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In this arrangement it seems the creator, Mark Hanson, sometimes plays the alternating bass with the melody note (in a pinch) and sometimes, randomly it seems, he plays the bass note then the melody using eight notes as highlighted in red below. 
My question is, does he choose the pinch versus bass then melody arbitrarily or is there another reason?
Traditionally, alternating bass is played in quarter notes, as in the example in your question. So, if the melody note happens to fall on the beat, it will be played simultaneously with the bass note, otherwise it will be played in between bass notes. It's as simple as that.
