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Xbox Support
https://support.xbox.com/en-US/help/hardware-network/display-sound/choosing-speaker-audio-output#:~:text=Select%20Profile%20%26%20system%20%3E%20Settings%20%3E%20General,5.1%20or%207.1%20uncompressed%20signals%20or%20bitstream%20formats.
Xbox 360 HDMI - Connecting Headsets for Game Audio ...
https://support.turtlebeach.com/hc/en-us/articles/223083827-Xbox-360-HDMI-Connecting-Headsets-for-Game-Audio
If your TV does not have a digital audio output, you will need to obtain the Xbox 360 Audio Adapter Cable to work alongside the HDMI connection. The HDMI connection will send digital video and digital audio to your TV, while the Xbox 360 Audio Adapter will give you the optical digital audio jack you need to connect from the Xbox 360 to your Ear Force headset.
How to Connect Xbox 360 to HDMI Monitor and Get …
https://www.youtube.com/watch?v=8be4T-mkhUY
This is my tutorial on how to connect and xbox 360 to monitor with a hdmi port and get audio through either speakers or headphones. What you will need: Xbox ...
Xbox 360 hdmi 5.1 sound - Microsoft Community
https://answers.microsoft.com/en-us/xbox/forum/all/xbox-360-hdmi-51-sound/2df16b9d-5055-4e04-9403-087ef4e5502b
Unless I'm very mistaken, the 5.1 sound should go fine through HDMI only. I believe the audio settings in xbox saying "digital output requires an optical audio cable" is referring to if you're using component cables, not HDMI. The best thing to do would be to try it and see what results you get. But I definitely think HDMI should be all you need.
Xbox 360 HDMI Setup - supercheats.com
https://www.supercheats.com/xbox-360/how-to/xbox-360-hdmi-setup
Connect one end of the HDMI cable to your console. Connect the other end of the HDMI cable to the HDMI port of your TV/ Monitor. If your TV has multiple HDMI ports, take note of the HDMI port number where you'll connect your console. Turn on the console and TV then select the correct input source. Reference.
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