Your comment about the 7th harmonic is right on target.
The even harmonics fall on octaves. The odd harmonics can be off relative to the piano scale (twelve root of 2 spacing). The third harmonic is -1.6 cent, the 5th harmonic is +13 cents, the 7th is +30 cent, the 9th is -4 cent…
I would think the 5th is bad enough, but the 7th wins the distortion race. You play A2 (110Hz) and the 7th fall on 770Hz but the piano closest note is 783.991Hz, almost 14Hz off (while the next note is only 46.7Hz higher).
Funny thing, about tubes: a non linear “curved” transfer function (say power amplifier) becomes more linear when using a smaller portion of that curve. In other words, you get more distortions (tube sound) at higher volumes (large signal peaks and deeps use more of the curve).
Reducing the volume way down will remove much distortion (tube sound). I wonder how many audiophiles know that simple fact.
Pianos are intentionally tuned to provide even spacing, but not even that, really. They are also tuned to spread the octaves, so that the right end of the keyboard is significantly sharper than the left end. And the strings themselves, as they interact with the soundboard, have rather complex overtones that cause a lot of intentional enharmonic blending to cover over the errors of equal temperament.
Musicians playing in ensembles do not try for equal temperament at all, unless they are compelled (in that moment) to be in tune with a mallet or keyboard instrument that is so tuned. When playing together, they will play with just intonation, which tunes the chords for maximum resonance. This requires them to play the same note with different tuning in different key and chord contexts, but that's the gig. Ancient wind instruments that lacked valves or tone holes were limited to the harmonic series of the instrument. They either avoided the overtones that were most out of tune, or found ways to use the effect musically. (Or they played solo, so that it wouldn't clash with other instruments.) Vaughan Williams actually used this when he asked the trumpets emulate a bugle call in his third symphony. The trumpets are specifically asked to play a seventh harmonic on the open instrument (rather than the usual first valve) so that it would play flat as it does on a bugle.
All of this intentionality can be screwed up by distortion products that rise to the level of audible, either singly or in combination. Even the so-called musical even-order harmonic distortion might drift away from a piano with its stretched octaves and undermine its resonance. There are plenty of places in the recording and distribution process where those distortions can sneak in, but as for me I don't want my playback system to add more.
Of course, very low distortion tube amps are entirely possible with good design and implementation, so this is less about tubes and more about bad tubes or inadequate tube-based circuits.
Another possible effect, though, is in the shape of a clipped waveform, which may differ more significantly between different designs and technologies. When an amp is clipped (and I think a lot of smaller amps are driven to clipping on peaks much more often than their owners realize), those clipping behaviors may also be part of the mix. A hard, square-wave clip is going to add a spike of odd-order harmonics to that peak, giving it a click that it would not have if the wave could be fully formed.
Rick "has a textbook on tuning that is three inches thick providing shelf ballast downstairs" Denney
