This will pertain to breathing and maintaining an open airway. I will start out with a concept that several may disagree with. All I ask is that you consider what I'm telling you. The diaphragm is called an involuntary muscle. It works without us thinking about it . It works when we are asleep. It can MAKE us sneeze or cough. We can however, exert some control over it. We CAN hold our breath , take a breath when we want, take a short gasp or a long deep breath. This indicates a measure of control. In as much as trumpet playing IS AIR and breath control then working on this major source of our breath is vital. There are several Yoga exercises that are excellent as is timed breathing while walking or jogging.
The airway must always be open both in inhaling and in playing. One problem is posture. I've seen many experienced players slumped over while jamming. I've seen them with their heads down our their arms against their ribcage. If we give this its proper importance then we see that these things WILL lead to a closed throat, shallow breaths and poor support. Breath deeply from the bottom of the lungs up.
If the jaw is pushed forward slightly this will cause the throat to be more open than it normally is. Try to move the jaw forward slowly and check if you can feel your throat open up. Think of the effect that can have on your tone. The more forward jaw position will also make your lower lip take on more of the workload. This increases endurance ( after you get used to it).
Another key feature in maintaining an open airway is a pivot. You could write hundreds of pages about this. But that's already been done. In a nutshell by raising or lowering the bell of your horn while you are playing you can maintain a more open airway and clearer tone. As you play higher and lower notes the air stream will slightly move in the mouthpiece. If we can keep it lined up with the throat hole the sound is better. The SLIGHT bell movement will produce an opposite movement or realignment of our lips to the mouthpiece. Now which way do you move the bell? Try this test . Play a low g 1-3. Move the bell up then move it down. One way should improve the sound. When you move to a lower note from now on always pivot this direction. The opposite direction will aid the upper notes. This is a good movement whenever you have to leap between notes.
The tongue arch has been used for years to speed up the air in order to play higher notes. Most people arch to the point where the sound quality is affected. Instead of arching up to eeee try aaaaa. This is a more open sound yet it still compresses the air slightly. After all the tongue arch cannot give you an extra octave. It is merely used for rapid note movement. The abdominals compress the air for your range. All lip trills , slurs and leaps are accomplished in part by using a tongue arch. If you have maxed out your tongue motion at Bb below high c how do you plan to continue going up? The tongue arch is like an elevator it should help you to compress and thereby speed up the air to achieve higher notes. Play a low c to second line g lip slur back and forth. Both of these notes are below middle c yet a tongue arch is useful in speeding up the excerise. Likewise if you are playing a high g and want to slur up if you are already in the extreme eeee position where do you go? My suggestion is to attempt to substitute a long aaaa when possible and save the extremes for a reserve.
Now for the full breath on every note or phrase. Have you ever had to play 1 note by itself to fill out a chord in a song? What about 3 or 4 measure phrases? These do not require as much air as a full 8 measure phrase. At the end of a very short phrase an inexperienced brass player will feel a need to exhale before he or she can take a breath. If this overbreathing continues for any length of time the player will sometimes turn red or gasp for air. No you didn't run out of air for playing however, your body really likes to have oxygen in your lungs. What has hapened is you took a full breath and used less than half. Now when you take a full breath you only replace half of the stale oxygen deprived air in your lungs. As this continues you end up gasping for air. Does this sound familiar? Overbreathing really is a kind of self suffication (in the extreme). In the upper register overbreathing becomes more apparent. Have you seen people get dizzy, lightheaded, or blackout. They were overbreathing. I know some people say if you release the pressure really slowly it will not happen. If you did not overbreathe and have so much leftover air under pressure it would not happen either.
Timed breathing is another aspect of playing. Some people always take a deep full breath. When playing in the upper register this creates tension. The upper register takes air compression and speed but not air mass. The low notes need the full breaths. Try a half or quarter breath before you play your next high g. This will allow your muscles to do their job.