Depends on the country, but in the US I think it's like anything before 1910 is not affected by copyright law (IE, they are public domain) and other than that it depends on when the work was created as to which laws apply to it. If you are looking to use stuff writting in the 1700s, you know you are safe. HOWEVER that does not apply to individual performances of the pieces. If Boston Pops performed the song, you can's just nab their performance and put it in your game, you would have to make your own performance/get someone to do it for you (So if you reproduce a song with a midi sequencer, that would be your performance)
EDIT: read your post closer, if one person does it via midi, and you do it via midi and they happen to be exactly the same, there's nothing they could really do about it since it still qualifies as a performance, and arguably there is only one way to perform it in that manner, so they can't claim you copied their performance. I think in order to claim someone copied your performance, you have to prove that your performance was significantly your work and not just playing the song. (IE, you can't get permission from Bob Dylan to play All Along the Watchtower and then play it like Hendrix, because Hendrix's interpretation was significantly different and was clearly his own interpretation)