Harmonic + chirp on a single note

Hi. First of all, I changed pianos: my Yamaha C3 had a thousand problems (including a cracked soundboard): psychologically, I didn't feel up to having it repaired; perhaps it can be brought back to life and someone else can benefit from it. However, I am very angry with the seller (a very, very well-known one). I purchased a Pleyel/Schimmel 178 cm (practically a Pleyel manufactured in Braunschweig by Schimmel, an Schimmel in short). I like it very much, both its sound and its action. It has never belonged to anyone and the dealer (this time a different one) took it directly from Braunschweig where it was on display. The only downside is that I don't like the tuning (done completely differently than when I tried it); no big deal: the seller will redo it for me exactly as it was when I first tried it, taking advantage of the fact that a month of settling in will have passed. As soon as it arrived home, it still held the trace of the tuning done a month earlier by the dealer with very little out-of-tuneness: at my house, a young technician came and the result was a different and unstable tuning. This is just to confirm what Paolo Ferrarelli often states: often pianos are not out of tune, but badly tuned! But my question is something else: as it went out of tune, a note (E4) began to produce a chirping sound and an overtone (which seems to me to correspond to a D). I deduce that, since it wasn't present at the beginning, it must depend precisely on the tuning (the tuner told me that on smaller models it is better to make the bass area more ascending, unlike larger pianos where it is more acceptable for it to be more descending, but it seems to me that it should always be descending in proportion to the disharmony, I ask for enlightenment!). Pressing the two lowest Ds slowly and playing the offending note loudly, the bass strings go into vibration producing the E (is it possible that the low Ds have E as an overtone??). I tried pressing the bridge and nothing changes. Initially, an overtone was produced on another note, but I realized it was a metal object placed in the living room; this one, however, produces an overtone and a chirp.