Solution 1:

Do you have a resource pack? Some resource packs remove all of the in-game music. Try turning it off and see if music comes on. You can also try deleting your current version jar file.

C:\Users\ {YOUR USERNAME HERE}\AppData\Roaming.minecraft\versions

Delete the folder for the version of Minecraft you are currently playing in. (Note that this method does not erase any save data, but if you are using a modded version of Minecraft, you will have to redownload Forge) Open the launcher up and the launcher will redownload the version you were using. If none of that works, try pressing F3 + S or FN + F3 + S at the same time. Finally you can try changing the Mipmap levels in game to OFF and then back to 4.

(ESC -> OPTIONS -> VIDEO SETTINGS)