Here is the full list of all 13 Music Producer achievements.
Play your first song
Successfuly perform with a song
Fail your performance with a song
Complete a song to 100 per cent
Discover Art Cafe
Gather all musicians
Increase level of a musician
Increase level of all musicians to the maximum
Get popularity of 50
Get popularity of 100
Earn $1000
Complete all songs in Subway to 100 per cent
Complete all songs in Art Cafe to 100 per cent