I'd just like to add:Odd harmonics sound bad and even harmonics are musical but both are added distortion.
Most Valve amps were push-pull, only really low power ones were single-ended, so they generated odd harmonics when driven into clipping.
A Valve amp, with greater linearity, and less feedback gradually runs out of gain as it approaches clipping, so the harmonics generated are of lower order.
A transistor amplifier, with more gain and more feedback, clips abruptly and generates odd harmonics extending to 9th and 11th an beyond.
The 3rd harmonic is an octave and a fifth above the fundamental, which is a concordant sound. 7th and higher odd harmonics are discordant.
It is possible to make a valve amp with a lot of gain and high feedback (but with four or more stages and a transformer inside the feedback loop would be a bit of a b***** to keep stable). It would generate high-order odd harmonics like a transistor amplifier.
I believe that a hi-fi amplifier should be "blameless" to quote Doug Self, and should produce as little distortion as possible. There's really no excuse for distortion products of a 1kHz signal to be above the noise floor. If users of said amplifier prefer the sound of a little bit of 2nd harmonic distortion, then they can add it with an effect unit.