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the hypnosis skill you get it by reading the book that the mc's father sends to him. Not so sure if you receive the book from the beginning of the game or if you need to play some days to receive it. But it will come to you automatically.

And to lower Harmony's purity there are only 2 ways, either you pick specific choices during her events that will lower it or use the hypnosis skill XP.

Here, read the wiki:

https://wiki.andrealphusgames.com/wiki/Harmony