What I love about good barbershop is its adherence to natural intervals so that, when well sung, the four parts create overtones and 5 or 6 notes can be heard.

I once visited some rehearsals of The Elmer Iseler Singers. One evening the director stopped them and said to the 2nd sopranos, "that C# is in a D chord, not an F# chord" whereupon restarting they tuned it up a smidge, a wee smidge that many people wouldn't even notice, and the chord no longer sounded excellent but truly exquisite. But that kind of detail is not possible with piano, organ or any tempered accompaniment.