Assuming that the music has had its copyright registered, you risk liability of $150,000, plus attorneys' fees, for each work infringed. Moreover, many licensors will not license to someone who has already infringed their rights.
Speeding up or slowing down the song does not change anything: if you are using the work without a license, that is copyright infringement. Don't do it. Get a license first. If you can't get a license, then don't use someone else's work.
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I would add that many content creators are artists at heart and would be happy to work with you on fees etc for using their work. Talk to the content creator, you'd be amazed at what you can accomplish.
If you want to include songs as part of your application you need to a license from (1) the owner of the copyright in the song's words and music [the "musical composition" copyright] and (2) the owner of the copyright in the particular sound recording of the song that you want to use [the "sound recording" copyright]. You need to acquire both licenses before you include them in a publicly distributed application. You have NO option other than speaking with a company who facilitates these licenses or an entertainment attorney.
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Yes you are engaging in willful, illegal copyright infringement. This is a very dumb idea. If you get sued, you could face statutory damages of up to $150,000 plus attorneys fees of tens of thousands of dollars. I seriously doubt if the app will make sufficient money to cover your exposure here. Get a license before you use this song or risk very serious adverse consequences.
Yes, No, No and No.
Yes it's copyright infringement unless you get permission before doing this.
No, merely altering play speed won't avoid the copyright.
No, you can't safely infringe and only later seek licenses. You might succeed, but you might get bludgeoned legally for that. You may have to grovel and plea for mercy and that might not be pleasant or profitable or pretty (except pretty ugly) .
No, you can't just pay once you get rich. Most copyright owners don't like that and will charge you much more since they have you in a legal box. The smart money pays to play and does not pay for past play in your situation. You get a better deal that way.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
I agree with the other responses by the attorneys. By using the music in the way you describe, you are engaging in willful, illegal copyright infringement. I'll offer another alternate option. Back when I was a web designer, there were legitimate software programs where you could essentially create your own music inexpensively.