When you say even sound, do you mean the same volume? Or just quality of tone?
Optimally, the only thing that changes to ascend to higher pitches is shape, primarily through a tightening of the corners while leaving the middle of our lips relaxed. Abdominal air support isn’t needed to hit high C.
Less air is required the higher we play, so the higher notes will be softer and I even back off a little, typically. If we do that, we don’t feel as much pressure, just engagement in the corners to keep the air from escaping the embouchure/mouthpiece, only as much as necessary for the pitch. The notes are softer, but they are notes–they feel and sound just as good as lower notes in the register. If we try to put more air through to make the note louder, then there is more resistance the higher we are playing.
But to get the pitch of high C doesn’t require a lot of air pressure, just the proper shape. As Greg demonstrates in one video, he releases all his air and just spits out what little left he has using his mouth and hits multiple (short) high C’s.
To be honest, I think I’m a little lazy with my air support and that’s something I need to work on to progress. Part of that, I think, is because I have focused mostly on passive air the past year or so to make sure I release any tension and am getting the pitch cleanly; I figured I could add air to increase volume later–which is just about now…
I can play up to high F reliably and am starting to get more and more double G’s. Not every day is as good as the last. Today doesn’t seem like a great day, just played a little so far. But I got the high F and I’m currently playing a 1 1/2C mouthpiece.
I did a little harmonic slurs (Schlossberg) up to high C (I think it’s #32 or something like that) after reading this thread, just to think about it. And I’m not feeling a lot of pressure in the body (i.e. stomach) to play high C, I tighten the corners and actually back off the air a little then kind of relax/tighten the corners a bit, open up the aperture a bit and give it a little more or less air to get the lips to vibrate / sound to resonate as much as possible. Even high D isn’t that hard to reach, in fact it’s the very next harmonic in open valve position from high C.
Not sure that answers your question, but hopefully helps, FWIW.