cantoresq…your posting begs the question…Which trope do you accept as nussach? The litvishe trope? the German trope? The Galitzianer trope? The Yemenite trope? (Probably the most authentic one)
In truth, none of us even how what they sang in the Bais Hamikdash and what was music two thousand years ago. And nussach changes all the time but them, we have argued over that in another posting….
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Firs of all, I really am limiting my comments to ashkenazi nusach. I’m not very well versed in the development of Sephardic chazzanut much beyond a very superficial exposure to the makamot. But to answer your question, first off, there is remarkably little difference between say, the Lithuanian trope system and the German one. Secondly, I refer you to the seminal research of Leib Glantz and Abraham Idelsohn before him into the origins of trope, their relation to Greek tetrachords, the distintcly Jewish adaptations of those terachords, the expansion of trope into nusach, with the addition of additional tetrachords, and the expansion of nusach into chazzanut. Indeed we do not know precisely how the shira sounded. That tradition has been lost to time. But we do know the basis for that shira and can trace how that basis developed into Ashkenazi chazzanut. For a good book on the topic, I suggest you read the recently published The Man Who Spoke to G-d; festschrifft puvlished by Jerry Glantz in tribute to this father Chazzan Leib Glantz.