No, there isn't a way to transfer songs to the default Music player. However, you can use DropBox or other third party apps to transfer songs wirelessly. The problem is they won't be integrated into the default Music app; therefore, you cannot use Siri to play songs & will lose other features.
I wish Apple would make it possible to transfer songs without iTunes or at least download songs from other places (Amazon, Artist's website) and have it play in the default Music app.
If you don't care about the player and just want to choose your songs, I know dropbox can play your songs. You just have to sync your songs to Dropbox and then you are good. You'll have any song you want. It just won't play on the default player.