First of all, I'm sorry I do not have top-notch knowledge to properly formulate the question, so I apologize if some of you do not get the point of my question.
It seems in western music, the de-facto standard for writing music is the staff, and the most fine division of note pitches is the 12-notes system.
Based on this system, we choose subsets of the 12 notes to create scales, and chords to create harmonic movement.
If I want to create an alternative formal system for writing music, would it make sense to just start at the scale level, so instead of referring to notes by their name (C, C#, D, ...) I refer to them by their grade in the scale (I, II, III, ...)? (so, in principle it is not possible to write notes that are not part of the scale, but we could introduce back the notion of accidentals, i.e. IIIb or II#)
What are the advantages and disadvantages of that?
I am aware that a musical piece does not use a fixed musical scale but rather modulates back and forth between different (perhaps similar) scales; this aspect could be as well part of the notation.
Is the standard system more complex than this? (in the sense that it allows more expressivity than this)