Sorry for the bump but I believe I have figured out the exact piece of these cameras that make this tuning fork 'ping' / clang / gong / tuning fork sound or tone.
I just received a Nikon FE from ebay and noticed the pinging sound right away. As
Ixtl mentioned, it makes the sound when the mirror goes up and makes it even louder when the mirror goes down. This can be confirmed by using the the self timer so that the mirror up and down are separated.
I googled and found similar complaints about the FE, FM and occasionally the FE2.
Anyway, I did some experimenting by placing light pressure or even just touching various parts of the camera as the shutter fire and it is the small metal aperture control as circled here in red
This is a small steel lever that acts like a tuning fork when it hits its stop on the way down or up.
You can test it a few ways
1 - Just take a finger nail and use it to 'pluck' the end of it, you will hear the tone that you hear when taking a photo
2 - Cock the shutter, set the self timer, then place a finger tip on the lever in such a way that the mirror won't hit your finger tip and then press the shutter release. When the mirror tries to pop up it won't because you have your finger on the lever. Gently let the lever move to where it wants to go and you won't really hear the tone from the mirror popping up. Keep you finger on the lever so that the mirror can't go back down. When the shutter trips
you may still hear a slight amount of the tone but far less than normal. Now let the lever go in the direction it wants to gently release it.
The underlined part may indicate that both ends of the lever, the end that connects to the lens as it slaps against its stops and the end deep in the camera near the shutter can produce the tone.
Unfortunately I don't have a solution, yet.