I'm no expert on ancient Greek music, but Pythagoras did not invent the three tetrachord genera. It does appear that his mathematical work heavily influenced others who defined ancient music theory.
Pythagoras was born in the 6th century BCE, and he was the first to work out ratios of string lengths that produced certain tonal intervals such as the "perfect fifth." The systema teleion, from the 5th century BCE, discussed tetrachords as they were understood at that time. Historians do not agree on who invented Pythagorean tuning.
Ptolemy and Boethius, both famous historians in their day, actually attribute the tuning to Eratosthenes, not Pythagorus. (See: https://en.wikipedia.org/wiki/Pythagorean_tuning)
But it was Archyta, a Pythagorean disciple from the 4th century BCE, who defined the diatonic, chromatic, and enharmonic genera as we know them today.
Also indebted to Pythagoras were philosophers Aristoxenus and Cleonides -- contemporaries of Plato and Aristotle -- who wrote the first treatises on harmonics in the centuries that followed.
More info here:
It's hard to say for sure, but it seems very likely that the many music schools that popped up in the 6th and 5th centuries BCE began to formulate music theory, and it's quite possible tetrachords go back centuries before that.