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I'm working through study 16, and it clearly focuses on changing chords. But other studies in the opus are very different. What are they "characteristic" of?

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Just as the etudes Op. 10 and 25 by Chopin - although probably a bit less difficult - these studies Op. 109 aim at the exercise of a broad range of technical skills. From a quick look, I would say that, for example, No. 2, 10, and 18 are good for exercises in scales, while some others (No. 4 and 6) may help you play more precise staccati. No. 1 certainly aims at precise broken chords in the right hand, while you may find No. 8 helpful if you want to improve the uniformity of fast notes with the hands changing.

I did not do a complete analysis of these pieces, but I think there is no such thing as an overall characteristic in a technical sense, except maybe that they are all useful if you want to improve your technique.

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Remember that though Burgmüller was German by birth, he lived in France (Paris) from age 26, so these will have been published in French. Curiously, the "Characteristic" title only seems to appear in English -- French Wikipedia for example just calls them "studies". But anyway, this suggests that each piece has some distinct character, rather than just being a "study in thirds" or whatever, and each study has a French title: this one is La séparation, which perhaps means two people splitting up, rather than the hands being further apart. You can read all the French titles on the IMSLP page http://imslp.org/wiki/18_%C3%89tudes,Op.109(Burgm%C3%BCller,_Friedrich) and if you look at the Schirmer edition, it includes English translations of the titles.

For reference, I learned much of this from Wikipedia: https://en.wikipedia.org/wiki/Johann_Friedrich_Franz_Burgm%C3%BCller

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