I don't want to turn off notifications for new emails (I use the stock app), just the audible tone...it it possible to silence the tone but still get the notification on top of the screen icon?
There should be a Notifications option in the Settings of your e-mail app that lets you select the ringtone used for incoming notifications. There, select the ringtone as SILENT (usually at the very top of the list).
I think disabling that option also removes notifications from the status bar. If you want to keep that, you can leave it checked with the notification sound set to silent like SpookDroid said. Here's my setup showing a silent mode while keeping a visual notification. I just wish they would add a feature to use the notification LED as well with the screen off.