Brandon- Last time I missed a note, my dad almost drew blood.
To "draw blood" means to cause blood to flow, even if only to a small degree.
"When you bang your knee on a chair it will usually hurt, but it doesn't often draw blood".
In your example, I assume that it's an exaggeration and he's just saying that his father got very angry. However, if dad is inclined to whack his son across the knuckles with a ruler whenever he misses a note then maybe it should be taken literally.
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