Here I am
I wish there was an easy answer for the OP but there isn’t. I can’t speak for Eastern traditions but though I’m mostly a performer than a researcher when it comes to chant, in the Western tradition there’s no easy answer. There were several chant traditions in the Western Church. Only three seem to have survived in current liturgical use: Gregorian (the predominant one), Ambrosian (in the diocese of Milan) and Mozarabic (diocese of Toledo in Spain). The latter only in very limited use.
Some that didn’t: Old Roman chant (Italy), Gallican chant (France), Beneventan (Italy), Sarum (UK), Celtic (British Isles).
The main body of Gregorian chant seems to come from the Carolingian era. Gregorian chant sort of died out by the 17th century… or rather it became adulterated and corrupted. It wasn’t until the late 19th century that the monks of Solesmes revived chant and researched ancient manuscripts to provide the chant we know today; the Graduale Romanum of 1908 was the seminal book of Chant for the Mass that was the culmination of that work. Similar work took place on the side of the Divine Office; in 1910 the Monastic and Roman (diocesan) Divine Offices parted ways with the 1910 Breviarium Romanum.
The chant we know today isn’t exactly how chant was in the Carolingian era. It is how the monks of Solesmes interpret how it was. Most early chant was written in neumes.
This introit taken from the Graduale Triplex (the Graduale Romanum with the ancient neumes superposed) shows the difficulty. The neumes were not written on a staff. The Gregorian notation between the two sets of neumes is not traditional, it’s a 19th century creation. Try interpreting the neumes (the top from Laon, the botton from St-Gall, from the period 10th to 12th centuries approximately) without looking at the Gregorian notation:
You have to agree it’s a tough job. The monks of Solesmes have produced a practical work for the performance of chant and took some liberties that drive musicologists nuts. It really is an interpretation. We have no exact idea what it sounded like. Moreover it’s a work in progress: here’s an antiphon that changed between 2008 and 2010 in two different chant books: “Les Heures Grégoriennes” and “Antiphonale Romanum II”.
Les Heures Grégoriennes, mode VIIIg
Antiphonale, mode VIIIc
As for Jewish chant, especially of the psalms, a Jewish expert would be better placed to help. All I know is that the Church did take an awful lot from Jewish liturgy.