An instrument that overblows at the octave has identical fingering for both registers.
Sax created an instrument with a single-reed mouthpiece like a clarinet, conical brass body like an ophicleide, and some acoustic properties of both the horn and the clarinet.
This enables a chromatic scale to be played over two octaves simply by playing the diatonic scale combined with alternately raising and lowering this one digit.
However, this keywork never gained much popularity, and is no longer in use.
All the instruments were given an initial written range from the B below the treble staff to the F, one space above the three ledger lines above staff, giving each saxophone a range of two and a half octaves.