planets (Sun in Pisces, Mercury in Aries, Jupiter in Cancer, and Saturn in Virgo) are not represented at all.
You can also see that, once you know the 2, the 5, and the 8 of each suit, you can derive the remaining correspondences from these. When the planets are put in their Ptolemaic order (as I have done in the chart above), the pattern is obvious. So the question is, How are the 2, the 5, and the 8 assigned? How is it decided which maps to which of its suit's three signs, and how are the planets chosen?
Well, when you tabulate the correspondences as I have done, with both the planets and the signs in their traditional order, the pattern becomes pretty obvious. Ignore the suit signs for now and just focus on the numbers, starting with 2 at Mars in Aries. Go left, then left again, and then diagonally down-left. Left, left, down-left. Keep doing that, wrapping around to the other side when you reach the left edge of the table, and you've got all your numbers. Then add suit signs based on the elements. Fire signs are Wands, Water signs are Cups, and so on. And that's that; that's the whole system.
The only remaining question is, Why start with Mars in Aries? My guess is that it's simply because Aries is the first sign and is traditionally "ruled" by Mars.
