I've used Nokia Play to to play music on my 360, although it doesn't play when I only have server/cloud music Pass tracks on my 1020 --I get a format cannot/error message. It plays when I have only WMA and mp3 tracks, and I guessed it was a DRM limitation. I also tried using SmartGlass at first, but it could never "find my console", although the 360 settings were On appropriately, and the 1020 did "see" the 360 on the network. I guessed that I needed Xbox Live Gold, but I just found the Play to and never pursued SmartGlass. There is a SmartGlass for the One, and isn't that another option?
I use the 360 almost exclusively as a Windows Media Center extender, but my music did sound amazing playing through the Xbox.