I would like to know a bit more about how the valves of trumpets (and other brass instruments) work than is here: Trumpet at Wikipedia.
I am ignoring issues such as it being a transposing instrument. I only interested in the relative pitches of the notes and not the absolute pitches.
The fundamental is not usually used so the first few harmonics are: C4, G4, C5, E5, G5, *, and C6. (I have skipped the 7th harmonic, its use might become another question).
There are valves to drop the pitch by 2, 1, or 3 semitones. So, I can use these to fill in the gaps between the harmonics. The biggest gap is between C4 and G4. Filling this gaps requires 7 of the 8 conceivable fingerings. Only valve 3 by itself seems to be unused.
Now, let's do some maths. The full length is 1.480m. So, to drop it by a well tempered semitone, this needs to be extended to 1.568m. So, you might expect valve 2 to add 88mm of tube. To drop by 2 semitones, the length needs to become 1.661m so an extra 181mm (note more twice that needed for the semitone). Now comes the problem: to drop by 3 semitones requires the length to be 1.760m so an extra 280mm which is more than the sum of the lengths added by the previous valves. My calculation is that adding 88mm and 181mm only drops the pitch by 2.89 semitones.
Is the valve mechanism more complex than I am assuming and using 1 and 2 adds more length than the sum of 1 alone and 2 alone?
The player corrects the difference somehow (e.g. embouchure)?
It is just a little bit out of tune?
There is the further problem that the gap between the 2nd and 3rd harmonics will be a just tempered 5th rather than a well tempered one. I'll ignore that as well for the moment.
Note: I have played a variety of woodwind instruments but no brass. My knowledge of the trumpet is primarily theoretical through reading.