

So try to pin down which is in operation in any given discussion. If there's a good reason for the first system, it's yet to reach me. All we need to know is what the tonic is.

The tonic chord of a minor key is i, it contains the minor 3rd above the tonic. The tonic chord of a major key is I, it contains the major 3rd above the tonic. And it's why I recommend the other system, one that aligns with the way we name intervals. What, you might ask, about chords that include the 6th and 7th degrees of the minor scale? Do we name according to the natural, melodic or harmonic form of the minor scale? All three often mix freely in the same piece.

In C major, E major triad is III, in C minor E♭ major triad is III. You can decide what key you're in and name strictly diatonically. There are two systems of naming chords by Roman numerals. Notice the difference in symbols depending on whether you give the key signature before the Roman numeral symbols.

The secondary dominant is sort of the standard harmony analysis way, changing the letter case to alter diatonic from the key signature is more of a jazz way to label chords.) Finally, you could prefix the number with a flat - ♭III - and that would mean the E is altered from the C major key signature to become E♭. To clearly convey that secondary dominant sense, you would write C:V/vi. (A small aside: that chord could be considered a secondary dominant to chord vi A minor. If it were given as C: III the chord would be E major. In case-sensitive style iii is used because the diatonic triad on the third scale degree is minor. in C major, just III could mean simple the third scale degree, E or any triad rooted on E.
