Mrtennis, Human hearing is all there is in audio. It is what is perfect, it is the audio equipment that is not.
IOW, the ear/brain system has certain rules it follows. The closer the audio equipment comes to following those rules, the more it will sound like music rather than electronics.
Now in the case of odd ordered harmonics, which as at the crux or our discussion, the ear happens to be very sensitive. In fact, this is one of the things that the ear is *most* sensitive to. So its to your advantage to not distort the odd ordered harmonics- the result will be a harder, brighter sound if you do. Now to emphasize the point: the ear is *so* sensitive in this area that modern distortion instruments have trouble measuring the area that is important- often the distortions are buried in the noise on the test bench.
It happens though that tubes tend to make less of these distortions overall, although a **lot** depends on the topology of the amplifier, and so if a designer knows what he is doing, he can do it with transistors too- its just a lot harder. To avoid this distortion, you have to keep the circuit as linear as possible, without loop feedback.
This is a difficult task, but you can see that if you really want to be about it, pentodes and transistors will have more in common than triodes and transistors will. Certain semiconductors are quite linear- approaching that of triodes, and to make the best use of them the circuit will have no feedback and will be class A. There are not many transistor amps like that. Those that are are either very expensive (+$100,000) or very low power (Nelson Pass First Watt).
Now its a lot easier to design the odd ordered distortion out of the amplifier if you work with triodes. I am not saying that they are superior, I am saying they are easier. But the fact of the matter is that there is a lot more experience in the field working with triodes in this way, so you might want to consider going the other way (towards triodes, away from transistors) if you are looking for the best performance with your Quads.