I’ve done it both ways depending on the circumstances and local custom.
CGZ—in 99% of circumstances, there would be an introduction, or do you commonly just start singing out of the blue? When we don’t have accompaniment in Lent, I still play the first few notes of everything to establish pitch and tempo. If it’s purely a capella, the congregation comes in 2-3 notes in.
When it’s a lectionary alleluia + verse with a “congregational” “refrain”: Organ plays it, cantor/choir sings it, people sing it, on to verse. In Masses without organ (outdoors or etc.), cantor/choir sings it, all repeat it.
When it’s from the gradual: Schola or subsection of Schola (“cantors”) sing to the asterisk, go back to top, all repeat it, Schola alone go on to jubilus and verse. All repeat Alleluia thereafter, straight through without a repeat.
At my parish there is an instrumental introduction and then the congregation enters right away, with no cantorial intonation. This is the pastor's mandate, and I don't recall ever being anywhere else where it was done this way. From this thread, though, I see that there are many parishes who skip the cantor's intro. Learn something new every day...
Alleluya should be intoned by the cantor and repeated by choir-people. What is comically redundant and silly is the widespread practice of having it played by the organ, sung by the cantor, and finally, sung by the people. This holds true, also, of the responsory of the responsorial psalm.
For the Responsorial Psalm (pax MJO), I play the respond on the organ, then the cantor intones, followed by the people. This is mainly because the chanting of the Responsorial Psalm is new* where I am, and is only done at the "High Mass" at 10:30: so the organ bit is more to remind the reader that the Psalm is sung and not to start reciting it. -- * New from when we switched to the Ignatius Pew Missal as our Missalette in autumn 2018 (Advent, year B). I tried before then to introduce the sung Psalm on some special occasions (mainly where the readings are ABC) from various sources, but I got pushback because they were "too hard" or "too long". The exceedingly simple responds (office tones) in the Pew Missal, while not ideal, are simple enough that people are catching on, and it is sung quite well now.
Toddevoss: It was recited. And still is, apart from the Sunday 10:30 and some Major Solemnities like Christmas and the Paschal Triduum; though all of the Psalms at the Easter Vigil are recited, except the Alleluya.
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