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 ?

I’ve been listening to the EL34 tube for over 35 yrs and have compared it to many power tubes. When it comes down to which tube is more musical, the EL34 tube is simply the better tube in my opinion. I own several very expensive tube amplifiers that operate strictly on the EL34 tube.  I really am not a fan of KT tubes except the KT66.  The KT66 I like very much and highly recommend it. I find the KT66 more musical than the EL34.  The EL34 tube is sweeter and gentler with the detail than many KT tubes.  Most KT tubes sound hifi- ish. I’ve found the EL34 to have better resolution also over the KT tubes.  Even though I personally do not like many KT tubes, I’ve heard them in circuits where they sounded fine. So the design is important and can make a KT sound good to my ears but still won’t have the same inner resolution that the EL34 tube has.  I purchased an excellent $5,000 300b amplifier 2 yrs ago from someone who preferred to use EL34’s on his JBL monitor speakers.