So the situation is not completely hopeless. Ralph, how these distortion parameters reflect the distortion of the harmonics of each degree? I think that now i know that i am looking for an amp with small first order distortion but large second and third order distortions, roughly. But, as I have also mentioned, i don't really know how large they can or should be, no idea at all.
In a nutshell, if there's enough lower ordered harmonic distortions, they can mask the higher orders. The higher orders otherwise cause harshness and brightness. Either the 2nd or 3rd can predominate, but if the 3rd dominates then it has to be at a lower level to begin with- about 1/10th what you might see if the 2nd were dominant in the design.
For example almost any SET makes 10% THD at full output- most of which is the 2nd harmonic. If you have an amplifier design that makes the 3rd instead of the 2nd (which can happen if the amp is fully differential and balanced, since even orders are cancelled throughout the design) then the THD will and must be much lower. This isn't that hard to achieve since a design that exhibits a dominant 3rd harmonic will tend to have less distortion overall anyway.
On top of that the distortion cannot rise as frequency is increased, which is a problem with most solid state amps made (and with tube amps employing feedback). Class D offers a solution for this- you can have the distortion vs frequency look like a straight line, and you can run a lot of feedback at the same time. In this way you can achieve distortion levels that are several orders of magnitude lower than you might see in a tube amp ye the actual distortion signature (2nd and 3rd vs the higher ordered harmonics) can look quite similar to a tube amp.
This allows greater transparency (since distortion obscures detail) yet the same kind of smooth mids and highs you associate with a really good tube amplifier- IMO, the best of both worlds.
In case you don't get my drift, I'm drawing a very straight causal line between the distortion measurements and how the amplifier actually sounds; IOW its predictable based on sufficient measurement! The tech to do this really didn't exist 40 years ago but it does now. A lot of audiophiles don't know that yet; like so many industries tradition plays an enormous role in this.