I have the feeling I miss / don't understand something conceptually. I would like to understand why lily refuses to engrave the slur of the first fis.

On the question of why the first tie does not appear: there is a note (a2) in the voice between the first and second fis1 that interrupts the tie. This can be resolved with tieWaitForNote.

\set tieWaitForNote=##t fis1 ~ \unset tieWaitForNote fis ~  etc,

as explained in https://lilypond.org/doc/v2.24/Documentation/notation/writing-rhythms#ties.

--
Timothy Lanfear, Bristol, UK.


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