The Master said there's nothing wrong mechanically with my beautiful new Yamaha Tenor Saxophone, it's just that I must make a slight adjustment in my Embouchure when I change registers.
I found out that the man is right!
HAL
I would like to suggest the idea that if the embouchure hasn't been "adjusted" i.e. tightened to produce the high register, then there is no need to make another embouchure adjustment for the sax to drop to the lower note or register when the octave key is released.
Essentially the entire range of the saxophone is played with the same embouchure except for the altissimo, and in some cases the upper palm key notes. I have my students do an exercise slurring Bb - F -Bb - F - Bb - F from low Bb up to high F and back down while concentrating on keeping the embouchure that produces a big and full low Bb throughout. Another technique I have found effective is to mentally be playing the lower note while fingering the upper note of a leap that slurs quickly downward.
A good test of a saxophone (and its player) is to slur rapidly back and forth between A above the staff and 4th line D. If this can be done with D responding clearly and rapidly, then the embouchure, throat, and saxophone octave mechanism are all doing their jobs. A common problem that makes this note change difficult or impossible is keeping the back of the tongue too high in the mouth. Singing "AH" and then blowing the airstream with this shape inside the mouth is the remedy for this problem.
Another exercise that can be very helpful is to play a low note as a long tone and have another person press the octave key so you don't anticipate the octave coming. This is a good way to get the sensation of producing high notes without making an embouchure adjustment. If the note stays high for any length of time when the octave key is released it indicates that the embouchure is too tight to begin with.
