When on the home screen in the game, you need to press R1/RB to scroll along the various menus until you reach the “Options” menu, then select the “Audio” menu and on this page you can change the music between “On”, “Off” and “Continuous”.
When the music is selected as “On”, it will play when the game is tense and there is more action going on around you, for example you might be shooting at someone or surrounded by enemies. When you select “Continuous” the music will be constantly playing so this is particularly good for a very atmospheric experience and with “Off” the music is completely off. This is a good option if you are live streaming and don’t want your video to get flagged up with a copyright claim on YouTube!