You can actually play all the songs to the commander in one go. When he is in his office, ask him whether there's anything else you can do for him. He will say play for me. Simply choose each song until the achievement unlocks.
There might be a couple of songs you need to obtain via quests.
One is from the two mice Gusto and Fatale. They will ask you to make up a song for them. This is part of the main story and is required for them to give you the guard helmet.
The other is obtained later on in the main story, it's called "Orphan Prince". When Silas reveals his secret to you, ask all available questions about his past life and you will unlock the song.