7. Advanced Intervals

There are occasions (such as when using odd scales like the Persian) where it might be correct to call a distance of 5 half-steps (normally a "perfect fourth") an "augmented 3rd", and by the same token 2 half-steps (Maj2) may wind up a "diminished 3rd". Seconds, thirds, sixths, and sevenths can all be diminished/minor/major/augmented if needed. Fourths and fifth are referred to as flat/perfect/sharp, but it's the same kinda deal.

A "third" can be thought of as a blanket term for the intervals between any notes 3 letters apart. The following can all be described as "thirds".
(Gb->Bb), (Gb->B), (Gb->B#), (G->Bb), (G->B), (G->B#), (G#->Bb), (G#->B), (G#->B#). They vary from 2 to 6 half-steps. I'm not positive, but I assume Gb->B# would be called a "sharp augmented third". This could potentially come up if your scale looked like this (F-Gb-A-B#-C#-D-E-F) or (1-m2-M3-##4-#5-M6-M7). Kinda out there, but it exists. If you have seven notes in your scale (as is the usual case) you want to have one representative from each letter group. It helps with the chord-building mentality of leap-frogging over letters.

Let's look at the G#sus2 chord from E Persian. (G#-Bb-D#)
There's a root, a note 2 half-steps up, and a note 7 half-steps up. Normally we would say (1-M2-5). It's easiest to name the chord "sus2".

If you want to get technical, within the context of this scale, that chord is really (1-dim3-5). You could call it "G#mb3" if you really wanted to, but no sense in confusing the musicians you're working with when you don't have to.

If we were in a scale with the notes (E-F-G#-A#-B-C-D#-E), the same chord would have the notes (G#-A#-D#) and you would definitely want to refer to the "A#" as a Maj2.