below you can find a fingering chart for quarter-tones on the saxophone.
due to the differences in built, different mouthpieces and different reeds, it is obvious that these fingerings only present a basis for your own search.
there is one quartertone that is relatively difficult to obtain: the one between g and g#.
I know of no other possibility to obtain it, than the mechanical alteration described below [illustration 1) and 1a)]
this little 'trick' allows us to partly open the g# key by fingering the normal g# and then pressing the d-key, that has been altered as can be seen in the illustrations below. apart from providing us with the quartertone between g and g#, this alteration of the saxophone mechanic provides us with a very good fingering for the g# above the normal register of the soprano saxophone (see: technique / high register ) . I don't know if this applies also to tenor and alto saxophones, I know it only from my own soprano.

[illus. 1]

[illus. 1a]
I use the French terminology for the fingering chart.
you can have a look at it below, or download it below:
pdf_french terminology for fingerings
you can also download the entire fingering chart as a PDF:
quartertone fingerings for saxophone
the symbol '(8)' means: octave key pressed


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