How can I open the Black Door to the west of Falkreath?
There is a black door to the west from Falkreath, when trying to open it a voice asks
What is the music of life?
As I don't know the answer I can't get through that door. Where do I get the password to that door, which quest does this dungeon belong to?
Solution 1:
It is the Dark Brotherhood's sanctuary.
To get through that door you have to
kill the owner of the orphanage in Riften and then a courier brings you a note. Sleep to be contacted by them, then they will give you the necessary information to join from there.
They may not contact you the first time you sleep so keep trying, eventually they will.
Solution 2:
Talk to the Inn Keeper in Riverwood and ask about any rumors he has heard. He will talk to you about a person in Windhelm trying to contact the Dark Brotherhood. After this you will have the quest line in your Misc. Quest Section. Just follow the quest line from there.
Solution 3:
All of the other answers seem to have the basics down, but don't really cover the whole story. Here's what needs to be done:
At some inns, you may hear rumors of a kid in Windhelm who's been trying to contact the Dark Brotherhood. Alternatively, you can talk to any of the kids in the orphanage in Riften and they will tell you about their friend who ran away to Windhelm. Perhaps one other "appropriate" way to start this quest would be to just break into the Aretino (as if you were just going to do a random burglary) residence in Windhelm and talk to Aventus there.
All of the above paths eventually lead to you having a chat with Aventus Aretino at his late parents' house in Windhelm. He'll mistake you for the answer to his prayers (a Dark Brotherhood assassin), and ask you to kill the owner of the orphanage in Riften. Go kill her and report back to Aventus.
The shortest way to do this though, is to skip all of the above.
Just go straight to the orphanage in Riften and kill Grelod The Kind. Then go break into the Aretino residence (I don't think I've ever encountered it unlocked) in Windhelm and talk to Aventus.
After following either path above, eventually you should have a random courier approach you.
He'll have a note that simply has a black hand-print, and the words "We know."
Once you've received your message, go home and go to sleep. (You may be able to go straight home without receiving the message, but I'm guessing that is why some others are claiming that the quest won't start on your first sleep.) Even one hour of sleep should be enough.
You'll wake up in a strange shack where you're addressed by Astrid, the leader of the Dark Brotherhood and your captor for the moment. She wants you to kill one of the three other people she's captured there. Kill any or all of those three, and she'll invite you to join the Dark Brotherhood.
At this point, you'll receive the password to the Black Door.
Useful references:
http://www.uesp.net/wiki/Skyrim:Innocence_Lost
http://www.uesp.net/wiki/Skyrim:With_Friends_Like_These...
Solution 4:
The answer is:
This is the door to the Dark Brotherhood. "Silence, My Brother"
But this answer isnt available untill you have completed the questchain mentioned earlier