I would think you instead would have copyright on the new bit that is mixed in, and because of that, the combined version as a whole still contains your copyright and therefore cannot be copied freely.

I don't think that would give you any new rights over the original work though.

But perhaps crucially would give you effective financial control via online tools that look for a musical fingerprint?

If the public have the rights but can't exercise those unimpeded then the erstwhile rights holders get to retain the financial benefit and prevent full effective entry to the pubic domain.

Bastards.

I’ve heard stories of people uploading their old works to YouTube’s ContentID system so it will remove them from an old account they can’t access anymore.

So you could upload science and own the inverted wave version of a song?