Antistrophe originally referred to a part in Greek drama spoken by the chorus. In Greek drama, a chorus of actors would move from one side of the stage to the other and speak the strophe, which comments on the action of the play and typically asks a question of some sort. The antistrophe was the answer--or response--to the question or dilemma presented in the strophe.
As a literary device, antistrophe refers to a repeated word or phrase that comes at the end of each sentence or paragraph in a text. Essentially, the repeated antistrophe is like an answer to a question presented in the text.
From Tolkein's Return of the King:
A day may come when the courage of men fails, when we forsake our friends and break the bonds of fellowship, but it is not this day. An hour of wolves and shattered shields, when the age of men comes crashing down, but it is not this day. This day we fight.
From Steinbeck's The Grapes of Wrath:
Then I'll be all aroun' in the dark. I'll be ever'where - wherever you look. Wherever they's a fight so hungry people can eat, I'll be there. Wherever they's a cop beatin' up a guy, I'll be there... An' when our folk eat the stuff they raise an' live in the houses they build - why, I'll be there.
From the Bible, 1st Corinthians 13:
If I speak in the tongues [a] of men or of angels, but do not have love, I am only a resounding gong or a clanging cymbal. 2 If I have the gift of prophecy and can fathom all mysteries and all knowledge, and if I have a faith that can move mountains, but do not have love, I am nothing. 3 If I give all I possess to the poor and give over my body to hardship that I may boast,[b] but do not have love, I gain nothing.
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