posted on Jan, 7 2010 @ 11:14 PM
Here this further explains the problem with the Golden Ratio:
The Golden Ratio change from A:B::B:A + B to A:B::B:A – B for Plato’s Academy fails to satisfy the algebraic solution of B squared even though it
can provide the Fibonacci number series. For example quantum chaos Professor Ian Stewart in his new about on how all of science is based on symmetry,
Why Beauty is Truth: A History of Symmetry (2007) states that the order of the symbols can be changed around in solving the quadratic equation, if
there is no concern about negative or positive values. But in music ratios, based on asymmetry, changing the order of the symbols has a direct
consequence on the meaning of the terms since C to G is 2:3 while G to C is 3:4. Kepler understood and rejected this changing of the order of infinity
because A + B is converted to A- B even though the two expressions have a different square value. Plato and Euclid relied on this concept of negative
infinity, using the quadratic zero, to justify the changing of the order, but in the Pythagorean natural overtones – the Law of Pythagoras - the
female principle comes first and one is therefore not a number. For Kepler, the method of exhaustion, derived from “Plato’s Theory of Numbers,”
(for the square root of two, the ratio of the triangle diagonal to the side as an infinite fraction series converging on the irrational, by
alternately greater and less ratios) exposes the inherent Pythagorean complimentary opposites of evolution – the male and female principles.
(“Plato's Theory of Number,” by Ivor Bulmer-Thomas, The Classical Quarterly, 1983). So for the Golden Ratio of A:B::B:A-B the result is A * A - B
equals 2, 3, 10, 24, 65, 168 while B squared equals 1, 4, 9, 25, 64, 169 creating the male-female (complimentary opposite) convergence. (“Kepler’s
Celestial Music,” in Studies in Musical Science in the Late Renaissance, 1975, Warburg Institute, Oxford University).
The Golden Ratio change does satisfy the switch to irrational geometric-based fractions which are not ratios based on the asymmetric Law of Pythagoras
overtone harmonics (again with C to G as 2:3 and G to C as 3:4 in violation of the commutative principle of symmetry). So the changing of the order of
the letter symbols of the Golden Ratio from A:B::B:A + B to A:B::B:A - B is in order to convert the Tetrad to the Freemasonic symmetric geometric mean
of the Golden Ratio. The Fibonacci Number Series 1,1,2,3,5,8 with 8:5 as the Minor Sixth and 5:4 as the Major Third is the “vanishing mediator”
from music theory so that now the Major Third or 5:4 equals the geometric mean as the cube root of two, from Archytas’ “doubling the cube” proof
that creates similar triangles. For example H.E. Huntley’s book The Divine Proportion (Dover) analyzes music ratios for the Golden Ratio but
erroneously considers 8:5 to be the Major Sixth, not the Minor Sixth.