I'm thinking about starting a business that uses written melodies as artwork (making a poster using the sheet music of somebody's favourite song, for example). I would be transcribing the tunes by ear and would only be using the melody, not the complete piece. Would it still be illegal for me to publish and sell somebody else's work if it's not intended to be used for performance?
If the tune is not in the public domain then your transcription of the tune would be an infringement of copyright. The fact that there's no intention to use the notated music for a performance doesn't change that.
For tunes in the public domain, there would be no infringement.