A quick custom sound replacement in your user library can adjust the default sound used in Notification Center.
Topher KesslerMacFixIt Editor
Topher, an avid Mac user for the past 15 years, has been a contributing author to MacFixIt since the spring of 2008. One of his passions is troubleshooting Mac problems and making the best use of Macs and Apple hardware at home and in the workplace.
When you receive a notification in Apple's Notification Center for Mountain Lion, the system will use the built-in "Basso" sound. If you do not want this sound to play whenever you receive a notification, then you have a couple of options.
The first is to disable the sound completely, which can be done on a per-application basis in the Notifications system preferences. Simply select an application and then uncheck the "Play sound when receiving notifications" option, which should silence it.
Unfortunately this is a relatively limited option and one that requires you to change it for every application. Additionally, Apple does not provide any settings for adjusting the sound to use; however, Erica Sadun at TUAW recently outlined a workaround to change the sound globally for all notifications, which may be useful to those who dread the dull, thud Basso sound.
To do this, simply create or copy a small sound bite in the AIFF format to your user's Library folder, name it "Basso.aiff," and then log out and log back in to your account. You can use QuickTime Player 7 to crudely edit and export a sound file to an AIFF to create such a sound bite (make sure it's short if you create one manually).
Alternatively, you can use one of Apple's system sounds in place of the Basso, which can be done by going to the /System/Library/Sounds/ folder and locating a sound file of interest (you can preview them by highlighting each and pressing the space bar to invoke Quick Look on them). With the desired one chosen, hold the Option key and choose Library from the Go menu in the Finder, and then paste the file in the Sounds folder.
With the file pasted, rename it "Basso" and you should be good to go once you've logged out and back in.
If you find this useful but would like to keep the Basso sound for other purposes, then you can likewise copy and paste it into this folder, but name it something like "Basso2" so you can use it as such, as the original Basso will have been replaced. Another detail to keep in mind is that these changes will only affect the current user account, so if you have multiple accounts on the system each user can make these changes to use custom sounds of their own.