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Here's an excerpt from the piece Chromatic Invention from Bartok's Mikrokosmos, volume 3, page 39. I don't get why at the end of the lower line the bass clef indication is repeated. Is this "for security" like it's sometimes done with accidentals? But why do that here? This line is not ambiguous at all, is it?

Mikrokosmos excerpt

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And why do the beams get carried across at beginning and end of the line shown? Phrase marks underlined? –  Tim Jun 1 at 9:28
    
I'm not sure I understand your second question but the answer to the first is that this line of music is surrounded by other lines in the original score. –  user10960 Jun 1 at 9:43
    
I merely thought that the longer beams may be emphasising the fact that certain dots were all together in a phrase. –  Tim Jun 1 at 11:41
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I'm curious to know if this piece has a lot of change of clefs involved. If so, then perhaps inserting the bass clef again to show that there has been no change of clef would make sense. just a guess since I don't know the piece –  Sazid_violin Jun 1 at 13:18
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Yeah, I think that clef is almost certainly a mistake, possibly the engraver saw the treble clef two systems above and forgot it had already been cancelled. It's also possible (more likely) that Bartok himself did that in the autograph sent to the engraver, and they just engraved whatever he sent them. In regards to @Tim's question, the beams indicate eighth-note groupings as always, it's just that these particular beams are being continued from the preceding line. Beams across barlines are quite common in Bartok's music. –  Pat Muchmore Jun 1 at 13:44

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up vote 11 down vote accepted

I found a motivic analysis of this piece, along with score, on youtube (the piece starts at 1:11, but watch the analysis before that as well):

For the question about beaming: This piece is built, like a Bach Invention, from the development of a single melodic motif. The beaming is used to make that motif clear, even when it gets rhythmically misaligned relative to the barlines, or split up by rests.

The bass clef is more puzzling. It doesn't seem necessary, and it may well be a typo, as others suggested. I note that it is not present in the analysis score that I linked above. However, its worth pointing out that this is the first time in the piece that the left hand goes that low in the bass, so it might be a friendly reminder that we aren't switching back to the treble clef again, and that the piece really does go that low. It's hard to say.

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