Note 3 beeps (melodically) off and on all day long?

Yeah, if you have connected somewhere, it will still automatically connect. I disable smart wifi and when I go to my mother's, my phone connects automatically. Smart wifi can cut your wifi off when it thinks you aren't using it, etc.

Sent from my Awesome Note 3

I think I know the sound. My wife's phone's sound is like when you use the S-voice. She doesn't know what she did.