When I was in high school, we did the musical "The Music Man," which includes a barbershop quartet. One of the music teachers told us that barbershop relies on a certain chord (which he named). I don't remember the form of it, but I'm pretty sure there was a sixth involved. To compose barbershop, you take a melody and have the notes of it be the top notes of one of these chords. One of the parts, I believe it was the second tenor, is a nightmare, because it is this man's job to put in the note which makes that strange chord.

Perhaps if you search barbershop sites on the Internet you will learn what chord that is. Another question is, does every note of the song need to be put in that chord? I believe the answer is yes.