I mean, it kind of sounds like you have the sheet music for the piece in question (as how else would you know that the “notes are all the same” (i.e. no sharps or flats). In that situation, it should be pretty easy to find the tonic note; if it’s an A, the piece is in A minor, if it’s C then the piece is in C major.
If you’re listening to the song, then the melody should give you a good idea. There is typically a strong sense of what the “home” note is.
If you’re simply looking at a chord progression, then there might not be a straightforward way – although many songs tend to resolve to the I chord so that might be a clue.
They have the same notes but their position is different. So, the first note, which we call the root, changes. This root note is like “home” which is to say that we want to resolve to.
Furthermore, a scale allows us to build the chords and completely different chords are constructed between the two keys.
This gives a happy or upbeat feel to major keys and a sad or solemn feel to minor keys.
I’m a bass singer in a church choir; I also play guitar fairly well, and fool around on an electric bass. With training and experience, you get a sense of what the root of a given chord is, in large part in the context of the piece of music as a whole. The most common four chords in the key of C would be C F G Am, but the most common four chords in a song in Am would be Am Dm E C.
Everyone else has answered it well, but I’d like to add that the notes are only the same if it’s A natural minor. There is also harmonic and melodic minor scale, which have a raised 7th note, or raised 6th and 7th.
So a piece in A minor could be identified by the use of F# and/or G#. Neither of these are likely to be in a piece in C major (though it’s not impossible).
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