While reading through Bach's Goldberg Variations, I noticed that at least the first few bars seem to be mathematically impossible.

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How would one play the first few measures of the bass when there is a dotted half, half, and quarter note there? They appear as if they should be played one after the other but it doesn't work out mathematically.

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    There are three separate voices in the left hand; one voice enters on beat 2, a second on beat 2, and a third on beat 3. – Richard Feb 26 '18 at 2:26
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    If anything, there are too few notes and rests in that bar: the upper voice should have another quarter rest so that the bar adds up to 9 quarters and not 8. But the editor considered aesthetics more important in this bar than mathematical accuracy. – Kilian Foth Feb 26 '18 at 7:46

The dotted half, half, and quarter notes in the left hand are in three different voices. You can identify this for example by the opposite stem directions, as well as the placements in comparison with the right hand stave.

Labelling the voices 1,2,3 from top to bottom, this is what happens in the three voices in each measure:

  1. quarter rest, quarter rest, quarter note;
  2. quarter rest, half note;
  3. dotted half note.

The first quarter rest is shared between voices 1 and 2.

| improve this answer | |
  • 2
    Virtual voices? – ex nihilo Feb 26 '18 at 2:42
  • @DavidBowling: nobody is opening their mouth... – Remy Feb 26 '18 at 3:10

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