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The music box song John was singing in eps 12/13 likely isn't going to be important/something that's linked back to considering it was improved.
However
"They lie and wait in the depths of darkness, they know just what you're thinking at all times and when they find you, oh they find you so cold"
"they have formed beyond the scope of human understanding, and when they find you there you will not be-"
"they have teeth that outnumber the stars, and they are just like razor blades, and if they bite down upon you, it's already too late"
"they know where you are and they are finding you, no Void can hide your scent from them, cause they are already on route John and Jane-"
Now, what in Uncommon Commons was in the darkness, what has teeth that bite down upon you, what shows up just a few episodes later.
That's right
The cats
Now what's more is that the music box said "DO NOT OPEN UNDER ANY CIRCUMSTANCE FOR THE LOVE OF ALL THAT IS HOLY" on it. Which means that something or someone doesn't want Doreen Wimbledon and her cats in the apartment. I don't currently have any spesific guesses for who or what that is yet tho.
Even "so cold" could make sense as the episode where Doreen Wimbledon is introduced and we hear the Doe's interact with a cat, is the same episode as "is it getting colder in here?" (ep 18)