How do I turn off mail notifications?

Norm1974, that's just what I needed. Thank you!

go to the mail app and go to the Mail>preferences and it should open up this tab

On the line where it says "New Message Sound" make sure it says "none"

- this will cure your audible "tap on your sholder"

On the line where it says "Dock Unread Count" make sure it says "none" also.

- this will make the red badge in your dock go away

Where is says "Check for New Message" I would not turn that to manual or the computer wont get new messages and you will have to fetch new mail yourself under the mail app "Mail>Get New Mail" and when you are in your mailbox you probably want to see your mail at that time. So leave the Check for new message on.

Norm 1974's solution sounds excellent, the kind thing i would want. But it seems that my OS, Mountain Lion, doesn't have those choices in the system preferences>notifications. Those choices look great. that's what i need. In mail>preferences, there's no way there to turn thise things off. I have it set to none, but they keep coming. this OS ***. i just got it a few days ago on my new computer. my old computer had snow leopard. Ther problems with mountain lion are endless. I was using Eudora mail since 1995 but Mountain Lion doesn't support rosetta which means no pre-intel mac programs, which means no more Eudora. So i'm trying to set up Mail to sort my messages. My inbox has thousands of messages in it. I want to apply my rules so that those messages can be sorted into the mailboxes i created. For every message that is transferred to a different mailbox, apparently, there has to be one of these notifications. That makes moving the messages take a very long time. Is there any way to turn those things off in this OS?

Thank you Norm1974 :-)

Thank you jaw444! This solved the problem. 😎

Thanks jaw444. Problem solved

thanks Jordankoop.88 disabling the sound in mail preferences solved my problem.

This worked. THANKS