I wonder how well it is known that the actually pitch for the E from a G major scale isn't the same pitch as the E string would be if the violin is tuned in perfect fifths.
When perfect fifths (C,G,D,A and E) are used the distance between the C and the E is nine-eigths squared, or rather two major seconds (the E is the higher part of the ratio the C the lower).
This is eighty-one sixty-fourths.
When, as is the case with a major scale based on the overtone series, the ratio is not based on fifths but on being a major third the ratio is five fourths or rather eighty sixty-fourths
81/64 perfect fifths, 80/64 major third
Thus the one with perfect fifths is higher
The same thing happens with the A string for a C major scale.
I was just playing on my violin improvising and noticed that I was above the open string E and wondered whether I had overcompensated and thus would be on the scale (well. Twice the "distance" between the two ratios above G)
Regardless, I think that's very important information. In the time of Plato or Josquin Desprez this was part of how they taught math I think (using actual occurences of natural harmony rather than numbers which are meant to add up to some quantity that is an illusion because it's just about possession).
Note the Ratio denotes frequency of the vibrations
In a major third, five fourths means the upper note has vibrations during the same time that the lower note has four.
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