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In notice that in some vocal works composed during the Renaissance, especially masses, long chunks of music (sometimes the whole piece) are preceded by a monophonic, often short chant which is sometimes written on a stave completely separate from the rest of the voices; I linked a few pictures below highlighting examples of what I'm referring to.

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My question is, were there specific ways in which chants were to be incorporated into these works? I've noticed that sometimes, a chant would be played at the beginning of a piece, then it would used as the basis for its cantus firmus, but that's the only real pattern I've found so far; and it only makes up a small portion of the vocal pieces I've observed that involve chants. Does anyone have any information about this?

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The examples you cite are settings intended for alternatim performance. In this practice, there are two choirs. One sings the first half of each verse of the psalm or canticle, and the other sings the second half. Sometimes, this is apparent in a polyphonic setting because, while both choirs are polyphonic, there are different numbers of voices. Allegri's famous Miserere is an example of this.

Another common approach was to alternate between plainchant and polyphony. In such cases, the source for the polyphonic setting typically includes only the polyphonic parts. The plainchant in these modern editions will have been added by the editor from some other source. This is apparent in the given examples from the prefatory incipits, which show the beginning of each part as it was written in the source. These all begin with the first notes of the polyphony.

Note that none of the examples you give is taken from a mass ordinary, but, as you note, the practice was applied to masses. For example, sometimes there are two settings of the Christe eleison. This may be explained as a ninefold Kyrie in alternatim, starting with plainchant:

  1. Kyrie eleison (chant)
  2. Kyrie eleison (polyphony)
  3. Kyrie eleison (chant)
  4. Christe eleison (polyphony)
  5. Christe eleison (chant)
  6. Christe eleison (polyphony)
  7. Kyrie eleison (chant)
  8. Kyrie eleison (polyphony)
  9. Kyrie eleison (chant)

More common in the mass, indeed nearly universal, is the practice wherein the celebrant sings the incipit of the Gloria and the Credo in plainchant. Most commonly, the choir then sings the rest of those movements entirely in polyphony.

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