That is a good and fair question John.
The answer is that it doesn’t matter if there is or isn’t an F in the melody.
The composer wrote a beautiful melody in the key of G major but found no use or need for the F note.
This is irrelevent as to the key in which the song is in. In this case it is indeed in the key of G major.
Not having an F in a piece of music that’s in G is more unusual than having one but it is indeed the case here.
If you want to analyze this music take a look at the notes used: more than any others, the most used notes in that song are G, B, and D.
when you lay out a G major chord what are the notes? G, B, D
and that is why you can’t just say it’s in the key of C.
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