Playing the modern french horn presents several special challenges:
The nature and cause of the last of these inconveniences are discussed here.
If a stopping mute is slowly introduced into the bell of the horn while a note is being played, the pitch drops (assuming the player follows the natural resonance pitch as it shifts downwards). The interval of the shift varies, depending on the starting pitch.
However, if the player stops playing an unstopped note, puts in the stopping mute, and then tries to play a note with the same pitch as before, the resulting note is always one semitone higher! (At least, this is the case for the F horn; for the B-flat horn the difference is greater; some horns have yet another valve to compensate for this.)
This seeming discrepancy has been the cause of considerable debate surrounding both the nature of the effect and its causes. John Backus, in the article "Input impedance curves for the brass instruments" (published in the August 1976 issue of the Journal of the Acoustical Society of America, volume 60, number 2) gives a good explanation of both. His description is summarized below, with a few new diagrams.
When a note is played on an unstopped brass instrument, vibrations from the player's lips travel down the tube, and are partly reflected back when they reach the large impedance gradient at the bell. The resulting standing waves are characterized by antinodes (points of maximum change of air velocity and minimum changes of pressure) at the mouthpiece and at the bell, and nodes (points of minimum change in air velocity and maximum change of pressure) between the antinodes, like this:
At least, that's how it's usually drawn. Another way to think about what happens past the bell is to consider that the walls of the room cannot vibrate, and thus they must be nodes, too. Actually, this "phantom node" is functionally much closer to the bell than the walls, as you discover when you play a note and walk toward a wall -- you have to get pretty close before the wall affects the pitch of the instrument. So, the "phantom node" might be drawn like this:
Each pitch for a given fingering has a different numbers of nodes; here's the set of pitches and their corresponding vibrational patterns:
If you had some way to restrict the volume of air in front of the bell, this "phantom node" would move toward (and into) the bell of the instrument, causing the other nodes to get closer together, arriving finally at the position of the mute:
This would result in the pitch going up (which is what you'd expect, since the effective length of the vibrating tube is getting shorter).
However, what actually happens when you bring a mute toward the bell is different, because the node which is affected is the first one inside the instrument. This node is thus drawn out, toward the bell...
...resulting in the pitch dropping. (The phantom node and its adjacent antinode are lost.)
How can we account for the difference between these two ending points? Simple: they have different numbers of nodes -- they are at different positions in the harmonic series:
The black lines show the drop in pitch that occurs when the mute is inserted; the blue lines show the pitch rise that would happen if the phantom node could be pushed into the bell.
So, the question of whether a horn's pitch goes down or up when a mute is inserted can be answered either way! (This diagram also shows how different-size downward shifts coexist with equal-size upward shifts.)