How do I delete the tone that sounds when a new email arrives in my inbox?

in addition to the steps set out by mrnonaustralianguy, you also need to go into Mail Preferences > General and set "New messages sound:" to None.

1. Go to System Preferences.
2. Click Notifications.
3. In the list, find Mail.
4. There should be a checkbox whether to play a sound for notifications for the app.

🏳️‍🌈

you're welcome. I'm always happy to help. :)