Well…. now things look different!
Harmonizer and Pitch Shifter are the same thing. They are nboth based on software modules (in Eventide programming) which use an internal pitch detector….as all non Eventide products do.
A small amount of detuning (a few cents) are not a problem. Audio will sound ok.
Octaves sound fine too, even though a coplex chord may show some artifacts, depending on different factors.
Eventide units have no problems at doing all you have described…. but regarding polyphonic sources…if they are simple, octaves and sometimes 4ths and 5ths should work. When the source is complex as the chord I mentioned….forget it.
Harmonizer (R) is an Eventide trademark.