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If the first D note (in the 1st beam) is flat, does that mean the next D note (in the 2nd beam) is also flat? (see photo)

enter image description here

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Yes, it is flat because it's in the same bar, and the same octave.

A D on a different octave, on the other hand, would not be flattened by that initial flat.

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An accidental in any bar will always and only affect the pitch concerned. So, here, any other D note in that bar only at that same pitch will be affected by the accidental (here, a flat sign). The very next barline will cancel any accidentals that occurred in the previous bar.

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