It's worth noting that the perfect fourth, when it involves the bass, is considered a dissonance in Common Practice music,* and even in the freer harmonic conte

It's worth noting that the perfect fourth, when it involves the bass, is considered a dissonance in Common Practice music,* and even in the freer harmonic contexts of twentieth century music it's a bit of a cipher: it acts as a dissonance in relatively consonant contexts and as a consonance in relatively dissonant contexts. * If you read Schoenberg's Harmonielehre, he has some pretty idiosyncratic but somewhat plausible theories about why the 6/4 chord is dissonant, involving the overtones of the bass being diatonic (say we're talking about C 6/4 and the relevant overtones being B and D) but also dissonant with regards to the nominal root of C. That the 6/3 chord is relatively consonant is explained by the fact that the overtones of the bass in that arrangement suggest a sufficiently distant key so as not to plausibly oppose the sounding fundamentals. I think the more likely explanation is the historical one, which treats the 6/4 as the reification of the frequently used motif of a suspension from the pre-dominant harmony over the dominant bass that then passes downward to the goal tones (in C major, the archetypal lines would be A -> G -> F -> E and F -> E -> D -> C over a bassline like F (or D) -> G -> G -> C). The 6/4 chord's dissonance, then, is merely a result of the fact that its appearance came to unequivocally imply a certain resolution.