There are important considerations to keep in mind when looking at intervals in scales. For example, the 2nd of some scales is a whole tone, but in other scales, it's only a semitone. This is the case for the 2nd of D major and E major, respectively. D to E is a whole tone and E to half is a semitone. For the sake of greater precision in describing intervals and their roles in specific scales with respect to specific degrees, we will need a new set of concepts and terminology:
Perfect - The unison, 4th, 5th and 8th. Suppose we have a C that is lower than a G. We can invert this interval, and the original perfect fifth becomes a perfect fourth. But why is this? We determine the number of an interval's inversion by subtracting it from 9. Therefore, if it's a 5th, we subtract 5 from 9 and we get 4. Therefore, it becomse a 4th (Prout, p. 9). Since unison relations lack a higher or a lower note, they technically cannot be inverted, but we do refer to a unison as inverted when one of its two notes is placed on a higher or lower octave (Prout, p. 9). Likewise, if we turn a relation in which a relative higher or lower note does obtain to a unison, we do technically refer to it as an inversion(Prout, p. 9). Prout summarizes further: "Perfect intervals remain perfect when inverted; major intervals become minor, and minor major; augmented intervals become diminished, and diminished augmented"(Prout, p. 9).
But why is this?
"The reason of the rule just given will become clear to the student if he observes that the inversion of any simple interval is the difference between that interval and an octave. Thus a major 3rd, C to E, and its inversion, a minor 6th, E to C, will together make an octave, either C to C, or E to E, according to the note of the 3rd of which the position is changed. A third of any kind taken from an octave must leave a sixth; and if a larger (major) third be taken out, a smaller (minor) sixth will be left; and conversely, if a smaller (minor) third be taken out from the octave, a larger (major) sixth will be left. Evidently the same reasoning will apply to augmented and diminished intervals"(Prout, pp. 9-10).
Because a compound interval is larger than an octave, we do not merely change its octave by raising or lowering it to produce an inversion. No such inversion will be produced. Instead, we raise or lower by two octaves one of the notes. We can also, of course, produce an identical effect by raising one note an octave and lowering its counterpart an octave (Prout, p. 10).
Major - The 2nd, 3rd, 6th, 7th.
Minor - We turn a major interval into a minor by raising the lower by a chromatic semitone or lowering the upper by a chromatic semitone. As Prout points out, in the case of C and E, we can turn this interval into a minor by turning the C into a C# or turning the E into an Eb. It's at this point that we begin to see how important precise notation is. If we write C# as Db or Eb as D#, we have the enharmonic equivalent, but it becomes a special kind of 2nd rather than a 3rd, since either such alteration will make both notes adjacent to one another on the staff (Prout, p. 8).
Augmented - The augmented interval is one which is a chromatic semitone larger than either a major or perfect interval. This involves raising the upper note or lowering the lower. As Prout notes, since C to F is a perfect 4th, making the F an F sharp or the C a C flat makes the interval an augmentedd fourth. We don't, however, used augmented 3rds or 7ths. We occasionally see augmented 5ths, but most of our augmented intervals are 2nds, 4ths and 6ths.
Diminished - This is when our interval is a chromatic semitone lower than a perfect or minor interval. A diminished 6th, for example, in the case of G to E, would turn the G to E to G to Ebb (this would be the enharmonic equivalent of a normal D).
Finally, we speak of an interval as inverted when we cause both pitches in the interval under consideration to be switched an octave. Either the lower tone becomes raised an octave or the upper tone becomes lowered an octave.
Prout, Ebenezer. (1889). Harmony, its Theory and Practice. London: Augener, LTD.