I'm going to write kinda long and then give this thread a rest for a while; I've two exciting new musical projects getting started this April.
OK, it has nothing to do with equal temperament; I mentioned that the two versions of tuning and the two ways to write the chords were enharmonic - they are all using the same pitches (frequencies) of equal temperament. The difference is between how notes and intervals are defined in standard music theory and how they are thought of informally within the general guitar playing community.
Standard music theory defines notes as the letter names of the space and line positions in the staff. It does not define them by pitch because those notes are not pitches, but subject to representing up to five different pitches if you limit the key signature to double accidentals. Notes are not pitches, they are staff location position names. The note does not change when accidentals are applied. All of Gbb Gb G G# and G## are G notes, or just "G", because they all reside for instance on the second line up from the bottom of the G cleff staff.
Standard music theory defines intervals as the distance between notes, so intervals are the distance between position locations in the staff, not pitch distances, because the notes can represent different pitches without changing position in the staff, therefore the difference in pitch distance can vary, but the distance between notes (staff positions) does not. This why the interval of C to G is a fifth, but so is Cb to G# and so is C# to Gb... as long as the notes (letter names of the staff positions) are five lines/spaces apart, and despite accidentals that change the pitch distance, the note distance remains the same and the interval is still a fifth (which will include a quality term to account for the pitch distance variance).
A guitarist that has been playing a while will have formed a keen sense of the pitch perspective of what is going on (the distinct pitches of the chromatic scale and the distinct pitch differences counted as semitone distance between pitches of the chromatic scale), but when he looks at theory he's liable to feel dizzy because his definitions of the basic things (pitches and pitch differences) are now mismatches to the theory's definitions of notes and intervals (which are not based on pitch and pitch distance) so the standard theory constructs won't make sense. Standard theory will present a completely different connected dimensionality that appears incoherent, paradoxical, and irrational - things won't make sense applying the pitch based definitions).
Some will say so what? Guitarists for practical reasons will use pitch based methods anyway and say it is all semantics. I myself don't use the standard theory definitions of note and interval whatsoever to play, compose, practice, perform, or do studio sessions - I don't use any named things at all.
But like you suggested, "theory" as in a Jazz sub-forum named "Theory" must certainly mean standard music theory, but within which it looks like the people here don't seem to be discussing that theory, or even the same other theory among themselves. I've mentioned before, many who claim they use theory aren't referring to the standard theory with the peculiar definitions of note and interval...
Recall that chord tones are 1 3 5 7 9 11 13 based on stacked thirds of notes, not pitches. The quality of chords comes from changing their pitches by applying accidentals to the notes which themselves do not change because they aren't pitches, they're staff locations. This preserves the stacked thirds interval structure across various chord types.
Similarly for diatonic scales that mysteriously always have each of the seven note names, none missing, none duplicated, that is a product of the concept of interval that ensures linear scale degrees in order to make diatonic scales form straight lines without the two half-step kinks on the staff in all keys (enforcing this with the application of key signatures). That is based on the note system of sequential lines and spaces, adjusted in pitch by the key signature.
Hope this all makes sense. The standard music theory is really profoundly coherent, amazing, and intellectually beautiful.
Campellone Deluxe 16" thin-line
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