The first stanza sets up the contrast between summer and loveless Emily, which the rest of the song develops. Summer is usually seen as one of the best times of the year, but despite that fun connotation, Emily's life is sort of dismal.
She watches her flowers grow
While lovers come and go
To give each other roses from her tree
But not a rose for Emily
Emily, can't you see there's nothing you can doAnd while much of the song talks about this by illustrating the contrast between Emily and the other lovers, it is also illustrated by the contrast in the weather: "Though summer is here at last / The sky is overcast." Summer is usually a time of enjoyable weather, but here "the sky is overcast." The though indicates the opposition between the first two lines.
There's loving everywhere but none for you
"The summer is here at last / The sky is overcast" does not have that opposition, and it seems choppy and contradictory. Though signifies a subordinate clause, which improves upon both of these shortcomings. It illustrates that the two lines are grammatically connected and intentionally opposite.
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