Tuning+&+Transposition+(W2D1)

__The tension between tuning and transposition__. We know tones have frequencies, but "in the beginning", there were only string lengths (which, luckily, were more-or-less proportional, inversely, to frequency). So the Greeks experimented with string lengths and these experiments, being systematic, involved simple ratios. We obtained, through such experiments, the intervals that we know today and still define the same way: the octave (p8), 2:1; the perfect 5th (p5), 3:2; the perfect 4th (p4), 4:3, and so on. However, the way to cash out the "and so on" turns out to be a contentious bit of business. One could continue to divide our string and get a major 3rd (M3) 5:4, and a minor 3rd (M3) 6:5, //or// one could engage in "the transposition game", which, as we learn from Nettl (1958) is a game that is extremely prevalent in musics all around the world and throughout history, and not at all restricted to the Greeks. But as it turned out, the Greeks liked to play it too, and if you play the transposition game with that very nice interval of the perfect 5th and use it as a "generator" to obtain new pitch material, you end up with what is known as Pythagorean Tuning (PT).

"The heartbreak of Pythagoras." No number of p5's will ever equal any number of p8's. But 12 of 'em come damn close ... within 24¢ (a "Pythagorean comma" ... if Pythagoras were around today we might call it the "Pythagorean kludge"). Thus a p5 is 702¢ (12 x 2 = 24) instead of 700¢ which is what it would be in a perfect world where 12 p5s exactly equaled 7 octaves (think about this ... we haven't even mentioned "Equal Temperament" yet).