Golden Section

Dante Rosati wrote,

>All the references to Phi or the golden section in music (that I have
seen)
>have taken 1.618... as the musical ratio that manifests phi as an
interval.
>Translated directly into cents, it is ~833c. But the "golden section"
is
>also called "division into extreme and mean ratios", which, when
sectioning
>a line (for example) means "the ratio of the whole to the larger part
is
>equal to the ratio of the larger part to the smaller", which divides
the
>line at 1/phi or ~.618..

Which is ~-833c.

>But if I compute a division of the octave according to the above
dictum, I
>get the following:

>The ratio of the whole (2/1)
>to the larger part (call it "x")
>is equal to the ratio of the larger part (x)
>to the smaller part (2/x)

>or:

>2/x=x/(2/x)

>this reduces to:

>x^2=4/x
>x^3=4
>x=cube root of 4
>also
>x=(cube root of 2)^2
>x=~1.58740105197

>This is 800c, or >exactly an equal tempered minor sixth<! Its like
dividing
>the octave in 3tet, and taking two parts (cube root of 2)^2.

>The only thing I can make of this is that 1.618... is the linear golden
>section, which applies to euclidian space and 1.587.. is some kind of
log
>golden section, which applies to pitch space.

>Is this correct, or am I making a dumb mistake somewhere?

You are making a mistake in that the "ratio" of two interval sizes is
not a simple division -- in fact, a simple division is the "difference"
of two interval sizes. To divide the octave logarithmically into the
golden section, use 1200/phi=741.6408 cents.