Some songs are meant to be sung unaccompanied, and would suffer if one tries to accompany them in any way.Greek and Turkish and Arabic and Indian tradition has several of those, and the British one has quite a few too. They often have either a very "lax" time signature and/or rhythm, or it varies so widely as to make accompaniment nonsensical. Try accompanying the Cornish Oxplough Song ("Come all you sweet charmers and give me choice, there's nothing to compare to the ploughboy's voice...").

Les, dust off my last album and listen to "One by One" and I think you'll see an example of a song that accompaniment can only spoil.