how do i manage memory when migrating from 3Tb hard drive to 512 Gb new imac

If you have more than about 400GB on that 3TB drive you cannot migrate.

You should get an external drive and then copy the bulk of your data to the external drive; then use the external drive with your new 512GB iMac. There is no way to fit 3TB onto a 512GB drive.

The OP could use the old Mac to move the bulk of the data to an external drive – as you suggest.

Then back up that external drive, delete the data from the internal drive on the 27" iMac, and migrate from the 27" iMac (or a new backup thereof). It would still be possible to take advantage of Migration Assistant in this way; just a bit harder.

If you are already using Music, Photos or TV on your existing Mac, just copy the relevant folders to the external drive. The default folders are ~/Music, ~/Pictures and ~/Movies. (Be sure to copy the entire folders as is; if you do that all your media will be copied along with the Library and no consolidation would be necessary.)

Then, on your new Mac, the first time you open Music, Photos or TV, be sure to hold down the Option key while you launch each app. (Do them one at a time.) You will get a small popup asking you to choose the Library you want to use. Navigate to the appropriate folder that is on your external drive and select the appropriate Library file. The app will open using that Library.

Note: It would be prudent to make a backup copy of each before you attempt any migration.

With Photos,

• You can have multiple Photos Libraries, though the program doesn't make it easy to move things between them. Hold down Option when launching Photos to get the dialog to create and switch between libraries.

• If you're using iCloud Photos, it's probably good to keep your system Photo Library on a drive that will always or nearly always be mounted when the Mac is running. (The system Photo Library is the one that will synchronize with iCloud. Any others are strictly local to your Mac.)

There are ways of moving your iTunes / Music / TV libraries, too. I haven't done that yet, because I still need to get an external drive for my "iTunes Library" (soon to be "Music" and "TV" libraries) to live on.

I think what you have to do with Music and TV is something like moving the file that represents "the library" – then telling the application to "consolidate" all of your media files from the old library location to the new one. Probably something you want to do on a copy of your library – not your only copy – in case something goes wrong.