Just a little note regarding this thread.
One of the most limiting things about the various analytical styles of teaching is that they very often become fixated on one aspect of playing, and the "musical" teachers often share that problem coming from the other end of the spectrum.
You have your "embouchure" teachers, your "air" teachers, your "tongue" teachers, open and/or closed aperture teachers, etc., and you also have your "Just play...it will get better" teachers as well.
I can't tell you how many students I have had w/serious embouchure problems...multiple unconnected settings being a specialty of many high range obsessed lead jazz trombonists particularly...who have been totally misled and then subsequently abandoned by teachers who specialize in air and/or "sound".
10, 15, 20 lessons about air is not going to help a player who has lost (or never possessed) the correct setting for a good, full sounding low range, nor will the demonstration of how it is SUPPOSED to sound if the student cannot manage to give up, at least temporarily, his original approach. It is POSSIBLE that logical embouchure analysis might help in such cases, but most often the forest of embouchure possibilities...a near infinity of combinations of angles and placements throughout the range of a horn...hides the few potentially valuable tress to be found w/in it.
Yet many teachers persist and persist in these approaches well beyond any hope of success.
For example...say we have a person who needs an embouchure change (or at least an adjustment of some sort)in order to become a better player. His current setup simply doesn't vibrate through all the necessary ranges. More (or better) air is not going to help this problem; he will simply sound bad in a more powerful manner, and even if he has the ability to mimic a "good" sound (I won't even get into THAT can of worms, thank you...what IS a "good" sound, which "good" sound, etc.) he will most often have to at least temporarily sacrifice his stronger areas in order to do so. Telling him to "change it to this way" is at best a hit or miss approach, and a drastic embouchure change will almost ALWAYS result in some weeks or months (or even years) of relatively bad playing, even if it IS eventually something that will work..
If he is in school, the pressure from his ensemble teachers and his peers will make it difficult for him to do these things ("Hey, Johnny, I'm counting on you for the fall concert...don't let me down..." "Man, you sound TERRIBLE ! What's up with THAT???") and if he is a working musician, the problem is even worse.
What to do, what to do...???
Once again, the only teacher I ever saw who effectively solved these problems was Carmine Caruso, because his approach identified the strong points in a person's playing, no matter WHAT they were...air, high, low and/or middle range, power, finesse, flexibility...and then organically extended those strengths in a gradual and seamless manner (so long as the student didn't get obsessive about the process...a big if in many cases, including mine at first...) into the player's weaker areas.
All w/out a WORD about specific physical actions, all w/out a played SOUND from Carmine (who was a saxophonist anyway).
I use this word VERY rarely, but there it is...he made a quantum leap past all these problems, one that is still widely unrecognized and misunderstood 50+ years after he started doing it.
I certainly don't know the teaching styles of all the brass teachers in the world...I'm just a provincial musician, really, only my province is NYC...nor am I familiar w/every teacher in NY, but to my knowledge the only two people here who have effectively continued (and to some degree advanced) Carmine's concepts are Laurie Frink and myself, really.
The world still hasn't caught up to Carmine...maybe it never will...