I must be making a mistake.....

(sorry is this is a repeat, my first send came back):

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..

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?

dante

I have stumbled over some fairly interesting properties of ~ 833 cents some
time ago, and I had no idea that it has anything to do with the "golden
section". Would be interested in hearing more about it; are there any
references?

Heinz Bohlen

On Sat, 20 Mar 1999, Rosati wrote:
> 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..
>
> 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)
[snip]

This confuses the ratio notation for intervals with interval size, which
is usually measured logarithmically. (Brian McLaren, whom we've brought
up recently, once did this in a discussion of "self-similar" scales,
which would have resulted in all ETs being self-similar--a usage which
would have diluted the term to near meaninglessness, IMO.)

Try it this way:

The ratio of the whole (1 octave)
to the larger part (call it "x")
is equal to the ratio of the larger part (x)
to the smaller part (octave - x)

i.e.
octave/x = x/(octave - x)

This gives

octave * (octave - x) = x ^ 2

, or

x ^ 2 + octave * x - octave ^ 2 = 0

Using the quadratic formula, we get

x = (- octave +/- sqrt(octave ^ 2 + 4 * octave ^ 2) / 2

which reduces to

x = octave * (-1 +/- sqrt(5)) / 2

or about 741.64 cents.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

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