My gut feeling says the Cantor starts the Antiphon for the Invitatory, but I can't find it written anywhere - I'm looking at page 1779 of the Liber Usualis. I'd like to make it, whoever gets there first starts the Antiphon, as I've heard that the Invitatory was added to give sleepy monks time to turn up for the psalms. That would be cool having the keen youngsters vying to get to the Chapel first to intone the Antiphon, but then would they know which one to intone? Where the Little Office only has the one, so that's all good.
Anyway, it feels like a very minor point, and maybe the real answer is that whoever usually starts the Antiphon starts the Antiphon.
Sandhofe's Nocturnale Romanum has rubrics for the invitatory, and doesn't specify by whom it is sung: it just says dicitur (p. XXII). (As always, dicitur includes singing.)
Well, I certainly don't know the official rubrics. But I can tell you what I think based on my year+ experience at a monastery. They would only sing the invitatory to chant (i.e. not recto tono) on 1st and 2nd class feasts. On the recto tono days, what they called the acolyte at Matins would sing it (one of the choir novices, or two if it was an obligatory memorial/3rd class feast or on Sundays). On both 1st and 2nd class feast, even those 1st class feasts on which the Abbot officiates, the cantors would always intone it. For Matins, the "cantors" were basically the same ones who would be in the schola at Mass (6 or 7 monks).
So I don't know who would usually/traditionally intone it in monastic or cathedral choirs, but based on what Clear Creek does, it is always just one of the cantors, so your gut feeling is probably correct. I would hazard a guess that it is not traditional to have a non-cleric or non-professed monk intoning anything in a clerical or monastic choir (as I mentioned happens at CC most days at Matins), but again, that is just a guess.
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