AOH :: PHI&RES.TXT The relationship between resonance and phi, as shown by Fibonacci numbers.
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|  File Name      : PHI&RES.ASC      |  Online Date     : 05/22/94        |
|  Contributed by : Joel McClain     |  Dir Category    : ENERGY          |
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PHI&RES.ASC

THE RELATIONSHIP BETWEEN RESONANCE AND PHI, AS DETERMINED BY
THE FIBONACCI SERIES OF NUMERALS

Joel McClain
April 26, 1994

Because the symbol PI is indeterminate, the Egyptians used PHI in building
the pyramids.  They did this so that they could "square the circle", and
create a square base which contained the same area as a circle.

PHI simplifies the math required to square the circle, and is expressed as
a constant of 1.618.  This has come to be known as the Golden Ratio.  To
Great Pyramid", by Peter Tompkins.

For the purpose of this paper, it is sufficient to know that this ratio can
be very helpful in determining the true value in Hertz of notes on the
diatonic scale.  The standardized (1939) frequencies were accepted based
upon the sound preferences, as opposed to the PI or PHI relationship of the
notes.

In  a  previous  paper,  I  extrapolated  the  harmonics  of standardized
frequencies, proving the validity of Brown's Constant for determining
harmonic values.  However, this did not take into account the PHI constant
from the Fibonacci Series, which gives us a more natural starting point.

Further, there exists a correlation with the Fibonacci Series, which also
produces PHI, and which can be used for reference.  A Fibonacci Series is a
list of numbers, where each number is equal to the sum of the two previous
numbers.  For example, 1-2-3-5-8-13-21-34-55 is one such series.

If you divide a number by the previous number, such as 55/34, you get
1.618, the Golden Ratio.  As the numbers increase in value, the ratio gets
closer to PHI.

We know from previous study that a note has its first harmonic at the
frequency of the note times the cube root of PI, which we have named
"Brown's Constant".  The numeric value of this is 1.3313.

Let's see how we can combine this with the Golden Ratio and with the
Fibonacci Series to create a diatonic scale that is based upon nature's
laws, as opposed to men's ears:

Start with the Fibonacci string of 144-233-377, and let's assign the value
of 233 to the C note and you get,

WHOLE   FREQ    RATIO TO PHI                    HARMONIC @ FREQ
NOTE                                            TIMES BROWN'S CONSTANT

C       233     PHI @ 377/233                   310 (F)
D       263     CD RATIO = CUBE OF PHI          350 (G)
F       310     DF RATIO = SQUARE OF PHI        413 (A)

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G       350     FG RATIO = CUBE OF PHI          466 (C)
A       413     GA RATIO = SQUARE OF PHI        550 (D)
C       466     AC RATIO = CUBE OF PHI          620 (F)

Now, we have frequencies which are balanced relative to each other, as well
as based upon natural resonance.  To check your answers, relative to PI,
consider that PI = 4 DIVIDED BY THE SQUARE ROOT OF PHI, so PI = 4/1.272 or
3.1447, based upon the 377/233 ratio

As with Brown's Constant, the numbers can vary, as long as the proportions
are held constant.  In other words, if you start your scale with a value of
377 instead of 233, and observe the same ratios, your chart will be as
viable as anyone else's.

Interestingly, the Fibonacci Series was understood by the Egyptians, and
has been mentioned as a means for deriving "Magic Squares", once again,
based upon the PHI ratio and relationship.

resonance based designs, and to use the frequencies thus derived in their
experiments.

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