I have one recording of my voice $x[n]$ singing a song. One thing I could do is independently record me singing the song again to obtain $y[n]$. Then, $z[n] = \frac{1}{2}\left(x[n] + y[n]\right)$ would sound like a "chorus" of some sorts (assuming that both recordings were done in perfect sync).
My question is, without doing a bunch of independent recordings, is there some decent way to go straight from a single recording $x[n]$ to the 2-fold chorus $z[n]$? More generally, I'd like to be able to go to a $k$-fold chorus (equivalent to averaging $k$ independent recordings).
My gut tells me one could simulate independent recordings by randomizing the phase in $x[n]$'s STFT somehow, but I'm not sure of specifically what should be done. In this regard, the only relevant method I'm aware of is Paul stretching, which doesn't really fit the bill, I think.
Also, is there a standard name for this particular effect I'm trying to create?