Think of a bugle in C. Bugles have no valves, and the notes you can produce on them are only the following:
C - G - C - E - G - Bb - C - D - E - ...
These match the overtone series of C.
On the trumpet, however, you have valves, which enable you to play additional notes. Press the second valve and the length of tubing increases in the amount needed to lower this series half a tone, as if you transformed your bugle in C to one in B:
B - F# - B - D# - F# - A - B - C# - D# - ...
Press first valve alone and the original series is lowered by a whole tone (bugle in Bb). First and second valves together: one and a half tone down. Two and three: two tones down. One and three: two and a half tone down. And finally one two three: three tones down.
This allows trumpet players to cover all the tones from low F# (below the treble clef), and theoretically unlimited upwards.
Now how to make the jump from the low C to the middle G, which are both fingered with no valves? Closing the embouchure does work in the lower register, but is a dangerous practice. The problem is that if you continue to close your lips more the higher you play, you'd have no air coming through after a while.
What I understand makes a difference is the air speed. The higher the speed, the higher the tone. It is similar to overblowing a bottle: with enough air speed, you can jump to a higher tone than the original one.
There are several ways to increase air speed through the trumpet:
- closing lips (smaller hole but same amount of air makes for higher speed). It has the aforementioned drawback of blocking air flow in the higher register,
- forming a smaller channel with your tongue. (it feels like it could have the same drawback, but I have never experienced blocking the air flow with my tongue doing this.),
- blow harder.
I think the correct way to go is a combination of the second and third bullet.
I also find that playing tones in an overtone series feels very similar to whistling. Try and whistle up and down from the lowest to the highest tone you can achieve, and observe what your tongue does. Try to do the same on the trumpet, and support with enough air.
An additional bullet point, which I now think is necessary for the double high register (G with four ledger lines and above):
- lip stiffness. It is similar to tightening a guitar or piano string to increase the pitch. This requires a lot of work out: you need the muscles to hold the embouchure without closing it, and compensate with larger air flow to start the vibration. Hopefully this gives range without sacrificing volume (which a smaller tongue channel could, at least in theory as mentioned earlier). I have been working out a bit for that, I am not quite there yet. Since I lack control, sometimes my lips get too stiff and no sound comes out at all. Also I suggest combining strength with suppleness, flexibility. If you only get the muscles and give up flexibility, you won't be able to play soft anymore.