Unfortunately, it is not my idea, but it seems that such alternation was intended by the composer.

I read that some masses of Dufay have the same shape: Kyrie eleison, Christe eleison, Kyrie eleison are each set only once.
It is written in Jackson's "Performance practice" that such configuration is evidence of chant-polyphony alternation.
Exactly the same situation is in the mass "Rorate coeli" composed by G.G. Gorczycki (polish composer from XVII/XVIII century).

So, the question remains: if we need to alternate chant and polyphony how to do it?

We ran into a similar situation this past Christmas while preparing to sing "Byrd 3" for the midnight EF Mass. David Hughes was kind enough to provide me with a setting in alternatim but starting with the polyphony. Therefore: Kyrie p/c/p, Christe c/p/c, Kyrie P/C/P, and not uncommon according to David.

OK – This is my third attempt to write a comment to this forum. I was trying to reply to Aga’s and Richard R’s queries and comments, but my previous messages got eaten by the Forum software. So this may be a day late and a dollar short, but anyway –

It is sometimes obvious that parts of the mass were intended to be chanted – for example, many Iberian masses from the Renaissance have two ‘Christes,’ indicating that there were chant parts and that the first Kyrie would have been chanted and the center part would have been a chant ‘Christe’ sandwiched between two polyphonic ‘Christes.’ I personally think nearly all of them were done in alternatim chant & polyphony (my opinion, not fact). For Solemn High masses we do it like that, whether starting with polyphony or chant, because at certain places in the mass more time is needed for the actions in the sanctuary to be performed without rushing. For a regular missa cantata we just sing the polyphony – usually there are at least three iterations of the text in each section.

A similar situation exists for the Agnus Dei. If there is only one setting and the text ends with ‘miserere nobis’ (e.g., in many Lassus masses), it seems to me that it was probably meant to be sung between two sections of chant. If the piece is not too long we usually sing it first and then sing a chant second Agnus, followed by the same polyphony, but we substitute ‘dona nobis pacem’ for ‘miserere nobis,’ which not only has the same number of syllables, but the accents even fall in the same places, making the substitution quite easy. If the piece is long, I sometimes edit it, resetting it with the complete text and all three repetitions included. This does take time and some ingenuity but is usually worth it.

As for choosing chant pieces to insert in the Kyrie or Agnus Dei, if you can figure out what mode the polyphony was written in, you can pick a chant of the appropriate mode from the mass ordinaries in the Liber. There is usually more than one in any given mode (don’t forget the ‘ad libitum’ section following the Credos). Finding the mode is not as hard as one might think – a lot of Renaissance masses are written in transposed Mode I (the key signature has B flat but not E flat, and they tend to cadence on g and have a ‘minor’ sound). Mode V has a ‘major’ sound and the key signature does not have the B flat but sounds similar to F major, cadencing on F. Some of the other modes are not as common in polyphonic writing, so it doesn’t have to be rocket science. If all else fails, you can look through for a piece of chant that just sounds right – much chant travels through several modes on its way to the final in any case.

The most important thing is to make sure you know what the clergy is comfortable with – some celebrants are not as concerned as others about the length of various parts of the ordinary - but we always have at least a moment of conference with the celebrant before the mass so there are no surprises. The music is always the handmaiden of the liturgy, not the other way around.

P.S. Re: Yurodivi’s comments about Machaut – that’s very true, but we still use pre-Trent masses – they just have to be edited. You can chant part of the text, using the techniques above. That takes a big chunk of time, but is worth it. Musicologists would probably protest vehemently, but this is not a concert, nor is it a seminar in grad school – it is a living, breathing mass, and I am certain that in the ages before the apotheosis of the “composer” in the 19th century, people were much more flexible in using music than we have been since. We have a number of pre-Trent masses that we do regularly, and various parts (especially the Sanctus) of all of them have had to be changed to fit the Tridentine liturgy, but they are all wonderful.