I've read that the amount of cross-pollination depends both on the variety, and how many AGGRESSIVE pollinators you have.
I recall that Joseph sought and found a variety that was very prone to cross-pollination, like it tended NOT to pollinate itself thoroughly before opening. He put a lot of those "promiscuous" genes into his crop so that the offspring would also cross more freely, to advance his "landrace" policy.
The variety had a funny name, with some numbers and single letters in it, if I recall.
