When does it get scary?

Just before Act II. Just play through the game normally and you'll see.

(Spoiler) 

The second time you go to Sayori's house. After the game sends you back to the main menu, from then on most everything is kinda creepy

I haven't played in a while, but play the game normally and I think after you read sayoris poem you'll find out

act 2 and its scary at the end but epic to because YOU DELETE MONICA FROM THE GAME

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when you choose who youre going to do the project for the school festival with, you choose to go to sayoris house, i believe saturday. the game has a different mood from then on.