Any system that doesn’t fit the tension and release cycle that is pervasive in functional harmony cadences is called non-functional harmony and it occurs when no chord “wants” to specially resolve to the next one.
In fact, such chords are avoided in order to undermine a tonic association. This type of approach to building a chord succession is commonly associated to the impressionist movement, but it can also be associated to modal jazz. The main point of chord succession is to create contrasts that can be more or less pronounced.
This involves making new chord combinations like quartal chord formation and/or using chords with more extensions that introduce ambiguity regarding their function because of a growing rootless feel; use of modes and exotic scales; or parallel chord motion; and the extreme use of chromaticism, or among other elements that undermine the sense of a tonal center.
More often than not, the goal is to evoke a mood, feeling, atmosphere, scene or a sound-landscape that should be appreciated for each moment that it creates and thus without a sense of departure or arrival:
As you see, non-functional chord successions can often be the result of voice-leading, chord transformations or the use of chromatic mediants. That said, those chords are the result of a step-wise motion in the various chord voices, that more often than not is chromatic and in various directions. Although chords built in thirds (triads, tetrads and their extensions) are the ones commonly used in tonal harmony, in the case of non-functional harmonic succession they are used to briefly allude to other tonalities.
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