Hi guido239,
If you have ample space on your iPhone for all of the songs, you may need restart your Mac and iPhone and sync again:
- "Connect your device to your Mac.
- You can connect your device using a USB or USB-C cable or using a Wi-Fi connection. To turn on Wi-Fi syncing, see Sync content between your Mac and iPhone or iPad over Wi-Fi.
- In the Finder
on your Mac, select the device in the Finder sidebar.

4. Select Music in the button bar.

5. Select the “Sync music onto [device name]” tickbox to turn on syncing of your music.
6. With the tickbox selected, syncing is set to transfer all your music to your device.

7. To sync a selection of music, click “Selected playlist, artists, albums and genres”.
8. Select the tickbox for the individual items you want to sync in the music list. Deselect the tickbox for any item you don’t want to sync.
9. Click Artists, Albums, Genres or Playlists to see your content organised by that category.
10. Select syncing options:
- Select the “Include videos” tickbox to include videos when syncing.
- Select the “Include voice memos” tickbox to include voice memos when syncing
- Select the “Automatically fill free space with songs” tickbox to have your Mac fill the free space on your device with songs when syncing. Use this option if you have a large music library on your Mac but don’t have room to sync all your music to your device. This option syncs as many songs as possible.
11. When you’re ready to sync, click Apply."
You can find these steps along with other information that may be helpful, here: Sync music between your Mac and iPhone or iPad – Apple Support
Restarting will clear out any minor software issues that may have occurred in the process.
Restart your iPhone - Apple Support
Shut down or restart your Mac – Apple Support
If you still have questions, reach out to Apple Support directly, as this may require closer guidance: Get Support
Cheers!