Read on to find out how to turn off the annoying sound when changing the volume in Windows. Two steps are necessary to make sure no beep will be heard.
The Windows system tray typically shows a speaker symbol in the system tray. Clicking on it allows setting or muting of all system sounds. If you don’t see the speaker symbol, then go to Start>Settings>Control Panel>Sounds and Audio Devices and select the check box “Place volume icon in the task bar” in the “Volume” tab:

When all settings are at their default, then a sound will play when the volume is changed. When the mute box is selected, no sound will play. To turn off the volume changing sound even when the mute box is not selected, two steps are necessary. Most other websites only mention the first one:
Go to Start>Settings>Control Panel>Sounds and Audio Devices, go to the “Sounds” tab, and select the “Default Beep” from the “Program Events” list:

Next, select “None” from the Sounds menu below:

This menu is also useful for turning off all sort of other annoying sounds, such the sounds when starting and shutting down windows, when connecting USB devices, etc.
Unfortunately, turning off the default beep may have quite the opposite effect. Depending on the setting in the “Volume Control” panel, the default beep may be replaced with a very loud PC Speaker Beep. Unlike the Default Beep, which was played at the selected volume, the PC Speaker Beep has only one sound level, and it is very loud, especially, when listening through headphones.
To turn it off, double-click the speaker icon in the system tray. If the mixer does not show a slider for the PC speaker, then you need to turn it on via the Options>Properties dialog:

Finally, mute the PC speaker, to silence the PC Beep for good:

I found the the tip for this crucial second step in this discussion.
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