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In my experience as a trumpet player I have found that many many players who have learnt or are learning experience playing high notes or playing in general using too much pressure. There are many articles on the internet available and there are many approaches. In general a pupil who uses to much pressure will only get a limited range of high notes and/or run out of stamina quickly. If a pupil can learn to play with less pressure then half the battle is already won.
Using lots of air is akin to fuel in a car engine. No steady flow of air=no sound and if a player has good lungfulls of air from proper deep breathing (filling up from the bottom of the lungs) and blows correctly (steady and strong) using correct support (good use of diaphragm muscles) and then uses less pressure, then they can begin to work on high note playing.
The muscles around the edges of the embouchure need to be secure (calisthenics, Cat Anderson method, charles collins lip flexibilities) to prevent the embouchure from being blown out of shape within the mouthpiece. There are some quite incredible forces used when playing very high and the embouchure needs to be able to withstand that.
Staying relaxed is also a problem. It's usually a psychological problem and a player needs to realise that in all aspects of playing, they need to stay as relaxed as possible. Grip the horn with the left hand without the knuckles turning white. The right hand (for the valves) should be as relaxed as possible in order to press them down efficiently.
Play a middle G as quietly as possible, with the teeth together for around 5 mins a day to start with and breath through the nose without taking the mouthpiece off the embouchure. You may find the sound thin and raspy but this is normal. If your cheeks/ embouchure etc start to hurt then stop playing and rest. It's important to rest as much as you've played the exercise for before carrying on. This probably contradicts most trumpet teachers advice but believe me it really does change a players sound, support and embouchure strength. Within a week there should be some strength and security top notes. Increase the time to 10 mins, then 15 mins and finally 20 mins a day. The Americans are very good at this sort of thing and there are lots of musicians to find out about. Bobby Shew is my favourite and he has lots of excellent advice about breathing and so on! Go and check out his website.
To increase lung capacity and breathing capablilities try (without the trumpet etc) sucking breath in for 4 counts, holding it in for 4 and then slowly breathing it out for 4 (through and embouchure sized hole) whilst marching slowly around. Increase the counts to 6,8, 10 etc. Also try unloading all your air at the curtains/blinds in your bedroom (through an embouchure sized hole) make them move and then take a big step back and try again and again and again! Enjoy as you feel the power of your blowing increase. Try it through the trumpet etc.
Remember any exercise like this is designed to compliment a practice regime not replace it. This is one of lots of methods available and I do try and incorporate it into my lessons.
