How to add iTunes to time machine to free up disc space

Bad idea. Where will your backup then be? What will you do when your backup drive fails?

When you put your iTunes data on an external drive, that drive is not your "backup" for the iTunes data. It's your primary copy of your iTunes data.

You can put your iTunes data on an external drive in TWO ways.

(1) If you want to keep your iTunes library database (the iTunes folder) on your internal drive, you can put just the iTunes Media folder (where your media files are stored and organized by iTunes) on the external drive. To do this, create a new (empty) folder called iTunes Media at a location on the external drive. Open iTunes preferences Advanced pane. Point the iTunes Media folder location to the new one on the external drive.

NOTE: The TWO checkboxes there for Keep iTunes Media folder organized AND Copy files to iTunes Media folder when adding to library should be checked, if they are not already.

Then, use the "consolidate library" command to tell iTunes to copy its media files to the new iTunes Media folder; from the iTunes menu bar, File -> Library -> Organize Library -> Consolidate files (checkbox). Your iTunes media files are now stored on and accessed from the external drive. iTunes does not delete the media files (now duplicated) on the internal drive; after confirming iTunes is accessing its media files from the external drive, you can free up the space on the internal drive.

(2) With iTunes not running, use Finder to copy the complete iTunes folder (located in your user account Music folder) from the internal drive to the external drive. Launch iTunes with Option key held down and open the iTunes library file in the iTunes folder (the one you just copied) on the external drive, as described in this document

Open a different iTunes Library file or create a new one - Apple Support

You "Choose" the file called iTunes Library.itl in the iTunes folder. After confirming Tunes is accessing its library and media files from the external drive, you can delete the iTunes folder on the internal drive to free up that space.

Method (1) can be noticeably more efficient, because iTunes is constantly accessing and updating the iTunes library file, and the internal drive is usually faster for data access. Media files are typically only accessed when they are played, so access speed is not as critical. Method (2) keeps all of your iTunes data in one place. I use method (1).

Either way, you need to have that external drive connected and turned ON when running iTunes going forward. This external drive should NOT be the same drive as your Time Machine archive. Instead, use a separate external drive for Time Machine. To back up the external drive with your iTunes data using Time Machine, open System Preferences Time Machine pane. Click Options. A list of items excluded from Time Machine backup is shown on that list. External storage devices are excluded by default; REMOVE the external drive with iTunes data from this list. Time Machine now backs up the external drive with iTunes data to the Time Machine external drive.