The simple modelling of the rich sounding effect of non-linear mouthpiece action on layers of the repetitive delays down the instrument tube, I found, can be applied to other pipe instruments  like this simple saxophone model. Reed instruments work a little differently to trumpets though...

To make the above saxophone patch sound, the whole ‘pipe-reed system’ is made to oscillate; that is the gain round the delay line, filter, VCA loop raised to one or just slightly higher. I’ve marked this path is a deeper blue in the diagram. Uncontrolled, above one the oscillations would get  bigger and bigger and it’s very hard to keep it at exactly one, so there must be some form of  amplitude limiting. In reed instruments,  this is a function of the limiting spring action of the reed in the mouthpiece. If you blow hard enough, it will sooner or later reach its limit of elasticity or even collide with the body. Electrically, in the model a soft amplitude limiting function built into the delay line for overload protection just happened to work nicely as a very crude reed elasticity limiting device. For once luck was on my side.

Looking at saxophones, they have conical bores while clarinets, for example, have straight cylindrical bores. This I all very well in reality, but with electronics, it is very difficult to construct a conical analogue delay line! With delay lines you can either have positive or, by electronically turning the waveform upside down, negative feedback from the output back to the input. Fortunately though with delay lines, both forms at unity gain will oscillate - when positive; at all harmonics of the fundamental, when negative; at odd harmonics only but an octave lower. In the saxophone, the conical bore does tend to act as an open pipe resonant at all harmonics, so in the model a positive delay loop is used as the nearest simple practical approach to this.

To make the model work, the overall gain has first to be set just below unity to make the ‘pipe’ live and then blowing energy (gain) is added to make it oscillate. To do this a second multiplying element (VCA) is used (bottom centre) in series with and before the (reed) low pass filter. Also a little filtered noise is added (top line) to add some breath sound and randomness.  Pressing a keyboard note raises the gain as well as selecting a note. At very very close to unity this results in a sluggish  breathy pipe sound, above that results in the normal saxophone sound. The remaining multiplier (CVP) introduces a small amount of tremolo to keep the sound alive. The filter frequency (‘lip-reed position’) is tracked along with pitch. Deviating from this relationship can cause the ‘reed’ to ‘over-blow’ when it goes higher than the pitch.

When performing on the model there’s a few things that do jump out at you....
       
    1) Like the trumpet model, if the system is allowed to become absolutely static the sound became fairly lifeless.
    2) Much less squaring effect is required; Z =  Y(X + M) for the multiplier; setting is  X<<M. So I was probably wrong about the effective size of the reed compared to the pipe as its action is more obviously subtle than I had first thought.
    3) The model  ‘OVER-BLOWS’ easily.

There are loads of knobs to twiddle. Here’s what some of them do....
The frequency and resonant peak of the ‘reed’ low-pass filter have a very marked effect. If they don’t track the sound becomes strained. You can hear this in the first sound clip below towards the end. Setting to a low gentle roll-off frequency, the saxophone sounds woolly, with higher notes sounding really strained while the tuning is also seriously effected. This is because of the filter’s added delay (analogue filters all add a little delay to the signal) . Set much higher, the saxophone sounds bright and crisp but unintentional ‘over-blowing’ becomes a problem. The very extreme high notes still sounded strained as the sound quality gradually changed with increasing pitch.  

With a large number of parameters to play with (24 knobs) many varieties of saxophone sound have been produced, some good some pretty bad! This one has a distinctly oriental flavour...