I think the playing of long tones is absolutely essential to a fine tone. There is nothing that I know of, that will refine your tone as quickly. When playing long tones, you take away all the other aspects of practicing, like music reading, keeping up with up a metronome, etc, and you strip it down to the one thing that we strive as musicians to do. That is, to produce a well centered, focused, full and pleasing tone. The playing of long tones should not be thought of as an endurance contest. It is not about how long you can hold a note for. It is about focusing on one note and trying to play that note as perfect as you possibly can while thinking about all the things that are going on, from your diaphragm to your throat, your sinuses and your oral cavity. Think of it as “marrying” the two tubes involved, your body, and the horn. Think of your air stream as a piece of string that is constantly coming from your diaphragm, past your throat and all the way through your horn, out into the cosmos. I cannot stress enough the importance of this concept.
You should hold long tones for eight to ten seconds each. Take the horn out of your mouth after every long tone. That way you have to reform your embouchure every time. Try to practice in a quiet room, playing against the wall, so you can hear every nuance. Close your eyes and concentrate. Again, breathe from your abdomen and focus your air stream on the tip of the reed, and blow all the way through the horn. It would be a good idea to go through a series of steps before actually playing the note, like this:
This is a good mental image to keep in mind while doing your long tones.
It is a little easier to keep your long tone practice in the middle of the horn. Say, between low G and G with the octave key. Work your way up and down gradually. One variation I like is to start on middle D. Alternating up and down by half steps, go down a half step to C sharp. Then up to D sharp. Then C natural. Then E (O.K.), then B, then F (O.K.), then B flat, then F sharp, then A, then G (O.K.), A flat, A flat (O.K.), etc. Like this:
The best book I have found that deals with this subject is Top Tones for the Saxophone by Sigurd Rascher. The fingerings aren’t much good, but the exercises are the best. For more usable fingerings, try Studies in High Harmonics for Tenor and Alto Saxophone by Ted Nash. These are the books I studied to achieve an additional octave and a whole step on the top end of my horn. That, and playing in plenty of loud rock and roll bands ;-). On a good night, I can get up to a G4. Having those notes at your disposal definitely comes in handy and there’s nothing quite like letting rip a screaming high note just at the right time!
While I’m on the subject of long tones and articulation, one exercise you should do right after you finish your long tones is tonguing exercises. Play a long tone and repeatedly articulate the note by saying ” tah, tah, tah, tah,” like this. Think of playing one long note, just interrupt the reed’s vibration with your tongue. The air stream should be one long string of air, not separate pulses of air for each articulation. Try to have as little extraneous noise as possible, and aim for a clean attack on every note, like this. The tongue is made of muscle, and like all muscles, it responds very well to regular exercise.