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I read the sentence
The food was complemented with an array of musical bands.
My question has two parts:
Can food be complemented with music?
The difference between complemented by and complemented with. Are both correct? If not what is the difference?
It's a passive construction. "Moscato wine complements chocolate" = "Chocolate is complemented by Moscato wine". However, I feel that in the context of your example, there's no intention to convey that the music was chosen to complement the food; the music merely accompanies it.
Both prepositions are about equally common in OP's context, and I see no reason to prefer one over the other (nor do I see any possible semantic distinction). From Google Books...
His report was complemented with information from (some reliable source). (456 hits)
His report was complemented by information from that analysis. (519 hits)
But as has been pointed out, most people probably wouldn't use the passive construction for OP's Can food be complemented with/by music? They'd more likely ask Can music complement food?
The idea underlying 'complemented' is completion. OOD gives this definition of the verb to complement: "Contribute extra features to (someone or something) in such a way as to improve or emphasize their qualities."
The writer of the sentence you quote believes that the music improved the experience of enjoying the food.
I don't see any difference between with and by in the context of your sentence.
