Dear Stefan,

The way I see it, there are two options. You might:

1) translate the fingering diagrams to the template/format provided by LilyPond 
(I am sure you’ve seen this, but: 
https://lilypond.org/doc/v2.24/Documentation/notation/woodwind-diagrams) and 
code your translated fingering diagrams accordingly. Or:

2) create your fingering chart using markup/scheme command. This one, while it 
will allow you to replicate the fingering diagrams as you see on the page you 
referenced, is definitely more time consuming. 

For 2), I have come across composer Piaras Hoban implementing the flute 
fingering diagrams done by flutist Carin Levine in her “The Techniques of Flute 
Playing” published by Bärenreiter. The code can be seen here: 

https://github.com/phoban01/phlib/tree/master/fingerings/flute-levine

As for the actual usage of this scheme code by Hoban, you might do: 

%%% BEGIN %%%

c'1 ^\markup {
 \levine-flute-normal-fingers  #'((cc . ( one two five1h ))                     
                   
                                 (lh . ())
                                 (rh . ())) } 
%%% END %%%

Of course, you will have to extensively alter this code in order to use it for 
these saxophone fingering diagrams. 
 
I believe 1) is less time-consuming to implement. I do think the work Hoban did 
is truly noteworthy, however, and it has been one of my sources of inspiration 
(e.g., see: 
https://lists.gnu.org/archive/html/lilypond-user/2014-10/msg00313.html)

All the best,
Yoshi

-- -- --
Yoshiaki Onishi
https://github.com/yoshiakionishi/lilypond-snippets






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