David W. answered 06/11/19
PhD in Music Composition with 5+ Years of Teaching Experience
You might want to investigate the music of Arnold Schoenberg, who invented a method of composition in the early 20th century called the "12 tone system," or dodecaphony. The principle was that all 12 notes would be treated equally, which I think corresponds to your idea of composing with the chromatic scale. To ensure that this worked, he worked out an order for all 12 notes—call a tone row—and made the rule for himself that he had to use all 12 notes of the row in that order before starting over.
It was more complex than that, of course. For starters, you could also use transpositions of the row—which would also have all 12 notes in them—or do them backwards, or inverted, or inverted and backwards. And he could use more than one form of the row at a time.
Sometimes Schoenberg and his musical descendants made up new forms for this music; sometimes they used older forms, although they sounded very different from the older versions.
Because all 12 notes were treated equally, there were no key relationships, and thus no tonics, dominants, etc.