William said:
Someone tell me more about this practice.
I'm unaware of any OO liturgical tradition in which the entire Psalter is recited daily in the Divine Office at present, but I don't think it's an exaggeration to say that it used to be universal.
Bar Hebraeus, IIRC, records a Psalter reading plan (similar to the twenty kathismata in the Byzantine rite) which was meant to be read in full every day at the Midnight Office in the Syriac Church. The late Fr Woolfenden describes this plan in his book
Daily Liturgical Prayer: Origins and Theology (he repeats a lot of what Fr Taft has already covered in his book on the Office, but includes details Taft omitted or of which he was unaware, so I usually recommend reading this after Taft). Traces of and references to this continuous psalmody still occur in the hymnography of the Office, but it has fallen out of use. If you know the structure of the Office well enough, it's fairly easy to include the psalmody in the Midnight Office where it most likely used to be, and I do this in a modified way in my own practice.
AFAIK, the tonsured monks are required to read the entire Psalter daily, along with the full Office and a canon of private prayer which is hard for me to describe without actually showing you some books. I suppose it is like reading the morning and evening prayer rule from the Jordanville Prayer Book, except that it is structured around the seven canonical hours (and not just two moments in the day), and is based on a weekly cycle. I imagine monks who serve in the world (e.g., as parish priests where there is need) have this rule adjusted to accommodate those demands, but I'm not sure. I've never been able to get a monk to tell me about his rule.