I grew up in the mountains of North Carolina with Scottish roots. It wasn't until I was in scotland listing to the gaelic words and tunes and playing scottish fiddle vs old time that I heard what that "wandering" on either side of the note was. It is really part of the gaelic/celtic pronucataion of the word to the tunes. It doesn't just have one sound. Singing the tunes without instruments they were singing the way it was played or sung and passed down with the Scottish clearances and Scots/Irish immegration to the south where so many ended up in the Mountains.
Then in an Alasdair Fraser workshop in Oct at Loon Mt New Hamphshire he also talked about playing gaelic... made us hum in "gaelic" before we could play the tune. The notes had severaltwists or stepts to one side or the other. Someone called it ornamentation and he said "NO" its the celtic soul of the song. Again I could hear where the old old mountain singing be it religious or secular came from. It was a fascinating discovery after all these years of hearing it.