Adding to this, today in Israel there are two major religious delineations each with their own Rabbinate and Liturgy: Ashkenazi and Sephardi. It is true that over the centuries the Sephardi and Mizrahi Rabbinates were much more closely connected than each other with the Ashkenazi one.
While studying Jewish music, distinction must be made between those two Eastern branches since their music shows important differences. The Mizrahi music is much more close to Eastern musical traditions while the Sephardi music is something like a bridge between East and West.