Quartal chords are used in so-called "modal" jazz. (I
realise that's a contentious concept...)
The idea - as I understand it - is that tertian chords have
too clear an identity, and remind us too much of
functional harmony. Any tertian chord we hear tends to
suggest a context, a role in a sequence. And modal jazz
is not about chord sequences.
Eg, in Miles Davis "So What", the harmonies in the head
are (mostly) quartal: Em11 to Dm11, each voiced R-4-
b7-b3-5. That's all 4ths bar the M3 on top - which
actually identifies the chord(s).
A fully quartal chord is impossible to clearly identify,
because (a) there is no clear root, and (b) the chord
language we have is all based on tertian conventions.
Eg, if you have a chord voiced E-A-D-G, what would
you call it? It could be quite a few things, including
partial chords. (Normally, context would determine the
answer.)
Another example from Kind of Blue is the piano chord
played over the intro to "All Blues" - a rootless 6/9
chord. The mode is G mixolydian, but the piano
doesn't pay a G7 chord. It (plays something like) D-E-
A, while the horns and bass spell out riffs from the
scale. So a "G7 sound" kind of emerges from the overall
mix of notes.
This is the idea - again as I see it: Modal jazz is melodic
in emphasis. Harmony (such as it is) is static. Any
chords that are played are more like random bunches
of notes from the scale than identifiable chord forms.
The only non-random element is the avoidance of 3rds
as much as possible. So chords can use 2nds and 5ths
as well as 4ths.
A typical phrygian modal chord is the susb9, voiced
(eg) 1-b9-4-5-b7 - quite a lot of 3rds in there, but in a
deliberately ambiguous mix.
I'm not sure how (or if) quartal chords can be applied
in functional harmony (key-based songs) - simply
because they don't have clear functions.
(OK, now those more knowledgeable than me can take