I have been playing recorder for two years now and have recently been using this chart, http://www.dolmetsch.com/efingeringchart.pdf for third octave soprano fingerings.
Playing the D or the F# fingering with a closed bell lowers the pitch a semitone, to C# and F, respectively. Playing the D# fingering with a closed bell instead raises the pitch a semitone, up to E. Why does closing the bell raise the pitch sometimes and other times lower the pitch?
EXCELLENT question!
The answer actually rather like "Why does stopped french horn sound up a half step when closing the bell lowers my tuning?"
Horn players routinely adjust their hand position to close or open the bell of the instrument, thereby changing the tuning. When they need to adjust themselves down, they close the bell, and when they need to adjust themselves up, they open the bell. However, when they close the bell completely, for a stopped horn effect, the pitch is raised half a step.
This is because the next highest partial is closer to the original note than the pitch that would normally be created simply by closing the bell. This is highly dependent on the harmonic series of the instrument in question. In the french horn's case, it's playing range is so high up in the harmonic series, that they just learn to transpose their fingerings down half a step for stopped horn because closing the bell is going to pop you up to the next partial in virtually all cases.
These recorder fingerings are doing nearly the same thing, except the harmonic series of each fingering is QUITE different from that of the french horn.
For the D fingering, the surrounding partials are far enough apart that closing the bell does not cause you to jump to one of those partials. A good exercise would be sustaining the D, and then gradually closing the bell to varying degrees. You will be able to gliss from the D to the C#.
For the D# fingering, the next highest partial is rather close. If you gradually close the bell on this note, you WILL hear the pitch go flat, but by the time it's fully closed it will have jumped up to the E.
