Lasse Thoresen (b. 1949) is a professor of composition at the Norwegian State Academy of Music. Thoresen has pioneered the use of microtonality in Norwegian contemporary music. His latest research project, "CONCRESCENCE", involves quartertone singing. This is an example of how he scores. One of his principles is to use symbols that differ visually in two ways from the normal accidentals.
His symbols are shown in these ear training exercises (written by Lasse Thoressen and Gro Shetelig Kruse) from the CONCRESCENCE workshop in November 2005. Kjell Tore Innervik played the exercises for the singers and made a rehearsal cd with 80 exercices.
Quartertone flat and quartertone sharp
Bjørn Fongaard (1919-1980) was the original pioneer of microtonality in Norway, starting his work in the late 1940s. Fongaard wrote a lot of exercises and short pieces in the 50s and 60s. He wrote for his electric quartertone guitar and for other instruments using quartertones. He also used quartertones in his orchestral works.
Excerpt from Piece for Quartertone instrument, Opus 34, February 1964.
The Yahoo group TUNNG, George D. Secor and David C. Keenan concluded that if quartertones were the only microtonality they would prefer the Tartini / Couper notation.
The Philharmonia user's manual: Peter Tornquist recommends this site to learn about orchestration. The site also includes information about quartertones and extended techniques. Peter also recommends How to write for Percussion by Samuel Z Solomon (2002) .
Harry Partch used the Just intonation system. Many of his microtonal instrumental scores use the conventional notation system and define the instruments according to the normal 12-tone reading system.
John Cage scored for Zoomoosaphone using these symbols including fractions.
For more details look at the work list and see how the composers have notated the quartertone marimba works.