Singing in Harmony

It's not uncommon to hear people seriously affected by certain traditions of worship object to singing in harmony in the church.  I suppose that the usual reason for this is that singing in unison (actually octaves) is supposed to be more emblematic of the unity of the church.  This idea is very old and goes back to the traditions of the post-apostolic church in its Greek philosophical environment.

The only problem with this, as I see it, is that to limit our singing to monophony is similar to limiting our gifts to be of a single kind.  Would it be a better expression of unity in the church if all of us were eyes?  I don't think so.  If the harmony of the different parts of the body is emblematic of the harmony of the gifts of the Spirit in the church, why isn't it reasonable that a harmony of voices in the church could be a chord, rather than an octave?