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#353860 - 12/17/06 11:20 PM Bartok and the Golden Ratio  
Joined: Dec 2005
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tomt6113 Offline
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tomt6113  Offline
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new york
Recently ive been reading about mathematics in music. ive come accross a few articles concerning barok's use of the golden ratio and the fibonacci numbers in pieces like allegro barbaro. unfortunately the articles dont describe how he used it, but only state that he did. i find this really interesting and would love to find out more. id greatly appreciate it if someone who knows about this subject could tell me about it. thanks.


im not really in a creative mood
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#353861 - 12/18/06 03:36 AM Re: Bartok and the Golden Ratio  
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btb Offline
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Fascinating subject ... 1.618 golden ratio ... the proportion which guides the designs of architects. Thought I'd throw in a diagram to show how the ratio is obtained . On a square ABCD ... draw in the median e-f ... then a diagonal through half the square e-c ... with circle centre at e expand the radius eD to meet BA projected at g. The overall rectangle now has
a proportion of the golden ratio of 1.618.
web page
The Swiss architect Le Corbusier used the Golden section in standardising measurement to and his Unite D'Habitation in Marseilles using his patent
version of the Fibonacci series ... his blue series ran 20.33.53.86.139.226.

Can't imagine how Bartok managed to use the Fibonacci series for his Allegro Barbaro ... the closest I've come to relating 1.618 to the keyboard was finding that the octave was 1.71 larger than the dominant interval.

#353862 - 12/18/06 04:26 AM Re: Bartok and the Golden Ratio  
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pianojerome Offline
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Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... add each number to the previous number to get the next number (1+1=2, 2+1=3; 3+2=5, etc). Divide any number by the one before it, and you get an approximation of this "golden ratio" 1.618 (not exact every time, but very close).

I'd recommend you read Erno Lendvai's book. It's great. He gives pages of examples. One example involves counting the total number of 8th notes -- and if you apply the golden ratio, you get *exactly* the 8th note that begins the recapitulation section.

I haven't studied Allegro Barbaro, but here are a couple of examples from the 6 Romanian Dances

4th movement: there are 18 measures. divide by the golden ratio: 18/1.618 = 11.124, and in fact measure 11 is the beginning of the 2nd section. (this movement has 2 sections: one for the first melody, and one for the second melody)

5th movement: there are 28 measures. Divide that by the golden ratio: 28/1.618 = 17.305. Hm, see the double bar at measure 17? That is where the first half (minus introduction) is repeated an octave lower.


It seems that the way he has done this is through the introduction. The 4th dance, if you take away this 2-bar introduction, has two sections that are both 8 bars long. But if you add the 2-bar intro, you get an approximation of the golden ratio in there. The 5th dance, if you take away the 4-bar introduction, has two sections that are both 12 bars long. But if you add in the 4 bars to the start of the first section, then you get an approximation of the golden ratio in there.


Sam
#353863 - 12/18/06 04:32 AM Re: Bartok and the Golden Ratio  
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pianojerome Offline
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There is actually a funny story about this... because, you know, this is really a big area of study in the Bartok world.

For a long time, there was no evidence at all to suggest that Bartok actually did all this on purpose, that he ever calculated anything. No papers, nothing.

Then, somebody found something. It was a paper in Bartok's documents that was just covered with numbers from the Fibonacci series. Different calculations and so forth.

It raised a huge controversy; kind of like the rosetta stone, as one of the professors here described it, of Bartok study.

Well, you know what it was? A shopping list. Bartok had just been adding and subtracting prices, and figuring out taxes, and it's just a coincidence that these prices happened to be fibbonaci numbers!

But read Lendvai's book. The examples I gave are so-so, and that's the only piece of music by Bartok that I've studied, but Lendvai really provides a lot of good examples, and above all explains the whole concept and various aspects of Bartok's compositional technique.


Sam
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#353864 - 12/18/06 10:18 AM Re: Bartok and the Golden Ratio  
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lol_nl Offline
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I know that in Bartok's Music for Strings, Percussion and Celesta Sz. 106, BB 114 there are a few of these amazing things in the first movement.

First of all, it's a fugue, but that's probably not very amazing.

The amazing things are:

-The theme is played in such a way, that each time the theme is in the key either a 5th above or below the original theme and repeated in this way, until they meet each other again and the circle goes backwards again. So:
The first time the theme is announced its starts on an A, then it goes a 5th up (an E), then a 5th down of the A (D), then a 5th up of the E (B), and so on.
So you'll get:
1 - A
2 - E
3 - D
4 - B
5 - G
6 - F#
7 - C
8 - C#
9 - F
10 - G#
11 - Bb
12 - D# or Eb -> Here the two meet!
Then the cicle goes backwards again and the whole first movement ends with an A.

Furthermore there was something with the Fibonacci in this piece too, and something remarkable about the climax of the piece and bar 78, where the Celesta starts playing. Also, it seems that all notes together have some Fibonacci structure or something, so you have to forget about the bars and only count the notes, which will lead to another mathematical cycle or something.

I forgot what it exactly was.


Yiteng

"Music is enough for a lifetime, but a lifetime is never enough for music."
-Sergei Rachmaninoff.
#353865 - 12/18/06 10:27 AM Re: Bartok and the Golden Ratio  
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argerichfan Offline
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Wasn't this referred to as the Golden Section also? I studied this at university, but am a bit dismayed to learn here that it's all a coincidence. So all this for nothing?

I worked on the Romanian Dances at the time- learning the music was a fascinating experience. Alas for the Allegro Barbaro. What an awful piece of rubbish- just a lot of pointless banging. Alkan's piece of the same name (of which Bartok may have heard Busoni play) is far superior, though in an entirely different league technically.


Jason
#353866 - 12/18/06 06:30 PM Re: Bartok and the Golden Ratio  
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whippen boy Offline
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Quote
Originally posted by pianojerome:
There is actually a funny story about this... because, you know, this is really a big area of study in the Bartok world.

For a long time, there was no evidence at all to suggest that Bartok actually did all this on purpose, that he ever calculated anything. No papers, nothing.

Then, somebody found something. It was a paper in Bartok's documents that was just covered with numbers from the Fibonacci series. Different calculations and so forth.

It raised a huge controversy; kind of like the rosetta stone, as one of the professors here described it, of Bartok study.

Well, you know what it was? A shopping list. Bartok had just been adding and subtracting prices, and figuring out taxes, and it's just a coincidence that these prices happened to be fibbonaci numbers!
laugh This calls to mind something eerily similar: The relic of the shopping list in A Canticle for Leibowitz - I highly recommend this book!

#353867 - 12/19/06 03:30 PM Re: Bartok and the Golden Ratio  
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tomasino Offline
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I'm not an anti-intellectual. I'm really not. But there is something in this thread that makes me scoff a bit.

It reminds me of a film I saw seven or eight years ago called "Pi," in which a Jewish intellectual discovers a series of numbers which are the mathematical equivalent of the unspeakable name of God.

It ends up with the Jewish intellectual taking an electric drill to his right temple.

Tomasino

AKA: 9879797 037850327983672234568888888564782664736


"Whatsoever thy hand findeth to do, do so with all thy might." Ecclesiastes 9:10

#353868 - 12/19/06 03:33 PM Re: Bartok and the Golden Ratio  
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pianojerome Offline
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Quote
Originally posted by tomasino:
AKA: 9879797 037850327983672234568888888564782664736
Shhhhhh!! :p

Quote
Originally posted by tomasino:
It ends up with the Jewish intellectual taking an electric drill to his right temple.
You mean to fix the temple after it was destroyed by the Romans in 70 CE? Seems pretty constructive to me.


Sam
#353869 - 12/19/06 03:53 PM Re: Bartok and the Golden Ratio  
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Ted Offline
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If a pile of data is big enough it will contain strangely "meaningful" patterns - hidden faces in photographs, messages in books, "precognitive" dreams and the like.

I cannot see why (5^(1/2)+1)/2 should be more musically meaningful than the square root of Einstein's waist measurement or the logarithm of the time in seconds it takes for my lavatory cistern to fill.


"It is inadvisable to decline a dinner invitation from a plump woman." - Fred Hollows
#353870 - 12/19/06 03:58 PM Re: Bartok and the Golden Ratio  
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pianojerome Offline
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The question is whether Bartok actually used these numbers while he was composing his music -- and there are a number of people who suggest that he did.

It might be coincidence that one can apply mathematics to particular pieces of music; on the other hand, it might not be.


Sam
#353871 - 12/19/06 07:12 PM Re: Bartok and the Golden Ratio  
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tomt6113 Offline
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tomt6113  Offline
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thanks for all the replies. they have been helpful. im not trying to be super intellectual here, im just trying to comprehend the intellectual aspect of bartok.


i understand that all big data piles contain some kind of pattern and that it is not always meaningful. but the same pattern (golden ratio, fibonacci etc.) has come up in a number of bartok pieces. the reason it is i think it so important to understand is because it supposedly has something to do with the form of the piece. and how can you express the musicality of a piece if you dont understand its form. it would make you understand the music better as well as play it better. if by meaningful you mean emotionally evocative, i have read that the most evocative moments of bartoks pieces sometimes fall on the "points" that create the golden ratio in a section. dont get me wrong i dont mean to suggest that you should view bartok's music as solely mathematics, but you also should not view it as solely emotional. the emotional and psychological depth of his music is astounding, but so are the formulas behind is music and i think that understanding both is essential to understanding the music.


im not really in a creative mood
#353872 - 12/20/06 04:14 PM Re: Bartok and the Golden Ratio  
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SBiv87 Offline
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Indiana
Quote
Originally posted by lol_nl:
I know that in Bartok's Music for Strings, Percussion and Celesta Sz. 106, BB 114 there are a few of these amazing things in the first movement.

Furthermore there was something with the Fibonacci in this piece too
The sections are broken down in measure numbers of the Fibonacci sequence. For instance, a new section begins at measure 55, another at 89, etc.

#353873 - 12/21/06 10:25 PM Re: Bartok and the Golden Ratio  
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tomt6113 Offline
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tomt6113  Offline
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ive ben messin around with this golden ratio thing and found this out. i heard a lot of pieces contain it without the composer being conscious of doing it, including chopin. so just for fun i counted the number of measures in etude op 10 no 1. 78. divided it by 1.618 and got 48. went to measure 48: at measure 48 the main "theme" returns in its original key. check it out for yourself. its pretty cool.


im not really in a creative mood

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