There are two fairly obvious ways a saxophone can be close miked. One is close to the mouthpiece, another is close to the bell.
The difference in sound qualityis tremendous. The same applies to all close miking. Small changes in microphoneposition can affect the sound quality enormously.
There are many books and texts that claim to tell you how and where to position microphones for all manner of instruments, but the key is to experiment and find the best positionfor the instrument – and player – you have in front of you.
Experience, not book learning, leads to success. Of the two saxophone close miking positions, neither will capture the natural sound of the instrument, if that’s what you want. Close mic positions almost never do.
If you move the mic further away, up to around a meter, you will be able to capture the sound of the whole of the instrument, mouthpiece, bell, the metal of the instrument, and the holes that are covered and uncovered during the normal course of playing.
Also as you move away you will capture more room ambience, and that is a compromise that has to be struck. Natural sound against room ambience. It’ssubjective.
The rule of thumb is that if you want to capture the natural sound of an instrument, the microphone should be at a distance of at least 1.5 times the maximum dimension of the instrument. Get closer, and you will have to experiment.