Recognizing that this is wholesale speculation, but also recognizing the value of having a "story" rather than just a collection of plausible assertions, let me get literary (a la Moakley's imaginary Milanese parade scene), and tell the tale of tarot.
In about 1420, someone associated with the Milanese court (although probably not a member of the family), was inspired to create a new form of entertainment with cards. We can picture him as a young man, perhaps not a serious academician, but comfortable moving in educated circles and used to entertaining people with his inventive notions and witty remarks. Perhaps he was a university scholar or young gentleman of means, with a penchant for diversions and a somewhat wry view of the society in which he lived. He loves the carnival processions, his set of pictures from Petrarch and dans macabre, and he knows Dante (though maybe never actually read the whole work) and a bit of classical myth. He's glibly cynical about the powers of the world, regarding the papacy as corrupt (loyalties having been disrupted by the Great Schism, still in living memory), and the Holy Roman Empire as an ineffectual puppet of self-serving German princes. He has an active imagination, and so is attracted to apocalyptic prophecies, magical secrets hidden in the lore of numbers, and the like.
His inspiration is to put the allegorical figures of the triumphs on to cards, so that the game table becomes like the world: a battleground where great rulers, passions, virtues, angels, devils, and the stars and planets all contend in an endless dance. The more he thinks about it, the more interesting potential it seems to have. By choosing the symbols with care, and setting the rules for which triumph over the others, he finds he can say many things in the language of pictures! The scope of the project soon becomes more cosmic and abstract. A little bit of everything must go in, from the lowliest man to the sphere of Heaven. So the guiding principle will be cosmographic, but he's can't leave behind the theatricality and story-telling drama of the poems and parades, so he settles on something like Dante, where the levels of the cosmos are visited in sequence, and one encounters the denizens of each layer along the way. The spheres of the cosmos are thus rendered in terms of their human consequences, rather than their physical characteristics. Perhaps that thought leads him to add in the humble Fool, the Lenten clown, as the pathetic protagonist of this cosmic travelogue, turning up everywhere but always at the mercy of his surroundings. How many triumphs will there be? 6, 7, 10--all significant numbers, but too small. 21, though, is nice. It is the product of two magical numbers, 3 and 7, and also a triangular number 1+2+3+4+5+6. It's also the number of combinations on a roll of two dice, and other things besides.
The general plan takes shape: The human world, with its rulers; then the allegorical powers that dominate their lives--some tyrannical, some beneficent. Then, as in Petrarch, the forces of Death and Corruption, which are the highest in the material world, at least to his cynical mind. Beyond them, though, lies the incorruptable heavens: the Moon, Sun, and Stars; the Angels and God.
Then the details come. He introduces the sequence with a master of ceremonies, a flashy entertainer, carnival king, roadside hawker. It's a selection which delights him. Not only does it suggest the beginning of the procession of powers, but it makes a nice counterpoint to the Fool--and also there must be a "common man" at the lowest wrung, before moving on to the figures of power. Yes, the disreputable wand-toter is a good place to start. Perhaps he even sees himself in the character. For the rulers of the world, he endulges some unorthodox philosophy or personal whim, giving the Emperor and Pope female counterparts. Perhaps this makes a political or theological statement, the exact significance of which will be lost on later generations. But there can be no doubt about the irreverence of it, and perhaps he's noticing the humorous possibilities. Thinking in threes, he places the Pope to preside over the pair of Empress and Emperor--perhaps an imperial wedding? While below, the Papess marries the puffed-up Mountebank to Folly!
He also knows how he intends to handle the Triumph of Death. This is the most exciting point in his cosmographic drama, so he milks it for all it's worth. Time (The Old Man) leads into Death, naturally enough, but between them, our designer inserts the Traitor, suspended by one foot. It's a pointed comment on the papal propaganda of the shame paintings. The irony is that the martyred "traitor" will soon transcend this mortal world, his soul rising far above the Pope and his politics. He is a traitor in Earthly terms, but not in Heavenly terms. Yes, the Traitor is the perfect introduction to Death and the episodes that follow in this imaginary ascent through the cosmos. After Death comes the Devil, which the soul must face in order to be finally released from the bondage of the world. The final signpost on the way to the heavens is the sphere of Fire, which the foolish people of Babel try to penetrate with their prideful tower. Now, leaving that place behind, the rancor of the world is at last dismissed entirely. The celestial spheres follow, then stands Gabriel, Angel of the Last Judgment, who alone can announce the final entry into God's Kingdom.
Our fellow has been keeping count, and 15 trumps have already been placed. What remains are 6 virtues and worldly desires. Recalling Petrarch, Love is made to triumph over the Pope. One might see Temperance or Chastity as triumphing over Love, but he reflects on the pattern of threes he's started with Papess and Pope, and decides that Virtues should appear in every third position. So after Love comes War, in the form of the Triumphal Chariot. Yes, that's good, another of his favorite images from the triumphs, this time doubling up to make for a timeless contrast: Venus and Mars. So whereas Petrarch's triumphs would give us a female Chastity on a chariot at this point, our Milanese designer makes sure he uses a male warrior instead. Now it's time for a virtue. He opts for Justice, being a natural to preside over the marriage of Love and War. Dame Fortune of course cannot be neglected! He hesitates whether to place her before of after Time, but opts for the latter because Fortune is the Empress of the World, and needs to come as close to the top of the worldly forces as possible. The triadic structure demands a Virtue between Fortune and the Traitor, though, so Strength is inserted there. The puzzle is almost in place. But there needs to be some fine tuning to make it all fit. There are still two virtues, Temperance and Prudence, that have to be placed. But we've already got 20 cards! Something has to go. He can't bare to part with the Bagatto, or with any of the four rulers, with their pointed symbolism. Love has to conquer the Pope, and that Love/War thing is too good to give up. The Traitor is also dear to his heart, being again a special, original bit of commentary. Death? The Devil? They must stay. Fortune and Time are too profound to lose. The Fire is the climax of the "act" that begins with the Traitor, and simply has to stay. There's no way out. One of the virtues has to go. Prudence is probably the least interesting of the four, so she gets cut. Temperance, now, would have to go in position 14, above Death, to maintain the threesomes. That's a little awkward, but there's nowhere for her lower down, and it's not easy to think of anything that could be added in the celestial reaches without looking odd. So in she goes, and it turns out not half bad, as everyone knows the Devil conquers Temperance on a daily basis! Counting by threes again, he decides to move the Star down below the Sun and Moon, since the Sun and Moon are an obvious pair and hence the Moon doesn't make sense as a triad-topper. That works too! For now there is a pattern of increasing light from Star to Moon to Sun, leading into the radiance of Heaven. And the Star now symbolizes the entry into the celestial reaches generally, not just the sphere of the fixed stars. With a little work, the star is made to double as an icon of the immortal soul, a fitting image to rise above the Devil and the Fire as the head of a triad.
It was a puzzle for him to work through, but everything comes out right! The designer's love of the Triumphs, his cosmographical ambition, his sense of story-telling flair, his numerological sensibilities, and even his political commentary and ironic humor are skillfully overlapped and integrated. He's sure there's no better way to fit everything together. He makes up a set of the new cards, and circulates them among his friends. The game quickly becomes popular, and is taken up even at court. Before long, cardmakers are printing them to meet the demand. Almost from the beginning, though, there is grumbling that the ordering is too odd. If there is a female pope, why isn't she next to the male pope, like the Empress is to the Emperor? And why aren't the Virtues together? Why doesn't Temperance conquer Love? In his immediate sphere of influence, our hero is able to persuade critics with some clever banter about the importance of the numbers 3 and 7, and other linchpins of his concept. After awhile, though, local card players reorder the lower cards in a pattern that's a little easier to remember, but Temperance remains stuck at 14, because there's no sensible place to put her without changing the number of Death. In Ferrara, the criticisms are met through a clever reorganization of the entire system. It not only makes the order seem less abstruse to the average player, but it emphasizes the Biblical theme by augmenting the Last Judgment sequence, and misses a little of the irreverence in the placement of the Papess. In Bologna, the pattern is further "simplified" by collecting the Virtues together, creating a numbering problem that is left unattended to, and may even have gone unnoticed until the cards were first numbered decades later. The Angel is promoted to the top, showing a clear departure from the original concept. The Bologna tarot is inherited by the Florentines, who eventually end up dropping the Papess anyway for political reasons, thus leaving the tarot one step further from the Milanese model. What the tarot loses in conceptual integrity as it moves south, though, it gains in stylishness!
Ironically, the original designer's order (or something very close to it), is preserved in France, where no one was paying attention to the logic of the symbolism at all! Whereas the Italians, to some degree, shared some of the original designer's conception of the project, and so felt compelled to tinker with it to suit their tastes, to the French the whole symbol system was rather "foreign", so they slapped numbers and titles on the cards to facilitate play and left it otherwise alone.
And the rest, as they say, is history (this surely wasn't!)