We could go to all sorts of crazy ends with the Antikythera mechanism, not the least of which would be supposing a proto-Babbage engine.
But lets stay a bit more grounded: Let us assume that some Greek artisan or philosopher that was familiar with the Antikythera mechanism (or one of its likely counterparts) decided to apply that craftsmanship to keeping track of smaller increments of time, rather than the motions of the heavens.
What might we see if clock technology advances to where it was in the 10-11th century AD, in the Hellenistic era?