The speaker envisions the American identity as Unity yet diverse or different.
The phrase “varied carols” tells that an America which shares a same vision but also has different types for achieving that same goal. Americans which are singing together, but every person is singing their own “varied carol” of the American song.
In the poem "I Hear America Singing", the working class is singing a song. Each one - the mechanic, the seamstress, the mason, the carpenter - sings a unique song, a song of their own individuality.
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