Ólafur Arnalds mixes strings and piano with loops and edgy beats crossing-over from classical to pop.
What does the writer mean by saying edgy beats? Does the writer mean it's not exactly classical, nor pop?
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In this instance edgy is an adjective that is modifying the noun beats.
In this context edgy means
having a bold, provocative, or unconventional quality
and is somewhat (although not completely) synonymous with cool as in being trendy.
So I agree with your assessment: the writer is trying to convey that the beats (or rhythm) in the music are giving the classical music sounds of strings and piano a pop feel, by having an unconventional (in the context of classical music) quality.
‘Edgy’ has been in use since 1976 to describe something ‘that challenges received ideas or prevailing aesthetic sensibilities; at the forefront of a trend.’ (OED) I fear my musical knowledge is insufficient to know exactly what is meant here, but the writer may mean little more than that the beats are unusual, a bit different.