Most trumpets' low C♯ and D are too low, and need correcting by pulling the 1st of 3rd valve.
My first understanding was that the distance between tones in the lower register is larger than between tones in the higher register, therefore the lengthening of the tube needed in the low register.
However, if all the tones in the lowest register (C down to F♯ with 3 ledger lines under the staff) are correct, the harmonics of this tones should also be correct.
My new explanation is that if you tune the valves so that middle F♯ (fingering 2), F (fingering 1), and E♭ (fingering 23) are tuned, then fingerings 13 and 123 are going to be too high.
This would imply that the whole harmonic series would be too high, including the lowest F♯. My feeling is that the lowest F♯ is in tune, while the C♯ (same fingering 123) is not. This does not make sense, maybe the low F♯ is also high?
Any explanation? Why is low C♯ so high but not low F♯? Is it even so?
If my reasoning holds, then low E (12) should also be a little high, if F (1) and F♯ (2) are in tune?
I wanted to put numbers on the 1+2 fingering not being in tune if the 1 fingering and the 2 fingering are.
Frequencies of the trumpet's low C and downwards, in equal temperament based on A at 440 Hz:
C: 261.626 Hz B: 246.942 Hz B♭: 244.082 Hz A: 220.000 Hz
C: 3.8223 ms B: 4.0495 ms B♭: 4.0970 ms A: 4.5455 ms
And their wavelength (assuming 343 m/s):
C: 1.3110 m B: 1.3890 m B♭: 1.4153 m A: 1.5591 m
And in procent of the reference (C):
B: 105.95 % B♭: 107.19 % A: 118.92 %
So if valve 2 is in tune for B (5.95 % length increase), and valve 1 is in tune for B♭ (7.19 % length increase), their sum (valve 1 and 2 down) gives a total length increase of 5.95 + 7.19 = 13.14 %. The theoretically tuned A should have a length increase of 18.92 % instead, so fingering 1+2 makes it a bit sharp.