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But A minor is the same notes as C major.

The chords in the key of C major will be the same chords in the key of A minor, although their patterns might be different.

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This is only partially true. The chords in A natural minor will be the same but it's much more common to hear a dominant fifth using the harmonic minor so you would change the Em to E especially in classical and jazz music.

nit: jazz is the melodic minor (raised 6th as well). Such a beautiful sound; and the modes allow you to get the altered scale (7th mode of melodic minor) which sounds good(?) over the dominant

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