Very often I'm asked by band directors, or see posts on flute forums about which Bb (Bb1 & Bb2) fingering is best to teach first. As frequently, I see passionate responses, as if they were dogma, about what one famous teacher or another said about that subject. I'd like to add my own thoughts, and hopefully present a valid argument to why and when we should use one fingering or another.
The Thumb-Bb Key
The Bb Thumb Key (also known as the Briccialdi B-flat Key) can be used for all notes where the thumb is depressed, with the exception of the B1, B2, B3 and F#3 regular fingerings. This fingering is best suited for flat key signatures since it allows for a quick transition to and from any other notes of the scale.
As a rule of thumb :-), I teach my students to use this fingering when playing in keys where flats are present in the key signature with the exception of Gb and Cb Major, and whenever the Bb is not preceded or followed by a B1, B2, B3 or F#3.
Note: I use this fingering, even though I shouldn't, in The Prokofiev Flute Sonata, 3rd Movement, #28 to 8 measures after #28 and 6 measures after #29 to 3 measures before #30 (IMC - Jean-Pierre Rampal Edition). I've made some attempts to avoid the thumb slide from Bb to B by using the Chromatic Bb or the Bb-Lever, but I'm so used to it that I always end up using it when I perform this piece. I encourage my students to use the Bb-Lever or the Chromatic Bb, which are better suited for this passage.
The Chromatic Bb (or A#)
This fingering and the Bb-Lever are best suited for chromatic passages, or passages where both A# and B are present. Either fingering should be used for F# and B Major, their relative minors D# and G#, and also B harmonic minor.
The Chromatic Bb (also known as the Long Bb) is better suited than the Bb-Lever for the following passages: D-Bb-B, Eb-Bb-B, E-Bb-B, F-Bb-B (or the reversed order of those 3 notes).
Note: Eons ago I had a teacher who had me play all the scales from the Taffanel & Gaubert Method (Daily Exercises #4) with the Chromatic Bb. If you have masochistic tendencies or too much time on your hands, try it. I do it once in a while to remind myself how hard it is, and how inadequate this fingering is for all-purpose use.
This fingering and the Chromatic Bb Fingering can be interchanged. In fact, I know some flutists that use this fingering almost exclusively.
This fingering has advantages over the other chromatic fingering in the following passages, which sound smoother: F#-Bb-B, G-Bb-B, Ab-Bb-B, A-Bb-B (or the reversed order of those 3 notes).
One additional advantage is that the Bb-Lever can be used while playing other notes without affecting them, while the Bb Chromatic fingering cannot.
An example of when to use this fingering is Frank Martin's Ballade m.73 to m.75 and m.314 to m.318.
Historical Note: Giulio Briccialdi (1818-1881) was an Italian flutist, composer and the creator of the thumb B-flat key.