In Episode 7, Nanami seemed to elude that when Shichika killed their father, the 6th Kyoto Ryu, his father was about to kill her, or she already knew that he was planning on killing her.
I am wondering, did Shichika's father try to kill Nanami? And for what reason? (Was it to ease her suffering, or to stop her from becoming more powerful from observing more of the Kyoto Ryu art?)
Also if he did try and kill her, did Shichika kill his father solely to protect Nanami?
Unfortunately it is never explained why Shichika killed his father, however from episode 7, we can get a pretty good guess that it might have been to protect Nanami.
Here is the dialogue from the second fight at 35:00:
Nanami: Do you remember when you killed Father?
Shichika: I guess I do...
Nanami: I haven't thanked you for that yet. But I'm not particularly grateful to you for that, so even now I don't intend to say so. Because I would have been fine with being killed by Father back then. I would have been fine with it. There's no point to me living after all.
What we can gather from this implicitly is that Shichika killed their father in order to protect Nanami, without any request from Nanami, e.g. simply sibling affection. In this sense, this means their father either mentioned that he would eventually have to kill Nanami and thus forced Shichika's hand or he attempted to kill Nanami and thus forced Shichika's hand. So whether or not he mentioned it or attempted it we don't know, however I believe it's most likely the latter due to Nanami's extreme potential.
As for the reason, what we know is that Nanami has the ability Migeika. For this reason in episode 4 Nanami recaps that her and Shichika's father neglected to make Nanami the Yasuri 7th generation family head, because this ability made her too powerful. Since she was too powerful because she could see any ability and learn it instantly, including the Kyotouryuu sword arts, that their father deemed her to be too dangerous with her potential and thus decided to eliminate her.
