What are the main ideas incorporated into the New Complexity movement (contemporary classical music, late 20th century)? What kind of notation, musical techniques, or theoretical ideas would composers incorporate into their music during this period?

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    You're going to need to be waaaaaay more specific. "Complexity movement"? In what genre? In what way is it becoming more complex? Can you provide some examples to show us examples of the new complexity? – user45266 Oct 16 '19 at 18:29
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    @user45266 New Complexity refers to a movement in contemporary classical music composition, sorta late 20th century. No need to be more specific. It is actually so specific that you -- and probably most people -- have never heard of it. The music of Brian Ferneyhough is a prime example. Check it out. It's... interesting. – ibonyun Oct 17 '19 at 16:16

There is a good summary in the Wikipedia article:

Though often atonal, highly abstract, and dissonant in sound, New Complexity music is most readily characterized by the use of techniques which require complex musical notation. This includes extended techniques, complex and often unstable textures, microtonality, highly disjunct melodic contour, complex layered rhythms, abrupt changes in texture, and so on. It is also characterized, in contrast to the music of the immediate post–World War II serialists, by the frequent reliance of its composers on poetic conceptions, very often implied in the titles of individual works and work-cycles.
Ferneyhough's Etudes Transcendentales, a song cycle for soprano and chamber ensemble, demonstrates many traits found in New Complexity music. In addition to being generally difficult to learn and perform, the pitch vocabulary makes heavy use of microtones—in this case, equal-tempered quarter tones. It also contains many tuplets of unusual ratios which are nested in multiple layers. Rapid changes, sometimes from note to note, happen in dynamics, articulation, and playing technique, including techniques such as multiphonics on the oboe, glottal stops for the voice, and key-clicking for the flute. According to Richard Toop, the rhythm for the oboe part in the first song is almost totally determined by a strict system with five stages of complexity, each governed by its own cycle of numbers.

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