A big topic, and I wish I had been keeping up with blogging about this, since I've been doing a lot of practice in this area in the last couple of years. I'm especially on fire about it because it is how Haydn would have learned solfège, and there's also some evidence that Bach was in favor of this system. Of course I often teach modern solfège too, but I'm really enjoying getting into this historical practice. To your questions:

1a. In general, one wants to switch onto Re going up and onto La going down no matter where you are in the hand and which scale you are doing. This is widely documented over many centuries. Singing one octave of the C major scale is DRMFSRMF going up and FMLSFMRD going down. If you keep going down, you try to switch onto La. In other words, the whole hand from G at the bottom of the bass staff up to E at the top of the violin staff goes like this, if you are using B natural: DRMFRMFSRMFRMFSRMFSL and down is LSFMLSFLSFMLSFLSFMRD. Notice that on the way down, only B=mi, which makes it especially significant. There are symmetrical FSL groups above and below. The same thing is true of G=sol on the way up, with RMF above and below. Interestingly, Do ends up getting used very little, which is part of the reason why it disappeared in English methods as well as in shapenote singing. Essentially, shapenote (four-syllable solfège) adopts the traditional way of singing the scale down, so there is only la, sol, fa, and mi.

1b. In practice, there is always a little bit of a grey area when singing with the syllables. The D at the bass of the middle fingers just is both Re and Sol. Sometimes you can justify singing one or the other. Usually what happens in my class is we agree on a point to switch, sometimes governed by phrases or groupings, but if we are singing something at sight there might be a few notes where both syllables are actually heard. I don't think this is a huge deal.

If you take my grouping idea, then the tract Domine Exaudi would start SFSFRFSFSSFRMFMFFMFRMFMFR. One could chose a different point for that first mutation.

In your example, Confitemini, you would actually just start on Sol, and there's no reason to mutate until "Domino." So: SSSFSFRFSFSLSFSFSLS // DRFMF etc.

1c. Dicit Dominus indeed uses the flat. Since this note is not in the hand, it is not musica vera; not part of the gamut. Instead, it's "made up" or musica ficta, by analogy with the B-fa of the higher octave. In practice, one sings Fa for all flats, including this B-flat. In singing this passage, you have to decide whether you want to have a whole fictitious hexachord on low F, so the chant starts LLFSRDFF, or whether you want to highlight that that note is not in the true scale but is a fa subbing for a mi, so the chant starts SSFFRDFF. You could probably find historical evidence for both practices in a case like this.

2. There is an old adage "una nota super la semper est canendum fa." If you only need the note a half step above the top of the hexachord, one can just borrow without actually switching hexachords. The last verse of Confitemini could start like this: FSLSSSLSLFSF.

Well, there's a lot more to say, but hopefully that is helpful.

https://www.youtube.com/watch?v=RlleweQuq14&ab_channel=ChurchMusicAssociationofAmerica

Dr. Mahrt sings so beautifully! Thank you for the video.

One really great thing about the old solfège is that it orients you toward the central half step (mi-fa), which is really the crucial and mode-defining part of the scale. Whatever hexachord you are in sort of puts you in the orbit of that particular point in the scale.

This idea of the half step being crucial extends to the clef notes, which are the key (“clef”) to the scale precisely because they are situated above the half steps. Even the b-flat sign was often considered a kind of clef historically, since it too signals where a fa is.

When you change hexachords, you also change which half step you are closest to. There is part of your mind that is calculating intervals around this, and the hexachords really bring this thinking to the surface.

One other advantage is that the transposed chants (e.g. mode 2 graduals) make a lot more sense because you see the relationship with the other clef.

When the system was applied to sixteenth-century music, pretty much any flat became a fa, although sharpened notes to make good cadences (what we now call leading tones) didn’t (for most writers) change the solfège but were given the same syllable they would have had.

The more I consider all these things, the more they make sense. Lots to think about.

Yes, and if you did the verses, you would see that the clef changes but the mode does not, because the verse at the pitch in which the Vatican edition puts the first part would require a high f-sharp. Compare a more recent edition which puts the whole thing in an F clef with low B-flats.