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 ...
Here's a link to the
Maurice Andre website which lists Maurice's opinions of the various piccolo trumpets he played over his career.
The most common design for piccolo trumpets has four piston or rotary valves, with valve 4 introducing tubing with the combined length of 1+3.
A less common design features three piston valves. Here's a picture of a beautiful ...