Here we go again.
Wire gauge doubles every THREE full integer wire sizes.
22Ga is then 19 Ga at three strands, and just bigger than 16 Ga
seven strands.
At six strands, it would be 16 Gauge equivalent.
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Omigod! You don't even know how to use a wire chart, or how to use
simple arithmetic. Sorry, I didn't know.
Here, let me help you. Starting with a piece of #22, to double its
cross sectional area you would have to go to #19, as you say, and then
to double _that_ area you would have to go to #16, again as you say.
However, doubling the area twice only gets you to 4 times the area of a
single strand of #22, so to get to 8 times the area you'd have to
increase the gauge by another three sizes, to #13.
But, seven strands of #22 doesn't enclose the same area as eight strands
of #22, they enclose seven times the area. Now, looking at the entry in
the wire chart for the cross sectional area of #22 and then finding the
entry which most closely yields seven times that area results in #14 AWG
being found. Got it?