Port 3689: DAAP
Port 3689 is DAAP, used by iTunes and the Music app to share music libraries over a local network.
What runs on port 3689
Apple's Digital Audio Access Protocol serves a music library on TCP 3689 and advertises it over Bonjour so other computers can browse and play it. Apple TV and Remote apps have used related protocols on the same port.
On a home network
Macs and PCs with library sharing on, some NAS boxes with an iTunes or media server package, and older Apple TV and AirPort devices may listen on 3689.
If port 3689 is reachable from the internet
There is no reason to expose 3689 to the internet. It would let outsiders browse and stream your music library and puts old media-server code in front of attackers.
How to check whether it is open on your network
Check the media or iTunes server package on your NAS and the sharing settings in the Music app. A LAN port scan shows which devices answer on 3689.
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