Your terminology has left me a bit confused. "Crossing the break" is an expression which is used on clarinet and refers to the change between "throat" A and what is called "long B" where the register key is added for the first time taking the clarinet from the
chalumeau register to the
clarion register.
I think you may be referring to the "change" on saxophone where one lifts each key one at a time to go up a scale until there are no more keys left to lift (middle C#). Then all 6 keys of the left and right hand are again added, this time with the octave key, and then the process of lifting keys one at a time begins again to go higher in the range of the instrument.
That said, I think I know what you are asking. Slurring a note which uses the octave key to a lower note which across a wide interval to a note which does not is a challenge on the saxophone, and other "conical" woodwinds. I think the reason has to do with acoustics. When an octave vent is opened it forces all of the notes using that vent to jump to their second mode which doubles the frequency measured in vibrations per second. "Shifting gears" in that direction is easy.
Where the difficulty begins is where we take a "mode 2" note and want to quickly and smoothly take it down to a "mode 1" note. For reasons I don't completely understand, a clarinet which sounds only the odd numbered overtones: 1 - 3 - 5 etc. makes these downward leaps effortlessly---something composers for this instrument take great advantage of. (see Mozart Clarinet Concerto).
I'm blabbering on and on because this is my favorite subject and I can't help myself.
Now I will cut to the chase and give some tips I have leaned along the way. I am certain the "professional players" Pete Thomas, David Roach, and Aldevis can give you even more good advice on this subject.
- As you are sounding the upper note "mentally play" the lower note which follows before changing the fingering.
- In some cases it helps to play the "mode 2" note without the octave key by voicing it as an overtone and then doing the slur.
- I was taught that lower notes take a more open throat and "warm air", but Harvey Pittel said Allard taught the opposite that high notes take a more open throat and low notes a more closed on. Now I don't know for sure which one to believe. Try it both ways and see what works for you.
- It often helps to "blow into" the lower note of the interval if the dynamics of the piece will allow you do that.
- I would use a legato tongue on the lower note as a "last resort" if nothing else works.
Thank you for posting that video. I had not heard that piece before. It is breathtakingly beautiful.
Any song that moves me to tears on the first listen is a "keeper".