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Hi Jim,
thanks for asking.
There are a number of situations that I can imagine where you'd hear a beep:
1) iTunes has a modal window opened (f.ex. an alert - or its preferences)
2) another application was active at the time you pressed the button (and therefore an other Behaviour, where this button wasn't assigned to an action). You can see which Behaviour is currently active when you click on the remote icon in your Mac's menu bar.
3) the Exit button isn't set up correctly on the Harmony side (please see "Step 3: Optimizing the Harmony configuration > Unlocking all buttons of the Smart Control" at https://www.iospirit.com/blog/article/184/-/)
If it's none of this, please send one screenshot of your action and another showing where you mapped it to info - at - iospirit - com.
Alternatively, you could also assign the action "Behaviours > Quit currently controlled application" at
a) Prefs > Mapping > iTunes > Exit if you want this functionality just for iTunes
b) Prefs > Mapping > Global mapping > Exit if you want this functionality available through that button for all apps.
That action uses a more robust way (Apple Events) to tell applications to quit, avoiding the complexities simulated keystrokes and OS X event routing sometimes have. Applications can still opt not to quit when receiving this event, though (which is exactly what iTunes does when you have a dialog or its prefs window open).
Hope that helps!
Cheers,
Felix