best way to back up Music

Assuming your library is neatly organized within one folder you can take a one-off snapshot of the library by copying the whole folder to another drive. Efficiently backing up only recent changes would require folder syncing software. In Catalina backing up your media has become more complicated as the single iTunes library has been replaced by separate libraries for Music, TV, Podcasts, and Books, with last two living in hidden locations within ~/Library. The Music artwork cache is also now in a hidden folder so typically wouldn't be captured in a simple backup.

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Time Machine will back up the same way it did before iTunes was removed. The source data, your local music files and artwork etc.. is in the same place it was before. The only difference is the app that "plays" the music is not the same as the app that "manages" the music.

Use another external USB drive to mirror your Time Machine drive.