1. In counterpoint, if at least one voice must sing Kyrie (or Christe) three times in each section, or if each voice can just sing Kyrie one or two times on a timing which is staggered from the other voices.

Of course, when the priest and server say it, and when a choir sings it antiphonally in plainsong, no one voice is singing Kyrie thrice. Also consider polyphonic settings that were written for double choir that toss texts back and forth. It would be so incredibly silly if this were a rule that this can’t be a rule, because not even the plainchant follows this rule.

2. If (and why) it is permittable to sing Eleison as a two syllable word (pronounced Eleson) instead of singing all three syllables distinctly as they are written in the official chant editions.

I assume this convention evolved from local variations in pronunciation. In some places it even seems that the intention was to elide the i and the e in Ky - rie.

This sort of thing is all over the literature, and I’m not aware of positive law forbidding it now - it isn’t an alteration of the text as much as an indication of how singers are to fit the text over given notes, as I see it. Metered music makes different demands on diction than the plainsong. It seems to me that, in metered music, an overheld “i” in eleison could do more violence to the pronunciation than simply including it right at the end of the “e” vowel.

Also, I heard once that new compositions of the Mass ordinary must be approved by the Bishop (or somebody) before being used for Mass, but I seriously doubt this to be true. Could somebody knowledgeable please confirm or deny this claim? I would imagine that one could just write a Kyrie one day, and use it for Mass, as long as it's reverent and agrees with the character of that particular Mass (the ideas being expressed in that part of the liturgical year).

With respect to the older Mass, which seems to be your area of interest given your Kyrie question... those Commissions which used to do this work on behalf of the bishop for Latin ordinaries according to the old norms have gone the way of all flesh long ago.

My point of contact within the hierarchy of you are concerned about this would be your pastor.

With the Kyrie, it may be worth looking at what the great composers did,
Have a look at the Byrd 3 part...
It would also be worth looking at Palestrina, what did they do?

With the Kyrie, it may be worth looking at what the great composers did,
Have a look at the Byrd 3 part...
It would also be worth looking at Palestrina, what did they do?

I have been doing that quite a bit. A while ago I asked our pastor about the way Palestrina repeated the text in a certain piece, which seemed to bend the norms slightly (I think it was an Agnus Dei), and he just replied "I'm not going to argue with Palestrina".

I was hoping there were specific rules somewhere in writing, since that would clear up any possible doubt. But if those sort of rules don't exist, I guess I'll just assume anything I see other composers do is fair game (at least the composers whose music is actually used at reverent Masses).

We are supposed to sing the text as found in the Graduale Romanum (citation needed).
So technically we need to sing Kyrie x3, Christe x3 then Kyrie x3... The Byrd 3 does not in anyway follow this, so what do we do. Well we have sung Chant, Byrd Kyrie 1, Chant, Byrd Christe, Chant, Byrd Christe, Chant, Byrd Kyrie 1 then Kyrie 2. We were told by a priest that this was fussy. We tried repeating the Kyrie 3 times, followed by the Christe 3 times etc. But that sounds silly.

Do the 3 voices count as singing the text 3 times? I am unsure what we should do. I suspect that Byrd just sung the whole thing once and did not worry about the repetitions.

I like the idea of Palestrina being a good authority, and it should work as a rule of sorts.

The problem with rules is when were they written and do they still apply. Would your bishop / parish priests understand the rule, or would they think something written say 120 years ago would no longer apply?

I still think that if the great composers did something and it was found in a number of pieces it should be fair game for us to do the same. Although I would recommend that examples should be on hand ready for anyone that complains.

We are supposed to sing the text as found in the Graduale Romanum (citation needed).
So technically we need to sing Kyrie x3, Christe x3 then Kyrie x3... The Byrd 3 does not in anyway follow this, so what do we do. Well we have sung Chant, Byrd Kyrie 1, Chant, Byrd Christe, Chant, Byrd Christe, Chant, Byrd Kyrie 1 then Kyrie 2. We were told by a priest that this was fussy. We tried repeating the Kyrie 3 times, followed by the Christe 3 times etc. But that sounds silly.

3 voices stacked singing identical text and counting as repetition seems a little too "telescopic" to me. That is not to say, however, that any single voice would sing the entire text -- which would not have happened even if we'd been singing straight out of the Graduale.

The rule is to do what established Catholic composers have always done.
Oh, but they don't agree?
There's your rule.
(I don't get this thirsting after rules. Isn't the Decalogue hard enough?)
My procedure is set the Kyrie with a clear ninefold structure, because that makes liturgical sense to me. If it's for the Novus, and somebody wants sixfold, I will tartly ask which member of the Trinity got demoted, and then proceed based on pecuniary interests.
I also set Ky-ri-e e-lei-son.
You are welcome to make my procedure your rule. Or not.