Re: "Here`s a history lesson:
http://www-math.cudenver.edu/~jstarret/tuninghist.html
But I`ll just tell you what you need to know.
"OK...lets say A=440, as in most cases.
On the violin:
If we`re playing a song in A-flat-major, with four flats including A(flat), A(flat)=422.4.
If we`re playing a song in A-major, with three sharps including G(sharp), G(sharp)=412.5.
On the piano:
A(flat) is a semitone lower than A, so A(flat)=440/2^(1/12)≈415.3=G(sharp).
So, on the violin, A-flat is a bit lower than G-sharp."
The web site you reference can no longer be found. I can`t check your tempered tuning (piano) arithmetic just now because I don`t have a calculator with me, and I can`t comment on your "violin Ab" because I don`t know how you derive it--more about that momentarily--but I can see clearly how you get your "violin G#", and I`m afraid you are in error here.
If A is 220, then E in both the Pythagorean and just systems will be 330, 3/2 times A. Major thirds in just tuning are 5:4, whereas major thirds in Pythagorean tuning are 81:64, which would make the just G# 412.5 and the Pythagorean G# approximately 418. Clearly you are assuming just tuning for the violin, not Pythagorean tuning for the violin. There is a chromatic Pythagorean scale (you simply continue to go along by 3:2 perfect fifths), but there is no chromatic just scale, which is why I can`t determine how you derive your "violin Ab" from A 440 (the Pythagorean Ab would be approximately 411). The just scale is strictly diatonic, and even the diatonic just scale involves anomalous intervals, including the infamous "wolf fifth" and one 32:27 minor third (as well as two kinds of major seconds, 9:8 and 10:9 major seconds).
You say, "So, on the violin, A-flat is a bit lower than G-sharp." This is true, but you give G# as 412.5 (the just G#), and Ab as 422.4 which would make Ab HIGHER than G#, not lower!
In fact, it was formerly typically asserted that strings play in Pythagorean (NOT just) tuning. Eventually someone actually tested this assertion scientifically and discovered that it is not true. Strings do not play in ANY codified tuning system. Rather, a string player will exaggerate his pitch in the direction of resolution. The reason a violin G# is higher than a violin Ab is that the G# will usually resolve upward, and the Ab will usually resolve downward.