Due to its very nature, there are several different frets for a given pitch. Has there been an attempt to remove the ambiguity in standard musical notation for similar instruments before?
You are asking about tablature notation, which indicates not only the pitch to play, but which string to play it on. Tablature was developed for the lute and is used to this day for the guitar.
Standard notation for classical guitar addresses this also, by providing numbers next to notes. Certain numbers indicate which string to play the note on, or sometimes also which fret position to place the hand to reach the note. The notations "p", "i", "m", and "a" indicate which right-hand finger to use to pluck the note.
I suppose a violin can be considered to have an "isomorphic fingerboard", as each string is tuned a fifth up from the one before (as opposed to the guitar with unequal intervals between the strings). There are many notes that can be played on more than one string on the violin. You will sometimes see an indication like "sul D", meaning "play this note on the D string."
Edit: Sometimes in violin music a single Roman numeral indicates which string should be played. Likewise in organ music I indicates Great manual, II the Swell, III the Choir or Positif. I think the violin strings are numbered from the top down. I is the E string. Similarly you could assign Roman numerals to your instrument's strings and indicate which string would be used for the top note of the chord.
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