towards a truly chromatic Horn

So here's a puzzler

Brief review of the physics of wind instruments

Your standard wind instrument is a pipe attached to some kind of sound source (lip or reed). Being of a certain length it resonates at particular frequencies and suppresses the others, ulimately being capable of producing tones that are multiples of a certain fundamental frequency f, 2f being an octave higher than f, 3f being a fifth above that and so on.

The main things to keep in mind about the multiples of f (the "harmonic series"):

1. going up by a particular interval means multiplying the frequency (or dividing the wavelength) by a particular ratio (2 ↔ octave, 3/2 ↔ perfect fifth, 5/4 ↔ major third, etc.…), and
2. intervalwise, the notes are getting closer together the higher you go.

Eventually, the notes get so close to each other that you can do scales of a sort -- this is how the natural (i.e., valveless) French horn works; the tube is so damned long that you're always playing in the 8f-16f range and then you just fudge the few notes that are out of tune with lip and hand-in-the-bell tricks. ... and likewise for why baroque trumpet parts are so insanely high-pitched.

But if you want to do anything useful down in the lower register, you have no choice but to mess with the length of the tube somehow, the two most popular methods being

1. (the woodwind solution) poke holes in the tube (and cover them with removable keys)
2. (the brass solution) insert valves or slides in the tube to change its actual length

We'll focus on the second solution since I really have no freaking clue what's going on with woodwinds. Oddly enough, the brass solution is simpler but it didn't come along until fairly late in the game. Not until the 19th century did metalworking technology finally get good enough that Heinrich Stölzel was able to construct the first actual valved instrument in 1814. Once this happened it didn't take long to catch on -- hence the 19th century explosion in the use of brass instruments in orchestral works (cf. Wagner, Mahler...).

So how do valves actually get used?

Let's first get clear what a valve is: ( Yes, we are going deep into the weeds. You knew this was going to happen )