it's purely a matter of design

@inna That's the answer.

If you want to hear a really musical KT88 based amp, the Harmon Kardon Citation 2 (properly refurbished) is a good place to start.

It makes a difference too if the amp is pentode or Ultra-Linear and also what class of operation its set up for. You can't get away from the simple fact that the larger the output transformer is, the more critical it becomes and the harder for it to make bandwidth. IMO this latter issue might be the most important.

Ralph, I see.

I have an EL34 headphone amp that sounds quite good.  Among pentode/tetrode tubes, my favorites are very low powered tubes.  I run 349 output tubes in pushpull that deliver about 4-5 glorious watts per channel.  This is my favorites tube aside from the truly crazy rare 252 tube.  My next favorite is the 350B (a variant of the 6L6 tube), followed by the 6L6 or KT66, and then EL84 (a tube often used in less expensive amps, but, I’ve never heard an EL84 amp I didn’t like).  
 

Of triode tubes, either pushpull or single ended, I particularly like the 45 and 2a3.  

Larry, what speakers do you run with that 349 tube based amp ?