Analyze the 1. Improvisation upon peasant's songs by Béla Bártok 
http://javanese.imslp.info/files/imglnks/usimg/5/58/IMSLP42556-PMLP04743-Bartok-Sz074ue.pdf (1st edition by UE, a jewel on Petrucci!!!!!)
carefully and apply the results of your analysis on a folk song of your liking.
I chose a Bolivian tune (called Akwi Akwi).

BartokBoliviana

One of my earliest compositions (written – of course – before I went to Ligeti), second half revised on three days in August 2011, finally I found the time to make a recording

Allemande

The small sister of Romantic Landscape 1. Composed and finished 1992/93 (before I went to Japan, and now it's up to you to speculate…) Even performed several times. The similarities and the huge distance between this and Romantic Landscape should be clear. Nevertheless, I think this small piece could be the starter for a series of many "Shortpieces" (and there exist sketches for about a dozen of them)… 

It may not be the most original idea to illustrate this piece with the best-known picture of Hokusai, but it was exactly this picture – and the experience of the sea at Kamakura in summer 1991 – that gave the inspiration for this piece. On closer examination, Hokusai's wave has a lot to do with fractals, and the piece also is a kind of miniature fractal – we discussed fractals a lot in the Ligeti class. 

CompositionStudy

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