Instrumental from an existing music

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So, what is the relationship between these two recordings?
An artist composed a music for a competition which he didn’t win. Later, he “changed a few parts (of the music) and added … lyrics” to make a new song with the music as the instrumental. The music is different enough when compared side by side with the instrumental to be called a remix but how could I relate the music to the song he created?

I would create two works and link them together with the “based on” relationship.