Resonant fingerings are a series of alternate fingerings used for the throat tone notes on a clarinet. They improve the tone and tuning and make the transition between registers more seamless (and easier as well).
This is a very variable register of the clarinet in terms of pitch and tone quality.
Since they are short tube notes (made with note many holes closed) any change in your embouchure, tongue position or air pressure will be much more noticeable than it would be with a long note (meaning a note fingered with all the fingers down.)
Resonant fingerings have the additional benefit of making these notes more resistant and therefore more stable.
Most of the resonant fingerings also lower the pitch of the throat tones, which are often sharp.
In the throat tone range the clarinet can easily go from being very flat to very sharp on any note depending on how loudly the note is played.
F notes will tend to be flat and P notes will be sharp. Unfortunately many of the other instruments have the opposite tuning tendencies so any pitch change will be very obvious. Throat tones include the notes from G to Bb. Notes just below ( e and f) are also sensitive and often require correction of their pitch. E and F notes are often flat at mf or f levels. Opening the C# key will raise the E and opening the right hand Eb will bring the F note up to pitch.
There are two considerations for your choice of fingerings, the pitch and tone of the note and technical facility. If the note is in a melodic context or last for a long time use the best fingering for tone and pitch. Always use the best sounding fingering that you can, even if it makes it a little harder. If the note it is part of fast passagework use the easiest fingering.
Though I am only talking about fingerings here don't forget not withstanding whatever fingering you use, having the optimum shape inside your mouth (syllable, throat and tongue position) cannot be underestimated.
000/103 (+ possible c key) seems to be the best for tone and tuning. This fingering focuses the note more. For passage work over the break the entire right hand can be down. All the fingers down and F will make the g flat on most pro horns so be careful that you do this only in very fast passages. We test by listening; can you hear the note in tune? Another good one to experiment with is /2 it's somewhat muted sounding so you can blow quite hard against it.
In a melody I often put fingers down (darkening by making the note more resistant* the tone but flattening it as well) and play with a firmer embouchure (focusing the tone and raising the pitch) to better match for instance the upper note that I am going to. I have had to play this note with the Right hand side Eb key open in FFF with brass.
*Yes, resistance makes it darker that's why many clarinetists fall into the hard reed trap. It is best to make the resistance by "blowing against yourself", think about that one.
Like g but tends to be flatter so ease up on the fingers down
023/023 E/B key (or C key if that's where you are going and it's quick)
I use this fingering a lot. Putting down the first finger on the right hand really mutes the sound, so unless it's fast leave it off. It's got to the point where I can use these resonant fingerings even in technical passages I think it shows.
Bb and thumb key 001/001 C key. This one rings the most. You can also play it like the A.