restore music library from time machine backup

The general process should be covered in Restore items backed up with Time Machine on Mac - Apple Support. What exactly do you need to restore? A specific album, the state of the .musiclibrary database, or the entire Music folder?

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Enter Time Machine, select a date when that album would still have been in your library, drill down to the folder that should contain it, e.g. ~/Music/Music/Media/Music/<Album Artist>, select the folder and click Restore. Exit Time Machine, then drag the restored album folder into the Music app. The precise location might be different if your current library was converted from iTunes, but you should be able to work out where your artist folders are stored.

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Your older machine may have had a library converted from iTunes, so its media files might be stored in ~/Music/iTunes/iTunes Media. You can use the right/ctrl-click > Show In Finder option in Music to reveal where files you have are stored, which should be a clue as to where the missing content used to be.

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