The pitch lowers gradually when the hand is placed in the bell and slowly moved inward. When the bell is completely covered (stopped), the pitch falls to a half-step above the next lower partial (harmonic). For example, playing a middle C (F-horn, open) and gradually covering the bell into stopped horn, the pitch will lower a major 3rd to A (or 1/2 step above G, the next lower partial). However, playing a 3rd space C (F-horn, open) and repeating the stopped horn, the pitch will lower a half-step to a B-natural (or 1/2 step above B, the next lower partial). The hand horn technique developed in the classical period, with music pieces requiring the use of covering the bell to various degrees to lower the pitch accordingly. Mozart's 'Four Horn Concertos' and 'Concert Rondo' were written with this technique in mind, as was the music both Beethoven and Brahms wrote for the horn.
Practically, it is too cumbersome to keep track of what partial is being played and what the "1/2 step above the next lower partial" would be. As such, when playing stopped, horn players over blow one partial while playing stopped, play the partial above the note then intended, cover the bell completely and thereby arrive at 1/2 step above their intended pitch, and then compensate by fingering a half step below the written pitch. Thus most horn players are taught that stopped horn "raises the pitch 1/2 step".
It is crucial to understand the difference (between practical application by the player and the acoustic theory behind it) because some modern composers have incorrectly notated that the horn is to bend an open pitch upward to a stopped pitch. This is impossible. The horn pitch can only be bent downward into a stopped pitch.
In the middle register, try F-horn fingers when playing stopped horn. In the upper register, however, experimentation with traditionally flat fingerings on the B horn (For example, 1st valve G) can yield more secure notes without sacrificing good intonation. Some B horns have an a stopping valve that compensates for this, allowing the player to use normal fingerings with the stopping valve.
There is also an effect that is occasionally called for, usually in French music, called "echo horn", "hand mute" or "sons d'écho" (see Dukas Sorcerer's Apprentice) which is like stopped horn, but different in that the bell is not closed as tightly. The player closes the hand enough so that the pitch drops 1/2 step, but, especially in the middle register, this is not closed as tightly as for stopped horn. Consequently, when playing echo horn, the player fingers one half step higher.
The difference between stopping and "echo horn" is a source of much confusion to younger players, especially ones whose hands are not big enough to close the bell all the way for stopped horn. Instead of stopping properly, they erroneously close the bell insufficiently and finger 1/2 step higher.
For more information on stopped horn see "Extended Techniques for the Horn" by Douglas Hill (ASIN: B00072T6B0) — Professor of Horn at the University of Wisconsin-Madison, http://www.geocities.com/Vienna/3941/stopping.html also has more information about stopped horn and the physics behind it; for more information on hand horn see A modern valve horn player's guide to the natural horn by Paul Austin (ASIN: B0006PCD4A).
Note that whenever one is using stopped horn, that the player should only use the F side. When one uses stopped horn on the B Flat side of the horn, the partials are not as accurate, even in ranges where a player would normally use the B Flat side. Also, the F side will give more of a full tone.
Also, the hand can be partially inserted into the bell in such a fashion as to lower the pitch of the horn one-quarter tone, an extended technique used in some modern compositions today.
Some less common transpositions include:
It has been speculated that one of the reasons Brahms wrote for horn in the awkward key of B(♮) was to encourage the horn players to use the natural horn; he did not like the sound of the new valved horns.
Another kind of multiphonics can be achieved by simultaneously playing two neighbouring notes of the harmonic series. A practical way of doing this is by placing the lower lip under and outside the mouthpiece, playing one note, and then gently, by increasing air pressure and adjusting one's lip-position, halfway slurring upwards to the next harmonic step. This might be frustrating at first, and the technique is quite an unstable one to perform in real-time, especially when compared with similar practices with other brass instruments, esp. trombone. This technique finds its occasional use in contemporary music where, successfully performed, it might evoke an interesting effect.
A and D on the B side tend to be sharp, especially as you get into higher octaves. One suggestion is to play these notes with the third valve (and trigger). Also, for the A above the staff (the horn A, otherwise known as concert D), a way to flatten it is to play it with just the thumb. However, all instruments are different, so you will want to check this against a tuner.
If you are unable to play the fundamental of the horn (the concert F that is an octave below the bass clef) with the traditional fingering of 0, then you can sometimes also play it T13.
A lip trill is a rapid oscillation between neighboring harmonics - used primarily for whole-step trills from second-line G up approximately an octave. Lip trills are possible both lower and higher, but much lower than G and the harmonics are too far apart for a whole step, and much higher and harmonics are to narrow. Many books give fingering charts for lip trills, but as with all things on the double horn, there are options. A good rule of thumb, however, is for a trill from G-A, use 1 & 3, on the F horn, A-B, use 2 & 3, etc. From third-space C-D, use 2 & 3 on the B horn, up to open B horn for f-g. In his book "The Horn", Barry Tuckwell also gives a fingering chart of possible 'faux' 1/2 step lip trills.