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Vox has a post up, This comprehensive Targaryen family tree explains Game of Thrones’ most complicated dynasty. I don’t really know if the family tree explains much, but it’s interesting. It does highlight two important dynamics which are probably important.
The Targaryen’s are notoriously inbred. The rationale for this is that their affinity with dragons is a heritable trait, and that is the basis of their power (or it was before all the dragons died, at least until Daenerys brought them back). If they mixed their genes with outsiders presumably they would dilute that ability. Of course that doesn’t take into account selection for the trait, which would diminish any dilution through outbreeding.
As you can see Daenerys’ parents and grandparents were full siblings. Using a pedigree method her Wright’s coefficient of inbreeding would be 0.375. For a point of comparison, the child of parents who were full siblings, but whose own parents were unrelated, would have a coefficient of inbreeding of 0.25. The terminal individual in the genealogy of the Spanish Habsburgs, often cited as a case study of how inbreeding can lead to the extinction of a lineage through sterility and imbecility, had a Wright’s coefficient of inbreeding of 0.254. Something really doesn’t add up in terms of the viability of the offspring of the Mad King and his sister-wife (to get a sense of what multi-generation inbreeding might do, please see the Colt family). But then, it’s fantasy, and genetics is one area where George R. R. Martin’s gritty verisimilitude gives way to flights of fancy.
This might surprise some, as the incestuous and closed-off nature of the Targaryen’s plays a huge role in the backdrop of George R. R. Martin’s novels. But incestuous marriage is actually a resurrected custom of the previous two generations before Daenery. Her great-grandfather, Aegon Vth, “Egg” from the novellas, was the result of two generations of outmarriage (so he was outbred). This pattern of outmarriage was not a conscious shift in the mores of the Targaryen royal house, but happenstance. Aegon’s father and grandfather both came to the throne because other senior claimants died. They were not expected to become kings. Aegon’s mother was a Dayne, while his paternal grandmother was a Martell. He himself married Betha Blackwood (a family of First Men background which still worships the old gods). Therefore, Daenery’s two grandparents were both 1/8th Valyrian, her parents were 1/8th Valyrian, and she is 1/8th Valyrian (at least that’s the expected value).