Of course, the sound that non-electrical instruments make is indeed dependent on the atmosphere in which they are played, assuming the same motions etc. You could probably make a violin produce the exact same sounds that it does on Earth playing it on Mars, but you'd have to play it differently than you do here. But even this doesn't apply to electrical instruments - Jimi Hendrix would sound the same if he could somehow play on the surface of Mars.
In that sense, I think any form of atmosphere, it will have its own music but it would probably not sound great in other atmospheres.
Also, any atmosphere can carry any sound wave for at least some distance. It takes more or less energy to do so, and if the medium attenuates rapidly, you have to start with higher frequencies so that the right frequency reaches your audience, but that's it. You need different instruments to play in different atmospheres, sure, but not different music. And electrical instruments actually work exactly the same regardless of the medium.