Interval complexity is a direct function of the distance between the lowest note of the interval as compared to the highest note of the interval with the closest note in the harmonic series of the lowest note of the interval. (Phew!)
Let me explain:
Poor Man's Harmonic Series: For the sake of this explanation, let's pretend the harmonic series represents these relationships. It does not, but this linear representation is roughly based on the harmonic series:
Octave, fifth, major third, minor 7th, major second, flat fifth, minor sixth, major seventh, minor 9th, minor third.
So if the root note is C, its sympathetic vibrations along the harmonic series are as written above. (Our musical system is not perfectly mapped to the harmonic series, so this is just an approximation.)
As you can see, the harmonic series firmly supports the simplest intervals (octave, fifth and major third), as well as those a whole step up or down from the fundamental (minor 7th, major second). As we climb further into the stratosphere we encounter dissonances not well supported within the fundamental, which would represent the more complex intervals of the flat fifth, minor sixth, major seventh and minor ninth.
Harmonics also imply their own inversions: Note that each interval also implies its own inversion, though, meaning that the perfect fourth, although not even present in this particular example, is well enough supported by the perfect fifth, the two intervals being so closely related.
Higher harmonics are also weaker: Also note that the higher the harmonic is, the more weak it is. So the fundamental is alot less related to the minor 9th than it is to the perfect fifth.
The 12-tone musical system does not relate perfectly to the harmonic series: many notes in the harmonic series are not too closely related to their closest relative in the actual musical scale. For instance, the flat 5th in the harmonic series is 49 cents more flat than the one in the 12-tone scale, a significant difference. This difference has meant that many harmonies which would be perfectly valid in a microtonal system are completely unusable in the 12-tone system we most commonly use.