Here is an example of Byzantine notation from the EBYPES project site on Byzantine chant notation. This ancient form has been used as music notation far longer than European notation and was designed exactly to express the subtile nuances of pitch found in eastern music. I don't read Byzantine notation but there are many priests and chanters who still read it today.

I don't know whether a system like this was ever used for secular music but there must have been some form of written expression as eastern music developed.

      - Mark