I believe the priest and congregation are suppose to sit. This is the one Proper of the Mass where the priest actually sits and listens instead of performing an action.
There are some contradictions between the general principle that, on one hand, Gregorian chant has first place, and on the other hand, the rubric in the GIRM prescribes that the alleluia "is sung by all while standing (¶62)." This rubric is evidently aimed at a rather simple, non-Gregorian antiphon (like the little three-fold alleluia from Holy Saturday) and a short verse, which is the usual practice in the parishes. The congregation is not capable of singing the entire Gregorian alleluia, yet these melodies are the summit of that art and reflect their own exquisite liturgical function; to rule them out absolutely would be a contradiction of Sacrosanctum concilium, which is a more authoritative document than the GIRM. Moreover, the Gregorian alleluias appear in the Gregorian Missal (1990 and still in print), which is a book prepared for parish choirs.
The liturgical function of the Gregorian alleluia is more complex than the GIRM prescribes (the congregation welcomes the Lord in the gospel and expresses their faith); it is at once a meditation chant which reflects on the reading just heard and an anticipation of the singing of the gospel. Likewise the duration of the alleluia is considerably longer than a simple gospel procession takes (except at Westminster Cathedral, where at the Pope's Mass the entire Gregorian alleluia was sung, and it lasted exactly the same time as the procession, which went about a third of the distance from the nave to the great pulpit); if people stand at the beginning of the singing of a Gregorian alleluia, they are left standing for quite a while, apparently for no purpose. If the alleluia is a meditation chant reflecting on the previous lesson, then it is more appropriate for them to remain seated.
In my own practice, the gospel procession begins toward the end of the alleluia verse, and the people stand approximately at the repeat of the alleluia. This fulfills the status of the Gregorian alleluia as one of the highest of the Gregorian forms, but it is in technical violation of the GIRM, since the congregation does not sing any of it. I have proposed a solution for those who wish to observe the GIRM strictly, that the congregation sing the repeat of the intonation of the Gregorian alleluia, after which the choir sings the jubilus. Congregations are able to repeat most Gregorian alleluia intonations without difficulty, and in doing so, they sing almost as much music as the little antiphon seen above [i.e., the Holy Saturday "triple" alleluia], and they listen to quite a bit more.
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