I have been wondering for years why, listening to any music (or simply to a single tone), better with headphones/earphones, if the volume is very low (almost inaudible: 20-30 dB SPL), each audible note of the same music sounds flatter than when played at a far higher volume (70-80 dB SPL), using the same equipment: I guess about 20 cents sharper, when far louder. It is like a "static" Doppler effect depending on loudness and not motion, but I am surely talking nonsense. Please, consider that I am absolutely ignorant of Physics, although I am a musician! Thanks for your help.
The perceived pitch of tones with a high frequency (above about 2 kHz) tends to increase when their sound pressure level is increased. This is a known psychoacoustic effect that agrees with your description. However, the perceived pitch of tones with a low frequency (e.g., 200 Hz) tends rather to decrease when their sound pressure level is increased. This appears to be due to an imperfect compensation for the more substantial effect an increased sound level has on the neural excitation pattern along the basilar membrane of the ear. The effect varies to some extent between listeners.
I cannot comment so here is an answer that might provide some guidance. First the ear is non-linear. You may already know that the ear will not respond with the same amplitude as driven, hence people perceive tones with different pitches and the same driving amplitude (volume) at the source as having different value of loudness. Hence we have maps from true physics amplitude to human perceived amplitude (Fletcher Munson curves).
Now you are quoting a different phenomenon pitch bending with higher driving force. I am not sure I can offer a closed answer to this but as guidance I'd look at a text like Physics and the Sound of Music by Rigden, or a text on Psychoacoustics (can't find mine).
One of the reasons I would not directly respond to the statement higher volume equals higher pitch is that you state that you hear the effect with pure tones and combined tones (music) and these might have different possible explanations.
I'd also ask, since you are a musician, how accurate do you consider your own pitch discrimination ability? I am sure your relative pitch is very good, but even musicians can be fooled by psychoacoustic phenomenon.
If the effect is not psychological in nature I'd guess it may be over-driving a set of coupled non-linear oscillators, i.e. the structure of the basilar membrane etc.