posted on Apr, 25 2010 @ 02:01 AM
reply to post by bigfatfurrytexan
Yeah that's a fascinating analysis of the Comma of Pythagoras. It's important to emphasize again the difference between the "divide and average"
measurement from China and Babylon instead of the "complementary opposites" which are noncommutative.
So for "divide and average" the measurement is to SQUARE the fifths and SQUARE the octaves. So you have
(3 / 2) to the 12th = 129.746338 in ratio to
(2 / 1) to the 7th = 128
So the octave is being squared. Logarithmic measurements comes from this concept of squaring the octave -- but it's based on measuring time as
distance. So it was published in the 1960s that Newton got his inverse square law straight from Pythagoras -- in the Royal Society of London. A
string stretched to 4 times it length using a weight of 4 times makes a frequency twice as high -- and THIS is why the octave or how the octave is
But in the Law of Pythagoras the wavelength is the INVERSE of the frequency -- not the inverse SQUARE as distance. In other words a string length
twice as long, instead of 4 times as long, makes half the frequency -- an octave lower. It's just based on length -- not on weight. So the octave
is DOUBLED not SQUARED. The Fifth is measured as frequency OVERTONE as 2/3 -- not 3/2.
This is a very subtle difference between overtone of the fifth as 2/3 frequency versus UNDERTONE of the fifth as wavelength -- 3/2. In the ORTHODOX
PYTHAGOREAN tradition the overtone was used not the undertone -- and it was JUST the Tetrad -- the fourth and fifth ratios -- as 1:2:3:4. But in the
Western interpretation of Pythagoras -- from Philolaus and Archytas -- this was changed -- so that the octave was doubled as being SQUARED.
I call this the "bait and switch" tactic -- so if you SQUARE the fifth you get 9/4 -- as the first squared fifth -- and then this is HALVED as 9/8.
But 9/4 was not allowed in Orthodox Pythagorean tradition NOR WAS 9/8. I document this using academic references in my blogbook DEEP DISHARMONY. My
research was VETTED by math professor Luigi Borzacchini and math professor Joe Mazur.
I know I'm way off topic but this also includes the ratio 8:5 -- which was the minor sixth as the GOLDEN RATIO -- again it's NOT ALLOWED under
Pythagorean Orthodox tradition -- but, again, this does not rule out the creation of spirals through the complementary opposite harmonics.
"Any who doubt that the musical ratios are all of greater inequality, i.e., that the antecedent or first term in each is greater than the consequent
or second term, should consult Archytas DK 47 B 2. This Fragment requires that the ratios be of this form if the assertions about the three means
[arithmetic, harmonic and geometric] are to be true. Accordingly, the ratios assigned to the octave, fifth, fourth and minor sixth, must be 2:1, 3:2,
4:3 and 8:5, and not 1:2, 2:3, 3:4 and 5:8, respectively, as Mosshammer and others would have them. Indeed, there is early proof deriving from the
Pythagorean school that intervals, such as the fifths, which are represented by superparticular ratios cannot be partitioned into any number of equal
subintervals because the terms of these ratios admit no number of geometric means. Consider now the question of the status of the ratio (8:5) in the
Pythagorean harmonic science that dates from the late fifth century B.C. to the time of Apollodorus. One should not expect that this ratio was
recognized as melodic by every school of Pythagorean musical theory. For example those who sought to derive all the musical ratios from the Tetrad of
the decad by compounding and dividing the ratios of the primary and most familiar intervals, the concords of the octave, fifth and fourth, would find
the minor sixth unascertainable. There is reason to believe that these were supplied by Archytas in the early fourth century B.C."