French Horn  Hand Stopping -- Why the pitch shift?

Playing the modern french horn presents several special challenges:

The most popular horn, the "double" horn, is two instruments in one, sharing the same mouthpiece and bell. A thumb-operated valve switches between a horn in F and a horn in B-flat (a fourth higher). So, the player must learn two sets of fingerings -- and decide when to switch between them as the music goes higher and lower.

Before the invention of valved horns, horn players played in different keys by adding lengths of tubing ("crooks") to transpose the fundamental pitch of the instrument to the desired key. With the benefit of valves, modern horn players don't have to use crooks to play in every key. However, horn parts are written for the old transposing horns, so modern players have to undo this transposition (by performing a reciprocal transposition) when they read music. Horn players need to read both treble and bass clef. When the "stopping mute" (or the hand, which the mute emulates) is inserted into the bell, the set of pitches produced for any given fingering is one semitone higher than it would be without the mute. To compensate, the horn player transposes the music down a semitone, in addition to any other transposition they're doing.

The Nature of the Pitch Shift

If a stopping mute (transposing mute); different from a normal mute which is non- transposing, 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.