The odd-ordered harmonics (above the 3rd) are used by the human ear/brain system to ascertain the volume of the sound being heard. This is pretty important to know if you want your system to sound like real music rather than a hifi.
If an amplifier has troubles with this, there will be two results- it will sound louder than it really is, and it will sound brighter than the music really is. All human ears are very sensitive to this!
The ear hears harmonic distortion as tonality. Electronics can have the fault of being overly 'warm' in sound, which is caused by the 2nd, 3rd and 4th harmonics. Quite often tubes get taken to task over this, but that really has more to do with the topology rather than the circuit being tube or transistor. If for example the tube circuit is fully differential in design, there will be no even ordered harmonics and so the circuit will have a more neutral presentation.
Conversely you can give transistor circuits a richer sound by building them single-ended- this will result in more of the lower-ordered harmonics.
Linearity of the devices themselves has a big effect on the sound of the circuit as well. The simple fact of the matter is triode vacuum tubes are the most linear form of amplification known. Additionally, it is easily demonstrated that even the most pedestrian tube amplifier will make less odd ordered harmonics than any transistor amplifier; a sine wave generator and an oscilloscope are all that's needed to demonstrate this.
This is why tubes are still around half a century after being declared obsolete.