When I play the G string on my violin exceptionally softly, among the sympathetic overtones I hear is a diminished fifth. Why is this? Is this specific to my particular instrument? Out of curiosity, I tried this on another (cheap) violin and I didn’t hear it.
Your ear could be off.
Even though the dim 5th is NOT an overtone of any vibrational frequency based on the standard accepted model of a string, there are a couple things that could cause this.
The response of the other strings to an input is not an ideal response right at resonance. All naturally occurring system have damping and this broadens the response cure. Ideally the response of a system at resonance is infinite and rapidly diminishes off resonance. When damping is present a system can respond off resonance with a relatively noticeable response relative to that on resonance. The peak response will still be on resonance but the neighboring frequencies will have a good response as well.
The ideal string model with integer harmonics, fn = n*f1, is not that accurate. Strings behave like very thin beams with boundary conditions depending on thickness and tension. The frequency spectrum of a stiff beam can have non integer harmonics, in one case the augmented 5th comes out. I've recorded this using a very low pitch tuning fork and analyzed. The overtones matched the theory fairly well. You are stating dim 5th and the case I am mentioning is a +5 but it is possible that for your soft driving force the violin string is not behaving like a perfect string but has some other physics in it. This phenomenon can be heard in pianos for some notes, usually at the high end of the piano.
A combination of the two concepts mentioned above but applied to the body rather than the strings. The body parts act more like plates with boundary conditions and these do not have integer harmonics. They can have all sorts of resonance modes with irrational ratios to the fundamental and exhibit stiffness as well as flexibility. This along with damping could be causing a plate harmonic to get excited by the string and driving force of the bow.
Also, it could be a red herring. But if you are interested in learning more about overtones in instruments that are not idea you can find this is several books by Fletcher and Rossing. They are fairly well known in the field of acoustics (either physics or engineering) and have spent their lives researching the physics of musical instruments. Some of their books are very dense with math but they do have a few that provide a survey of the facts without derivations.
Let us know what you find out.