Non-tertian harmony
Looking for new harmonic vocabulary, composers in the 20th century explored harmonies based on intervals other than thirds.  This non-tertian harmony includes quartal (based on 4ths), quintal (fifths), and secondal (2nds).

Some composers went so far as to construct vertical sonorities in extremely dense clusters or sound masses.

Below are some examples of these types of constuctions.
Quartal Harmony

Quintal Harmony

Secondal Harmony
If the selection below is analyzed, it becomes apparent that its secondal harmony, as is often the case, is simply quartal / quintal harmony with inverted voicing.  The example below contains exactly the same chords as in the example immediately above, with one voice displaced by an octave.

Cluster Harmony / Sound Mass
The effect of a cluster or sound mass is the minimization of individual pitches in favor of a dense timbre.