I play the game with my own background music on, but things like the New Day music, Meditation music, Musicblock music, and all other music (that I've found) aren't diminished when I set the music volume to 0.
These are events, so they don't really relate to the ambient music. If you want these turned down then turn down the slider with the headphone icon. This will turn all SFX and event triggered music. And! we now have a mute button!