Brass Instrument Valve Order

Question

Brass instruments such as the Trumpet and French Horn have three keyed/levered valves. Some related brass instruments such as the Piccolo Trumpet have four keyed valves. The Tuba has four or more.

Pressing down a key opens a valves, which adds extra length of tubing to the air column inside the instrument, thus lowering the pitch.
Roughly speaking, the first valve lowers the pitch by two semitones, the second valve lowers the pitch by one semitone, and the third valve lowers the pitch by three semitones. When present the fourth valve lowers the pitch by about five semitones.

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Why are the valves values in this order: 2,1,3,(5) semitones;
and not what might seem more logical: 1,2,3,(5) semitones?

Answer 1

The first valved horns, made by Heinrich Stölzel in 1814, had only 2 valves (arranged the same as modern 1st and 2nd).

On one of these horns, if you play the notes of a C major scale from C4 to C5 (or from G3 to G4) (8 notes): four are open, two use the tone valve on its own, one uses the semitone valve on its own, and one uses both together.
F major has three open notes, does not use the semitone valve on its own, has four tone valve only notes, and one note with both.
G major uses four open notes, uses the semitone valve alone twice, the tone valve alone once, and both together once.
So in these three scales, the semitone valve is used less than the tone valve.

To my mind this makes Stölzel's arrangement a little easier to play than if 1st and 2nd were reversed. He was a horn player so it may be design but there is not much in it and it may also be chance.

The 3rd valve would always be played with the 4th finger as this finger is weaker and that valve is used less.

Now that the fingerings are established there would need to be a significant advantage to changing them, and no such advantage has been found.