Mac – Notification from FaceTime won’t disappear!

I found this extremely frustrating as well. I spent an hour trying to record a silent ringtone in GarageBand but it consistently delivered noisey ringtones despite no sound on the track. I was about to give up when I thought someone else besides us must have been annoyed and figured it out. A Google search yielded the silent ringtone below. I tried it and it works great. Good luck.

http://www.codingmerc.com/blog/silence-ringtone-available-for-download/

I seems that I use filters in GMail to skip inbox, and the sound is actually come from Mail.app. As a result, no new mail arrives to the Inbox of Mail.app, but to the Archive of the Mail.app.

In the preference of Mail.app, one option is to choose "Inbox Only" for "New message notifications". But it seems that this option will not mute the new messages sound when new message arrive to other mailbox, only skip the notification alert that shows up in the upper right corner of the screen.