Actually yes, and it's made easier by making a resource pack and only making it have different sounds (so you don't need to make textures or anything too)
Read it well, and read over if needed. It's not that hard to make with the instructions.
You could also search google for "Adding sound to Minecraft resources pack" or something similar. Youtube videos are particularly useful for a visual aid.
Best wishes, i hope you can get your own music to be played