Why would an arrangement be in a specific key if the sharps aren't used?
I've been sight reading through a book, "easy piano classics", part of the Bastien older beginner piano library. In a few of the arrangements, the key signature indicates that there will be sharps or flats, but those notes aren't used. For example, there is an arrangement in g major but it never contains the F#.
So why not just think of it as a mode of c major and keep the sharps out of it?
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Key signatures don't indicate that there will be # or b, as you found out.They do what they are called: they give a sign denoting the key. It's quite unusual with #s that there is no leading note in a tune, but maybe the 'Easy piano' is a bit of a clue.
It's more of a technical thing having a key signature, and often gives the reader lots of clues about what to expect - not just which notes aren't white!
Yes, some pieces could be regarded as being modal - in fact, Ionian and Aeolian are commonplace.
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