I would argue that Hedda experiences both a modern and a classical tragedy. A classical tragic hero is a man or woman of high status who experiences a downfall because of a tragic flaw in his/her personality. Audiences are meant to learn from the experience of the tragic hero, to feel pity for the fallen and learn moral lessons.
Modern tragedy tends to implement social flaws as part of the tragic trajectory that the characters experience. Arthur Miller used this idea in both his modern tragedies, The Crucible and Death of a Salesman. In the first, John Proctor is flawed, but it is the society that persecutes him. In the second, Willy Loman makes many mistakes, but he is led astray by visions of the American Dream as enforced by his society.
In a classic way, Hedda is a woman of high status with a tragic flaw. Much reference is made of her father, the general: "General Gabler's daughter! Think of the sort of life she was accustomed...
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