Hi there, welcome. Good post from Ron above, as usual. I thought I would add my $.02 FWIW:
What do we move to change pitch?
That is, perhaps, the million $ question.
FWIW, my understanding is that Shape determines pitch. Air is for volume and long notes. As such, we must separate Shape and Air when determining how to achieve pitch.
Windworks has various exercises which use Passive Air–air Released, not blown, into the mouthpiece/instrument.
But exercises performed a certain amount of times or in a certain order do not result in success.
We must relentlessly ask ourselves WHY? Why am I doing the exercise I’m doing? What is the point?
Personally, I spent a lot of time experimenting solely with Passively released air to determine how high I could go. I agree with Ron to take it slow and WindWorks gradually builds up, but Passively released air can get you up to High C, perhaps a bit beyond–we are all a bit different.
Recently, things have been going great with my experimentation with Harmonic slurs. Previously, I treated these exercises as more Process focused exercises and was not mindful of the quality of the sound; I treated them more as endurance exercises to build strength. Recently, I took a new approach–I patiently worked through Harmonic Slurs to focus on the quality of the sound at each pitch. Using good air flow, but passively released air, I experimented with Shape (aperture corners, tongue arch, etc.) to obtain the next higher pitch and waited until I got that pitch with a quality sound before moving upward. I took breaths as needed to maintain quality air.
Moving up and down with Harmonic Slurs, ensuring I had good sound on each pitch, I determined an optimal Shape for each pitch.
When things are going right, there is tension Surrounding the lips (not within the lips themselves) toward the outside of the MP and in the aperture “corners” and the lips are left to vibrate freely within the mouthpiece like a drumhead vibrates, the only tension surrounding the drumhead.
The air being released through the lips is all it takes to play above the staff up to a high C and a bit above that; eventually, actively supported air is necessary and it’s necessary to actively support notes to make them louder.
Separating Pitch from Volume is very beneficial when separating Shape from Air. Learning what Shape determines Pitch, independent from Air, is helpful; we can then work to increase air flow separately to increase volume.
Bottom line–yes, experimenting trial and error and playing WindWorks exercises helps.
Focusing on Passively released air and paying close attention to how Shape changes Pitch helps.
Playing Clarke I, but ignoring his dynamics and instead playing softer as you ascend helped me. As did Schlossberg (i.e. Exercise 31 Page 8).
Speed is important as it ensures we’re being efficient. But I suspect that Quality of sound is perhaps even more important than Speed–for years, I focused more on speed and cared less what it sounded like. Focusing on what it sounds like and moving slowly has helped me recently hone in on precisely what to do when.
Hope that helps, FWIW. Godspeed!