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Here's sheet music for chopin's 2nd ballade. On page 6, in the 2nd measure in the 4th line of music, there's a D# and a D natural, with two lines coming out of them and connecting to the double bar (for the sixteenth notes). I was wondering how to play this bar. I think you have to play them at the same time, but I'm not sure.

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Yes, that's a "play both notes". See http://musescore.org/node/14449 for a note on the standard from the US Music Publisher's Association.

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I'd like to supplement the reasoning behind this:

  • Note heads take space, and chords are typically sharing the stem or in absence of one are placed above each other.
  • If the note heads would overlap (due to very small interval between them) the first remedy is, to keep the note stem common and imagine one note head pushed to the other side of the stem (see last stem in the first 16th group). In case of whole notes lacking a stem (multi-voice) they would be placed beneath each other and the spacing has to be expanded to make clear, which notes belong together. (For an example see this question.)
  • If there would still be a collision (typically by an accidental, which would overlap with the stem or one/both note heads), the stem is split.

A special case (see this question) for split stem is, when the same note head needs to be notated twice, but with different accidentals. (All alternatives look even more weird.)

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It's funny I saw the notation and recognized the piece right away. I can't play it very fast, but I recognized it!

Yes, it means to play both at the same time, but that's not all you want to do with this passage. If you notice there's a phrase in there that resides on the top notes. You want to balance the right hand so the top and bottom notes sound a bit louder than inner notes.

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