The handing out of communion often takes a while. Strangely enough, the gregorian tune sung on that occasion is the shortest one of the whole mass. Actually, it is not a responsorial chant like the gradual, which is a response to a lecture, but a communion antiphony, which was formerly chanted in alternance with a psalm.
Contrary to the Offertory tunes which grew richer with time, most communion antiphonies have kept their original caracter, i.e. a relative brevity and restraint, combined with a greater attachment to modality.
Hence the logics of sticking to the largely accepted practice of chanting the communion psalm in alternance with the antiphony. Unfortunately, gregorian parish books (780 and 800) do not contain the correct psalm, even though each mass has its own psalm.
The following downloadable chart contains a list of Temporal Sundays and Feasts with their respective psalm, as well as a link to the corresponding pages of the gregorian parish book (780 and 800). Also indicated is the gregorian mode of the communion antiphony, i.e. the psalmody mode, which is great help. Great but limited, as singing is still YOUR task!