See paragraphs 86-88 of the GIRM.

Some celebrations are entirely without singing.
But if it's a celebration with singing, there should be a Communion song (that is, sung during the Communion reception). And there can be a “hymn after Communion” as well.

So in answer to your question, the only thing excluded is to have a post-Communion hymn but nothing sung during Communion.

(The question of who sings, choir or soloist or the faithful as a whole, is a different question.)

^ The GIRM frequently says dumb things.

"The GIRM frequently says dumb things."

Except this is not one of those. The requirement underscores the unity of the Communion rite. It also, according to someone here years ago, reflects a requirement for the Communio in the ritual books for the preconciliar solemn high Mass.

“Seems a bit odd”, yes, it is odd, but in fact no less required. If it seems odd that's because hymns are odd as replacements for the proper ritual text and music. An oddness which is found elsewhere, too.

The current "position" is:

86. While the Priest is receiving the Sacrament, the Communion Chant is begun, its purpose being to express the spiritual union of the communicants by means of the unity of their voices, to show gladness of heart, and to bring out more clearly the “communitarian” character of the procession to receive the Eucharist. The singing is prolonged for as long as the Sacrament is being administered to the faithful. (cf. Inestimabile Donum 17) However, if there is to be a hymn after Communion [cf. no. 88 below], the Communion Chant should be ended in a timely manner.

Care should be taken that singers, too, can receive Communion with ease.

87. In the Dioceses of the United States of America, there are four options for singing at Communion: (1) the antiphon from the Missal or the antiphon with its Psalm from the Graduale Romanum, as set to music there or in another musical setting; (2) the antiphon with Psalm from the Graduale Simplex of the liturgical time; (3) a chant from another collection of Psalms and antiphons, approved by the Conference of Bishops or the Diocesan Bishop, including Psalms arranged in responsorial or metrical forms; (4) some other suitable liturgical chant (cf. no. 86 [above]) approved by the Conference of Bishops or the Diocesan Bishop. This is sung either by the choir alone or by the choir or a cantor with the people.

However, if there is no singing, the antiphon given in the Missal may be recited either by the faithful, or by some of them, or by a reader; otherwise, it is recited by the Priest himself after he has received Communion and before he distributes Communion to the faithful.

88. When the distribution of Communion is over, if appropriate, the Priest and faithful pray quietly for some time. If desired, a Psalm or other canticle of praise or a hymn may also be sung by the whole congregation.

45. Sacred silence also, as part of the celebration, is to be observed at the designated times.[Sacrosanctum Concilium 30; Musicam Sacram 17] Its nature, however, depends on the moment when it occurs in the different parts of the celebration. For in the Penitential Act and again after the invitation to pray, individuals recollect themselves; whereas after a reading or after the Homily, all meditate briefly on what they have heard; *then after Communion*, they praise God in their hearts and pray to him.

Even before the celebration itself, it is a praiseworthy practice for silence to be observed in the church, in the sacristy, in the vesting room, and in adjacent areas, so that all may dispose themselves to carry out the sacred celebration in a devout and fitting manner.

[emphasis added]

* * *

The GIRM does not have one-stop shopping, topically.

Of course, Liam. Maybe I should have drawn a distinction between "sacred silence" as mentioned for example in GIRM 43 and my single scarequotes 'sacred silence', which I suppose some literalists might concede is satisfied by the pause between the two pieces of music.

When he visited an anechoic chamber where his pulse and nervous system were very much foregrounded, John Cage wondered whether there was such a thing as silence. He adopted a working definition of "the absence of intentional sounds". If we do the same, the question then becomes whose intention: is it a requirement that after returning from Communion one wait for the schola to finish before they "praise God in their hearts and pray to him"? For now I'm going to remain skeptical about that, perhaps because I'm just as likely to be distracted by IDing the birdsong outside the stained glass as by my usual pleasure in music.

The practice at the parish which employs me is as follows:
1. The proper is sung as the priest receives
2. Organ interlude/improv as the choir and people receive
3. Congregational chanting of a psalm to a psalm tone during the ablutions

Does a strict reading of the GIRM prohibit this? It seems like it would. Does a strict reading of the GIRM preclude having the choir sing a motet during communion? What about the proper and then a hymn? The wording doesn't seem to account for the many musical possibilities that communion down-time presents.

"Hymn sung by the congregation (ALWAYS Eucharistic in Nature - one would think this is given, but I've learned it's not.)"

So long as "Eucharistic in Nature" includes psalms of praise and gratitude for God's feeding of and care for us (Ps 34, 78, 145, et cet. - consider the sources for the Communion Propers), and scriptural canticles like the Magnificat, then I understand it being "given". If it more or less means benediction/adoration hymns, then that's too restrictive to be "given" IMO.

I had the choir sing the English "communio" for the day while the priest received then shifted into a subdued organ piece while the choir received. The choir stayed in the loft and did not process downstairs but couldn't seem to keep singing and receiving together at the same time. When they were back in their seats I began the communion hymn.