Wikipedia
Alternatim refers to a technique of liturgical musical performance, especially in relationship to the Organ Mass. A specific part of the ordinary of the Mass (such as the Kyrie) would be divided into versets. Each verset would be performed antiphonally by two groups of singers, giving rise to polyphonic settings of half of the text. One of these groups may alternatively have consisted of a soloist, a group of instruments, or an organ. The missing even- or odd-numbered verses were supplied by plainchant or, perhaps more commonly (to judge by the organ masses of Hans Buchner), by improvisations on the organ. The verso became a particularly prevalent genre in Baroque Iberian organ music.
A large amount of musical repertoire was specifically written for alternatim performance, with Heinrich Isaac and Charles Justin (1830-1873) as notable composers. Alternatim performance of the Mass was common throughout Europe in the seventeenth, eighteenth, and nineteenth centuries. A similar tradition of alternatim performance existed for example also for Magnificat compositions.
In the Catholic church, the practice was banned by Pope Pius X in his 1903 Motu proprio Tra le Sollecitudini. The practice did, however, inform the works of Olivier Messiaen who wrote pseudo-versets for his many liturgical organ works, especially his Messe de la Pentecôte (1950).