The GIRM is pretty clear on this. "Unity of their voices." Congregational singing is preferred.
86. While the priest is receiving the Sacrament, the Communion chant is begun. Its purpose is to express the communicants’ union in spirit by means of the unity of their voices, to show joy of heart, and to highlight more clearly the “communitarian” nature of the procession to receive Communion. The singing is continued for as long as the Sacrament is being administered to the faithful. If, however, there is to be a hymn after Communion, the Communion chant should be ended in a timely manner.
87. In the dioceses of the United States of America there are four options for the Communion chant:
(1) the antiphon from The Roman Missal or the Psalm from the Roman Gradual as set to music there or in another musical setting;
(2) the seasonal antiphon and Psalm of the Simple Gradual;
(3) a song from another collection of psalms and antiphons, approved by the United States Conference of Catholic Bishops or the diocesan Bishop, including psalms arranged in responsorial or metrical forms;
(4) a suitable liturgical song chosen in accordance with no. 86.
This is sung either by the choir alone or by the choir or cantor with the people.
88. When the distribution of Communion is finished, as circumstances suggest, the priest and faithful spend some time praying privately. If desired, a psalm or other canticle of praise or a hymn may also be sung by the entire congregation.
We have the Mass of Thanksgiving for our Cathedral's newly ordained priest who plans to chant the entire Mass (including the Roman Canon).
I don't know whether this is a real omission, but I would have liked to see the Our Father among the sung parts of Mass on Matthew's fine program.
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