What’s a trope? See the introductory post.
|Kyrie, Rex æterno posse superno, cunticreator, eleison.
Kyrie, laudat dignum turba benignum tota polorum, eleison.
Kyrie, nunc præsentes respice gentes dona petentes, eleison.
|Lord, eternal King of lofty power, creator of all, have mercy.
Lord, the entire multitude of the globe praises thee, worthy and merciful, have mercy.
Lord, behold now the people here desiring gifts, have mercy.
|Christe, prævenias morbis nostris nunc Conditor orbis, eleison.
Christe, peste triumphata sumens hæc nostra peccata, eleison.
Christe, sanguine qui digno præservas hoste maligno, eleison.
|Christ, mayest thou now keep us from our ills, founder of the world, have mercy.
Christ, having triumphed over pestilence, taking up these our sins, have mercy.
Christ, who by thy worthy blood savest us from the evil enemy, have mercy.
|Kyrie, fac tibi clerum psallere verum pectoris hymnum, flamine fultus lumine vultus vivat in ævum, eleison.
Kyrie, nunc populorum Rex miserorum cerne precatus, flos pie florum fonsque bonorum terge reatus, eleison.
Kyrie, suscipe rursum, dirige cursum, corde rogamus, scandere sursum vivere cursum quo valeamus, eleison.
|Lord, make the clergy sing a true hymn of the heart to thee, may they live for aye, borne by the spirit, a brilliant countenance, have mercy.
Lord, now, King of the wretched peoples, behold the prayers, O righteous flower of flowers and fount of good things, cleanse our guilt, have mercy.
Lord, take us up anew, lead the way, we beseech thee from the heart, by which we might have the strength to climb up and live the way, have mercy.
The Kyrie de Angelis, Kyrie VIII in the Vatican edition, has proven the most enduringly popular of all the Gregorian settings of this first part of the ordinary. Part of its appeal rests in its agreeableness to the modern ear: unlike the other Kyries in the Vatican edition, whose mediæval tonalities can sound mildly alien to-day, the Kyrie de Angelis is actually a major-scale melody. Renaissance music theorists who first described the major scale, dubbing it the Ionian or eleventh mode, already noted its popularity amongst their contemporaries.
Of this [mode, the eleventh] there are many chants in the ecclesiastical books, such as the Mass called of the Angels, the antiphons Alma redemptoris mater and Regina Cœli lætare Haleluiah. Amongst the moderns this mode is in so much use and is so loved, that, induced by its sweetness and its beauty, they have changed many chants composed in the fifth mode into the eleventh, by inserting the note b-flat instead of b-natural. 
The Graduale romanum dates this Kyrie melody to the 15th-16th century, but the musicologist Amédée Gastoué argued it is more ancient, having discovered a version of it in a 14th century gradual from the Cathedral of Rouen, to be sung ad libitum on solemn feasts. It quickly became popular throughout France, and seems to have been often chanted in solemn votive masses; it became particularly associated with the votive mass traditionally celebrated on Monday in honour of the Holy Angels, and thus acquired its present moniker. A 15th century gradual from Rouen labels it de Angelis, and a Celestine gradual from the same century De sanctis Angelis. From France it spread to England and Italy, and acquired the hold on the Catholic musical imagination it retains to this day.
Pope St Pius X helped cement this popularity when, in 1904, he personally chose this Kyrie to be sung during the memorable Mass celebrating the 13th centenary of St Gregory the Great in St Peter’s Basilica. It was the first time in centuries that a Mass was sung entirely in Gregorian chant in St Peter’s.
The Kyrie de Angelis is one of the few Kyries in the modern Vatican edition that is not identified by the incipit of a trope. Indeed, given its late origin it was not often farced, but Dom Joseph Pothier did discover one troped version thereof in a manuscript gradual from the diocese of Toul from 1622, with the heading Super Kyrie de Angelis. Interestingly, as Dom Pothier explains, this gradual was prepared for the use of a simple country parish: a valuable indication not only that into the 17th century sung Masses were celebrated even in small churches, with full Gregorian chant, but that tropes survived the liturgical reforms spurred by the Council of Trent. In fact, the diocese of Toul kept its own liturgical use after the promulgation of Quo primum thanks to the intransigence of the cathedral chapter, which stood athwart their bishop, Cardinal de Vaudémont, who, pressured by the Jesuits, sought to impose the Roman use.
The composer of the Kyrie Rex æterno trope has taken the liberty of modifying the original melody, which is relatively unusual. That this was done deliberately, and is not a case of regional melodic variation, is shown by the fact that the manuscript also contains the untroped Kyrie de Angelis in essentially the same form as in the Vatican edition. The exuberant short melismas in the last three Kyrie verses are also noteworthy.
The striking metrical character of this Kyrie trope betrays its late origin. As Dom Pothier notes,
The first three Kyrie have tropes always composed of three verses, the first of four long syllables, i.e. a double spondee, and the two others of five syllables, forming a dactyl and a spondee, otherwise known as an Adonic verse; giving a total of fourteen syllables.
The three Christe also each have a trope of fourteen syllables, but these are contained in only one verse, viz. a hexametrical verse, of the sort that are called Leonine, where the two hemistichs rhyme together.
As for the last three Kyrie, these are also in verse, but now exclusively in Adonic verse. The tropes here, moreover, are doubled, not only at the last Kyrie, which is the usual rule, but for all three. The only difference in the case of the last Kyrie is that the vocalise, which is in any case very short, is repeated as a sort of echo.
Gastoué criticized the Kyrie de Angelis for its jocund melody, which is better suited, he says, to an Alleluia than a “cry of supplication” like the Kyrie. Nevertheless, he admits that
in this chant, we find a melody of ‘joyful’ supplication; the reason for this is, perhaps, if we consider the paintings and the sculptures of the Middle Ages, that the contemporaries of that time conceived the pure heavenly spirits praying God for humanity, with this calm joyfulness.
The text of the trope entirely supports this supposition. Throughout, the tone, despite a frank acceptance of the sinfulness of men, is one of confidence in God’s mercy, for he is the flos pie florum fonsque bonorum: an entirely suitable idea to ponder as the holy sacrifice of the Mass begins. The last verse is, moreover, a felicitous prayer for the start of the Mass: suscipe rursum, dirige cursum, corde rogamus, scandere sursum vivere cursum quo valeamus.
In the recording below, the Kyrie Rex aeterno trope is sung by the Capella Antiqua München.
 Di questo si trovano molte cantilene ne i libri Ecclesiastici, si come la Messa, la quale chiamano de gli Angioli, le Antifone Alma redemptoris mater e Regina Cœli lætare Haleluiah. Questo Modo da i Moderni è tanto in uso, e tanto amato; che molte cantilene composte nel Quinto modo, per l’agiuntione della chorda♭ in luogo della♮, hanno mutato nell’Undecimo; indutti dalla sua soavità, e dalla sua bellezza. (Gioseffo Zarlino, Le istitutioni harmoniche, 1558)
Gastoué, Amédée. “The ‘Missa de Angelis'”. The Caecilia, Vol. 60, no. 12, pp. 375-8.
Martin, Eugène. Histoire des diocèses de Toul, de Nancy & de Saint-Dié. A. Crépin-Leblond, 1901. Vol. 2.
Pothier, Dom Joseph. “Kyrie des Anges avec tropes”. Revue du chant grégorien, xiii (1904-05), pp. 81-88.