Tragedy literally means
a goat's song or ode.
Is there any indication in mythology of why a goat song would be sad or dramatic?
As far as I know, Pan, Faunus, Silvanus... all part goat deities are more in tune to celebrating life to the hilt...
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The German Wikipedia article on tragedy explains this well:
The word "tragedy" stems from ancienc Greek theater and denotes the "he-goat's song" or "song for the he-goat's price" (gr. τραγωδία, tragodía). In the Dinoysos cult, a "Kosmos" (gr. κῶμος kōmos) was staged, a procession with song, the participants masked and wearing the skin of a he-goat (gr. τράγος tragos), to impersonate the god himself or his accompanying satyrs. The theatrical form of tragedy developed from a myth sung in chorus, ... The choral parts of the theatrical dramas are a rudiment of this ancient form, the dialog parts a later, secondary addition. (my translation)
No source is given on Wikipedia, but all this is in tune with what I remember from studying literature and learning of the origin of ancient Greek theatre.
A tragedy, in ancient Greek theatre, is not "sad" in the contemporary non-theatre-related usage of the word (e.g. "What a tragedy!"). The Greek tragedy is defined by a fated conflict of the protagonist: the failure of the protagonist is unavoidable, because of the constellation he or she are placed in. Think of Oedipus: He did nothing wrong, it was his fate to kill his father and marry his mother. In contrast to a modern hero, there was nothing that he could to to avoid that fate. Modern heros on the other hand either succeed or fail because of their own deeds.
The ancient Greek comedy also derived from the Dionysian procession (see "komos" in my translation above). But while the tragedy took the mythic part of the procession and told stories of gods and men, the comedy took the festive part and became a satire and critique of (then) current politics and society. The ancient Greek comedy was more like today's political satire, than today's comedies.