@richardbrand wrote: "To discover what the first overtone sounds like, move up one octave and play that key. If the two notes sound as if they are the same frequency, then you cannot hear the lowest fundamental. But my experience on a Kawai upright piano is exactly the opposite, the lowest note is clearly an octave lower and is clearly audible."
I don’t think this is the proof you think it is, and I hope you don’t mind me posting a somewhat counter-intuitive contrary opinion.
Even if we cannot hear the lowest fundamental, those two notes will NOT sound as if they are the same frequency because the interval between the harmonics is TWICE as wide for the key that is one octave higher. The ear/brain system PERCEIVES the missing fundamental by inferring it from the interval (or spacing) between the harmonics. Please Google "missing fundamental", as this perceptual phenomenon is very much in play at the far left-hand end of the piano keyboard.