The Wikipedia article is remarkably short for such a weighty and historically important concept, though, isn't it? This might be because there are a few problems with the whole idea. The two most significant ones are, first, that it is based on a misunderstanding of musical structure. As every harmony text informs us, the development of every new harmony in music history came about as a result of voice-leading! The harmonic series, though beloved of some theorists, actually had little to do with it. The third came about through a descent from the fifth to the tonic. The seventh in diatonic harmony is NOT an accepted dissonance, but a voice moving to the third of the tonic. Voice-leading demands that it resolve. The augmented sixth chords came about by means of a chromatic progression within a Phrygian cadence. Here is the relevant section from Aldwell and Schachter's Harmony and Voice-Leading:
You cannot understand harmony without understanding voice-leading, hence the title of the text. Using the harmonic series as a justification for more and more harmonic richness is just bizarre. Harmonic richness comes from voice-leading, that is, the conduct of the separate voices is what both creates and resolves dissonances. Trying to connect this with the harmonic series is just theoretical mumbo-jumbo. What was really going on in the early decades of the 20th century can be seen from a lot of different perspectives. But the one perspective we need to be particularly wary of is the one sold to us by the composers arguing for it: chromatic harmony as "emancipation of the dissonance." As I have talked about here, much of the aesthetic ideology of modernism is really disguised marketing. Such is the case here.
This brings us to the second problem with this concept: it is only a metaphor and a flawed one at that! Chromatic harmonies are not "emancipated" from anything! Emancipation is the granting of rights to previously disenfranchised groups of people. It is a moral concept having nothing to do with music. But it was a useful metaphor as it made dissonance seem moral, healthy and, most of all, historically inevitable. Once all the intervals were emancipated, around the time of the First World War, then the pinnacle of music history had been reached--as far as harmony is concerned. So what has happened since?
Alas, history has not cooperated with the emancipators. If you view harmony from the point of view of emancipating all the intervals, like so many slaves of the lords of tonality, then there really is nowhere else to go. Sure, for a while composers tried serializing everything they could think of, but this didn't last long and audiences didn't care much for it. So now we are in a post-historical phase where, surprise, surprise, consonance has been rediscovered.
Let's end with a couple of pieces. First, Structures for two pianos by Pierre Boulez, an example of total serialization from 1952-61:
And second, the de-emancipated return to consonance of the music of Enojuhani Rautavaara. This is the Symphony No. 8 from 1999:
History moving backwards? Or just a bad metaphor?