Eventually I will have the famous "tuning rings" for saxophone.
I'm not sure I quite understand what you mean by "tuning rings". Do you mean something that will go inside the edge of a tonehole in order to lower the pitch? Curt Alterac's
Tuning a Saxophone With Crescents uses cork which is easy to cut, comes in various thicknesses, and can be sanded to make fine adjustments. The crescent is placed on just the mouthpiece side of the tonehole, and does not go completely around its circumference.
From Acoustical Aspects of Woodwind Instruments by C.J. Neverdeen p. 50
"We assume 10 cents (0.006) to be the threshold of perceptibility . . . From the two equations we than find that this corresponds with a 1% change in distance to the (acoustical) top of the instrument, a 10% hole diameter change or a 20% hole-length change."
"Hole-length change", of course, translates to tonehole chimney height as it applies to saxophones. By reducing the tonehole diameter in order to lower the pitch, you are also taking away from the venting of that opening. By placing a "crescent" on the "upstream" side of the tonehole you are increasing the distance of the tonehole from the top of the instrument without significantly changing the note's ability to speak clearly.
The one tuning variable not mentioned is the key height above the tonehole (including the protrusion of the resonator). Again, there is a point at which lowering the key height to bring down the pitch has a negative effect upon the venting of the note.
