I can attest that microtones are a part of chant performance practice to the present day.

The thesis sounds interesting from the POV of musicological recreation. From the POV of a parish musicians, it's very 1962.

Almost all correct intonation is on the low side of the organ's pitch. The only exceptions are a few notes, flatter than the key signature.

On the artful side of this question, there’s much one can learn about the tuning of chant from groups such as Marcel Pérès’ Ensemble Organum and Portland’s Cappella Romana.

On dealing with the non-artful side, aka flatting and pervasively poor intonation in ensemble singing, Peter Phillips (Tallis Scholars) published an extremely apt article “Singing polyphony” in the Winter 2014 Musical Times. It addresses a wide range of vocal and stylistic issues, especially the crucial role of chant in shaping Renaissance polyphony. He strongly suggests that only immersion in the Gregorian repertory can adequately prepare singers for the music of Victoria, Palestrina, and Byrd. He also holds that most professional voice training renders singers unable to shape and tune music before 1650.

Amen, and amen.

There are things worse than singers who tend to go flat: singers with a pronounced bias towards sharp. They are rarer, but worse, because they don't raise the rest, just more uncertain and therefore more flat.

I once had the distinct experience of listening to a quartet of singers at a church west of Rochester NY: two leads (one male, one female), each with a microphone, and two backups. The lead female had a noticeable sharp bias, the lead male trended flat. The backup singers (behind, and without mics) tried to weave a pitch between Scylla and Charybdis. It was ... memorable.

I should think any evidence of microtonal singing in the West would be negative, that is, a medieval theorist telling singers NOT to use them, especially in conjunction with fixed-pitched instruments used with the chant. I've never seen such evidence. Nor is it clear that instruments were not always used somewhere with chant, so that whatever microtonal singing existed in ancient times would pre-date western chant practice by a good bit. I'd be curious to know what research in the tuning of ancient instruments might suggest along these lines. Studies in ethnomusicology make a good case for microtonal singing, but as far as I know, the direct link between Gregorian melodies and ancient Semitic sources remains speculative.

Typically when we hear Microtones it is performed by a single cantor in melismatic style. I don’t think it is desirable (or possible) with two or more voices.

Nice stuff Simon. Are you intimating there are microtones in this?

Francis: Yes, there are microtones in the sample - quarter tones. In Europe, microtones – intervals smaller than a semitone – have been associated with oriental cultures. The idea that these ‘Arabic sounds’, notated in medieval liturgical manuscripts, could be part of Gregorian chant, the ‘foundation of Western (Christian) music’, has been too shocking for many to accept. Read the introduction of the paper.

I was interested in one of the comments in the link to the Coptic choir that Simon provided a few comments above here. The person said that when St Mark took the nascent Christian religion to Aegypt he, to make conversion and memory easier, took some of the songs of the Aegyptian religion and put Christian words to them - which, if true, would lead one to the conjecture that today's Coptic music is a relic (or a preservation) of pagan religious music from Romano-Aegyptian times. Can anyone here verify this interesting and not unlikely historical development?

While I listened to the choir and the leading singer with interest, I was really offended by the clueless use of a microphone. It seems that they just sing everything by rote with very little of the musical refinement that we would expect in western choirs of any confession. For sure, they represent a fundamentally different musical aesthetic from our western one.
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And, while this is not about microtones (it's actually about whole tones), I read somewhere long ago that in mediaeval Sicily (and presumably up into modern times) parallel organum at the second was sung there during Lent. Has anyone else encountered this?