How do I authorise my Mac for Apple Music?

I subscribed to ITunes Match yesterday but when I click the “Add this Computer” button in the Music app on my Mac, type in my password, it then shows a message about “Accessing iTunes Store” briefly, the “Add this Computer” button gets grayed out and that’s it.

If I close the Music app and launch it again, I get the option of adding this computer again. My Mac is authorized and I’ve signed in and enabled “Sync Library”

I tried this on another Mac and a Windows PC running the latest version of iTunes and it does the same.

I tried deauthorizing all the computer and then reauthorizing just one but it does the same. I also signed out of everything signed back in.

Any ideas?

[Re-Titled by Moderator]