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Author Topic: The Golden Mean/Ratio  (Read 3295 times)

neilep

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The Golden Mean/Ratio
« on: 07/09/2006 20:55:51 »


I have just been watching a DVD of Jon Anderson (he's the lead singer of my all time favourite group 'YES' )...anyway...he is well into this thing called the Golden Mean or the Golden Ratio.

Now, I know there is a fair bit on the net about it but I wonder if someone in the know here can explain it in laymans terms so that people like me can fully understand the meaning of this.

Also...what is this special number Fi (pronounced like Pi) 1.618 ?...how is it interwoven into the Golden mean ?

I think this would make a good subject for here.

Thank You..

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bostjan

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Re: The Golden Mean/Ratio
« Reply #1 on: 08/09/2006 01:15:30 »
You mean in the musical sense?

There are millions of tuning systems out there for setting up musical frequency intervals.  The most common in western music is to set apart each interval so that the distances are logrithmically equal.  To get from one note to the next tone in the series, you multiply the frequency by the twelth root of two.  In ancient europe and in India, the tones are divided into unequal intervals from a root note.  An octave was defined as double the frequency, a fifth was three halves the frequency, etc.  The older system sounded very clear and consonant over a fixed pedal frequency (or root note), but sounded awful through changes in key, so modern composers changed to the new system of equal intervals- they decided, eventually, that there should be twelve tones per octave.  (Although early Renaissance and late modern era composers generally use 17 or more tones.)

There have been many attempts to bridge the convenience of equal intervals with the clear consonant sounds of Ptolemy's "Just Scale" of unequal parts.  This has led to a revival of the older mean-tone tuning with new ideas applied.  The frequency ratios can be anything you want them to be, but the "golden mean" seems to be the best there is.

The practicality of all of these tunings is ratherlimited, however.

neilep

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Re: The Golden Mean/Ratio
« Reply #2 on: 08/09/2006 03:29:24 »
THANK YOU Bostjan......actually, I wasn't referring to music specifically  but to the mathematical properties........... and  then it's use in all types of geometry , Art AND also Music....apparently Bartok and Debussy used the Golden ratio to great effect.......very very subtle and only visible to people who know how to recognise it....it is found in tetrahedraon dodecahedrons and all over the place in nature too....

I am hoping that someone will explain the math(s) about it.

....and THANK YOU again for your wonderful post.



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Mjhavok

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Re: The Golden Mean/Ratio
« Reply #3 on: 08/09/2006 05:46:01 »
The golden mean is the ideal moderate position between to extremes.

The golden section - the division of a line so that the whole is to the greater part as that part is to the smaller part (i.e. in a ration of 1 to 1/2(#8730;5+1)), a proportion which is considered to be particularly pleasing to the eye.

1 to 1/2(#8730;5+1)) = the golden ration, approx 1:1.61803

This ratio is thought to occur throughout nature.

Da Vinci suggested that the human body has proportions close to the golden ratio.

The golden ratio is also closely related to the Fibonacci Sequence.

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Mjhavok

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Re: The Golden Mean/Ratio
« Reply #4 on: 08/09/2006 05:46:50 »
#8730 is supposed to be the square root sign but it doesn't want to show itself.

chris

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Re: The Golden Mean/Ratio
« Reply #5 on: 19/10/2006 09:52:07 »
quote:
Originally posted by neilep



I have just been watching a DVD of Jon Anderson (he's the lead singer of my all time favourite group 'YES' )...anyway...he is well into this thing called the Golden Mean or the Golden Ratio.


Is this "Tour of the Universe" ?

I ordered it recently. It's very good, but I'm not sure about the fat radio producer type people wandering about spouting stuff. I'd prefer more to watch something that isn't a total promo for XM -

Incidentally I'm off to See "Jon and Rick" play at the Corn Exchange here in Cambridge tomorrow (Friday) - are you going to see them on their tour?

Chris

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syhprum

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Re: The Golden Mean/Ratio
« Reply #6 on: 19/10/2006 12:08:00 »
I cannot see any way to produce a square sign unless you have microsoft office (119.00) may I suggest      "^0.5"

syhprum
« Last Edit: 19/10/2006 12:25:31 by syhprum »

DoctorBeaver

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Re: The Golden Mean/Ratio
« Reply #7 on: 19/10/2006 12:25:14 »
quote:
Originally posted by chris

quote:
Originally posted by neilep



I have just been watching a DVD of Jon Anderson (he's the lead singer of my all time favourite group 'YES' )...anyway...he is well into this thing called the Golden Mean or the Golden Ratio.


Incidentally I'm off to See "Jon and Rick" play at the Corn Exchange here in Cambridge tomorrow (Friday) - are you going to see them on their tour?





I hope you enjoy the gig :)

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Rapax

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Re: The Golden Mean/Ratio
« Reply #8 on: 19/10/2006 13:12:07 »
It's not just pleasing to the eye, there is a mathematical base to the golden ratio.

Take a line of unit length, and divide it into two non-equal parts. The larger is called the major (we'll call it x here), and the smaller is called the minor(y, or 1-x).

Now, the point of the golden ratio is that if you divide your line with exactly the golden ratio (GR), then the ratio of x:y is the same as the whole distance (1) to x. Or put mathematically:

1/x = x/y
1/x = x/(1-x)

this can be transformed into

(1-x)=x2
x2+x-1=0

using the standard formula for quadratic equations

x=(sqrt(5)-1)/2  [yes, there's a second solution, but it's a negative length in this case, so we can disregard it]

x=0.618...
y=1-x=0.381...

and the golden ration is 1/x, or x/y, which is both ~1.618

Mr Andrew

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Re: The Golden Mean/Ratio
« Reply #9 on: 28/10/2006 17:10:57 »
Another interesting thing about the golden mean is that it is the limit of the ratio of one fibinacci number to the previous one.

1/1, 2/1, 3/2, 5/3, 8/5, 13/8...Fn+1/Fn = phi

Pretty cool if you ask me.

Oh, and also, the golden rectangle (with sides of the ratio phi:1) can be used create a spiral found many places in nature.  It's quite fun to draw in a really boring class.

 

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