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Hello and thank you for your time! My knowledge in music theory is limited, so excuse me. I don't know where to look for the answer.

The goal is simple: harmonize a upper melody with chords stacked below the melody so that it sounds like big band brass hits. I heard in a YouTube video to stack thirds below the melody to achieve that jazz brass sound.

But the problem is that such a generated chord below the Cantus firmus results in a chord with a different root than the note that is actually to be harmonized.

Or is the simplest way just to put inverted seventh chords below the melody line?

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First harmonise the melody. Decide what chord goes with each melody note. Then stack notes below the melody using the notes of THAT chord.

Big Band style will, indeed, probably use a lot of 7th, 9th and 13th chords, secondary dominants and the like. And a tonic function chord may well be spiced up with an added 9th (2nd), 6th or major 7th.

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