Why are C, G and C all played open on the trumpet? Why are C, G and C all played open on the trumpet?  I would expect C, F# and C to be produced by adjusting embouchure as they're evenly each a half octave apart.  What am I missing?
 A: The notes that can be produced by a simple linear resonator, like a plucked string, or the tube of a trumpet, form a harmonic series, that is, the frequencies are small integer multiples of the fundamental frequency.  
For a wave to resonate on a string, a whole number of waves equal the length of the string, so there's a wave node at each end of the string. For a cylindrical bore tube, like a trumpet, it's slightly more complicated because you can have an antinode at the bell end, because the resonating tube is open, and the resonating air column extends slightly beyond the body of the instrument. Please see the Wikipedia article for more details, and diagrams.
The upshot of this is that you can get an octave, which is 2× the fundamental frequency, or a twelfth = octave + perfect fifth, which is 3× the fundamental. So if the fundamental is C, you can get the C the next octave up, and you can get the G above that.
To get the F#, a flat 5th, is mathematically difficult. In the modern equal temperament, that interval is exactly $\sqrt2$ times the fundamental, and since $\sqrt2$ is irrational, you cannot achieve it exactly with any ratio of integers, although of course you can approximate it as closely as you want, if you use large enough integers, eg 17/12. But I doubt you'd have much luck trying to blow that on a trumpet without using valves. ;)
