FYI, some "Extended Helmholtz-Ellis JI” accidentals [1-2], in fact designed
quite recently, but a nice input. There is also a Unicode font at [3]. Notation also
mentioned at [4].
The arrow accidentals that LilyPond has, are used for syntonic comma 81/81
alterations (staff system in Pythagorean tuning). LilyPond does not have those
for double sharps and double flats. In addition, there are accidentals with
double arrows, for double syntonic comma alterations, which LilyPond does not
have. (And even triple arrows.)
Some traditional quartertone accidentals end up on the 11-limit rational
interval 33/32, which seems to be a good idea: in E72, it is approximated with
E24 quartertones.
1. http://www.newmusicbox.org/assets/72/HelmholtzEllisLegend.pdf
2. http://www.marcsabat.com/pdfs/notation.pdf
3. http://www.marcsabat.com/
4. https://en.wikipedia.org/wiki/Just_intonation