In Macbeth, is Macbeth a tragic hero?

If you go by the Aristotelian prescription, the tragic hero must
be a person belonging to high rank/station. Macbeth, the most admired General of King Duncan, is
one such towering personality. But Aristotle suggested that the ideal tragic protagonist, though
not all good, must be generally inclined to the side of goodness. In his view, no villainous
character can ever be tragic, for the overthrow of a villain can never arouse the emotions of
"pity" and "fear" to achieve a "catharsis" of those and suchlike emotions. Macbeth, sometimes
called a "villain-hero", does not conform to this Aristotelian requisite. Macbeth rather
illustrates the paradoxicality voiced by the witches at the outset of the play: Fair is
foul and foul is fair. Divided within his own self between his "vaulting ambition" and
imaginative conscience, Macbeth is both"fair" and "foul"; both a saviour and a tyrant; both a
poet and a murderer. Macbeth is a Renaissance protagonist who, by his own characteristic moral
failing, invites his own fall. His decline from prosperity to adversity is the restoration of
order and vindication of justice. Even then, Macbeth does not wholly alienate our sypathies
though he goes down to defeat and death.