Another day, another mis-spelled song title. Par for the course when it's me doing the posting.

The Three Marys are of course "Mhuire." I was careless when posting the OP, copying from the Hyde-edited anthology; the Gaelic print in the anthology follows the convention of a dot above any given consonant, which indicates that the consonant is followed by an aitch. Unless you know what the dots are there for, the name on the page looks like "Muire" but of course it isn't.

Actually, looking up the printed sources and the numerous recordings, some disagreement over the title itself surfaces. Joe Heaney, it is stated, used a different title for this particular song, and if I read his commentary right, he learned it orally, not from a printed book.

An earlier recording than Heaney, or for that matter Ní Ghuairim, is the 1938 recording for HMV by Máire Ní Scolaí but, as Jim Carroll has kindly pointed out on the thread about the latter singer, she was not a sean-nós artist as such, but a concert performer whose repertoire included arrangements of traditional material. This recording, originally a '78 single, is of a concert-hall arrangement for voice with the accompaniment, imitating a harp, for a piano; in which regard the performance sounds that much more European-classical than traditional. I don't see an arranger credited on the label; the pianist is Duncan Morrison, but that doesn't mean he wrote it down himself. From my experience with classical songs with piano, it seems safe to presume that the pianist is playing, not by ear, but from a written or printed piece of sheet music on the piano's music stand: this is the convention in classical art song, that even if the pianist knows the music by heart, that music score is up there on the piano in performance. In that the performance convention differs from solo classical pianists, who centuries ago had the score in front of them in performance, but over time decided to perform from memory instead.

Anyhow, back to the title. The song title, as identified in the thread title (mis-spelling and all), appears in the Hyde edition and on Ní Scolaí's recording; I don't know, but the connection could be a direct one, as in, whoever wrote Ní Scolaí's arrangement might have been working directly from the Hyde version. In any event, a precedent was set, and more recent recorded performances seem to agree on the "Tri Mhuire" title.