This is another difficult topic. Circular breathing has taken on mythical meanings but the one truth which has remained is that it is hard to do. Saxophones, particularly, are not an easy instrument to breath circularly on. It is possible to get the idea in a few minutes, but it can take many years to be functional in performance. The basic idea (try this with a straw), starting with a half-full lung, is:
1) Breath out through the straw until nearly all lung air is gone
2) Fill mouth with an air pocket, while still breathing out from lungs.
3) Switch over cleanly from lung to mouth air, i.e.. isolate mouth from nose&lungs, and force air out from your mouth with your cheeks.
4) Breath in through the nose fast! :) Keep mouth air coming out.
5) Switch back to the now-full lungs
"As you're blowing air into the saxophone, fill your cheeks with the air (to store it.) Then, while you are inhaling through your nose, depress your cheeks, forcing out the stored air. Therefore, you are making music and breathing at the same time - your cheeks sort of act like the bag part of bag-pipes. Good luck, it's hard."I refer once again to Shooshie who has mastered this art:
"You've reached the point that takes anywhere from six months to six years to perfect. That little Step #5...it's the crux of the whole thing. Now here's a tip:
The more resistance you have on your instrument, the easier it is to circular breathe, because an instrument with more resistance makes Step #5 less tedious. Try holding your finger over the other end of the straw, rather than blowing bubbles. Give it some pressure so that you're blowing against some resistance. Now try that Step #5 and see what a difference it makes. Still no piece of cake, but you can probably see the potential. The smooth transition from "mouth air" to "lung air" requires that you have your lung pressure - diaphragm - equalized behind the little valve in your throat before you release the valve.
Seriously, that is the one little operation which takes you a while to learn. Part of the reason that it can take a while to learn is that different registers of your horn respond with different pressure, and you have to learn each of them, and be able to cross between them.
Let me make it easier: If you are doing the mouthpiece exercise, you have learned about diaphragm pressure. Diaphragm pressure enables you to pressurize the airstream rather than the instrument - so it feel as if you are playing your airstream and the instrument is just out there on the end of it - and this makes the entire horn respond more evenly throughout all the registers. This will greatly enhance circular breathing. If you have followed me up to this point, maybe now you will see the validity to my claim quite a while back that the mouthpiece exercise even helps circular breathing.
The less resistance you have, the more perfect you have to make the transition. The easiest instruments to circular breathe with are oboe, trumpet, French horn, and bassoon. Next would come clarinet, other brass instruments, and soprano saxophone. The larger saxes are progressively more difficult. Last, and most difficult to effectively circular breathe with is the flute. But it can be done. Galway apparently did it for the Paganini Moto Perpetuo (Man With The Golden Flute - RCA) - which is 5 minutes long without a break. Of course, the recording may have been spliced, but I can do it well enough on flute to believe that Galway actually recorded it that way.
Step #5 will come in time. Practice it until it becomes a subconscious maneuver, something which you can do in your sleep. I didn't pursue it seriously for many years, but I practised it daily as I played in bands or orchestras during tutti sections, or while practising etudes. One day, I realized I was doing it without any problem. I then took it seriously enough to practice it so that I could do it in solo performance without anyone realizing it. Once I got serious about it, I was able to use it in concert within six months or so, but I had been doing it for at least six years before then."