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Harmony in the contemporary period (20th-21st centuries) is different from the common practice period. Composers sought to create new sounds, breaking away from the tonal system of the past 300 years. Harmony became an easy target for composers given that so many intervals chord types did not fit the tonal systems of the past. Builing chords based on 2nds and 4ths became common. Various scales led to the use of chords considered "difficult" in the common practice period. Whole-tone scales led to more broad use of augmented chords. The octatonic scales led to the use of diminished triads, sevenths and ninths in ways that were not possible in a tonal setting. The use of multiple chords, and even keys, at the same time became common as well.
Note that many of these examples do not use key signatures. Because of the extended use of chromatic notes, it became easier in many cases to just add alterations, even when a tonal center or mode is implied.
Harmony built on seconds - These are often referred to as cluster chords. They are just stacked seconds. These are commonly found in the music of Henry Cowell, Charles Ives, and other early 20th century composers.
(Audio examples to come)
Harmony built on 4ths (and by extension, 5ths). Quartal music is built on both perfect 4ths and tri-tones. It can be used in both melody and harmony, as shown below.
Polychords are when two different chords are sounded at the same time, such as E major and B-flat major. A famous example is the "Petrushka chord," which is an F# major chord over a C-major chord. Other possibilities can occur as the examples below show.
Polytonality occurs when two or more tonal centers are played at the same. For instance, a Clarinet playing in concert G and a trunpet playing in concert D-flat (see Ives).