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Hidden Geometry in Music
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PostPosted: Wed Dec 03, 2003 9:43 am    Post subject: Hidden Geometry in Music Reply with quote  Mark this post and the followings unread

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Hidden Geometry in Music of Bach and Schoenberg: Reflection, Rotation, Proportion
Jonathan Saggau

Abstract
Musicians often equate the composers Bach (1685-1750) and Schoenberg (1874-1951) with a nearly obsessive relationship to numbers and mathematics. We think often of Schoenberg as a father of the numerical music organization systems of atonality and dodecaphony while there is an oft cited encoding within Bach's works the numerical and musical representation of his surname. Analysts often limit these composers' creativity to numerical surface details in the case of Bach or the tabulations of 12-tone row forms in the case of Schoenberg; this paper illuminates elegant architectural structures so prevalent behind such musical edifices. It comprises an architectural, geometrical, and statistical walking tour of two beautiful constructions, namely Schoenberg's Op. 19 No. 2 for piano and the Courante from Bach's Suite for Solo Cello No. 2. Through graphic measurements taken through modeling each piece of music onto a pitch-time two-dimensional complex plane (after Cogan and Escot), the paper analyzes internal architectural and geometric proportions of these works illuminating a consistent use of arithmetic, geometric, harmonic, and golden mean proportions amid these composers' works. It also provides graphical illumination of various pitch and time bilaterally symmetrical structures within the Schoenberg. Statistical contour correlations and oppositions are also found between ordered pitch data sets obtained from equal-length sections of the Bach using Spearman, Pearson, and Kendall data correlation methods. Finally, the paper compares the statistical distribution of pitch classes within the Schoenberg underlining his use of statistically lowest total duration pitches as contextually unique information at architecturally significant moments.




Read the rest at http://www.sapaan.com/vol1/saggau.htm
This is insanely interesting stuff. It will keep you guys awake for days.

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PostPosted: Wed Dec 03, 2003 11:06 am    Post subject: Reply with quote  Mark this post and the followings unread

to those interested in "Geometry in Music" I remind you of a previous and related thread.
Check that too.

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PostPosted: Wed Dec 03, 2003 11:17 am    Post subject: Reply with quote  Mark this post and the followings unread

Right on! Very Happy

The Solomon papers! Crack that whip, Indiana! Very Happy

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PostPosted: Wed Dec 03, 2003 4:09 pm    Post subject: Reply with quote  Mark this post and the followings unread

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The Golden Mean
The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.
The decimal representation of phi is 1.6180339887499... .

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PostPosted: Wed Dec 03, 2003 4:13 pm    Post subject: Reply with quote  Mark this post and the followings unread

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The Mathematical Magic of the Fibonacci Numbers
is formed by adding the latest two numbers to get the next one, starting from 0 and 1:

0 1 --the series starts like this.
0+1=1 so the series is now
0 1 1
1+1=2 so the series continues...
0 1 1 2 and the next term is
1+2=3 so we now have
0 1 1 2 3 and it continues as follows ...


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

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PostPosted: Wed Dec 03, 2003 4:17 pm    Post subject: Reply with quote  Mark this post and the followings unread

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Leonardo Pisano Fibonacci
Born: 1170 in (probably) Pisa (now in Italy)
Died: 1250 in (possibly) Pisa (now in Italy)

Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller..................

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PostPosted: Wed Dec 03, 2003 4:23 pm    Post subject: Reply with quote  Mark this post and the followings unread

Use of the Fibonacci Series in the Bassoon Solo in Bartók's Dance Suite.
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PostPosted: Wed Dec 03, 2003 4:26 pm    Post subject: Reply with quote  Mark this post and the followings unread

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The Harmonies of Nature
Béla Bartók is seen in our day as one of the twentieth century’s major composers, but we all too often forget that he first became known as a pianist and ethnomusicologist. Accompanied by his friend and colleague, Zoltan Kodály, he travelled the byways of Eastern Europe from 1907 to 1918, gathering thousands of folk songs, work that was to have a decisive influence on Bartók as a composer. In particular, he wove into his own works the modal forms and irregular rhythms typical of many of the folk songs gathered at this time.

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PostPosted: Wed Dec 03, 2003 4:29 pm    Post subject: Reply with quote  Mark this post and the followings unread

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Formal Considerations in Bela Bartok's
Fourth String Quartet

Béla Bartók wrote:




"I cannot conceive of music that expresses absolutely nothing."
 
"What is the best way for a composer to reap the full benefits of his studies in peasant music? It is to assimilate the idiom of peasant music so completely that he is able to forget all about it and use it as his musical mother tongue. "
 
"The sacredness of church music, the joyfulness and soulfulness of folksongs are the two pivots around which revolve true music."
 
"A nation creates music--the composer only arranges it."
 
"Risk is a crucial element of communication through music. Just as truly creative musicians must be willing to be open, to examine their inner life, to throw aside familiar comforts and plunge into the unknown, so they must take risks during performance."
 
"If there is to be communication with the listener, the musician's doors leading inward must stay open. Through this opening, the listener is invited into the reality of the musician. This involves risk for the performer. The inner world of the musician, the creative fount, is personal and sacred."
 
"In a performance setting, there can be no monologues. Every note, every breath, each moment sends messages between musician and listener. To ignore this two-way flow of feeling and meaning is to relinquish artistry for ego-gratification. "
 
"Competition is for horses, not artists."
 

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PostPosted: Wed Dec 03, 2003 4:30 pm    Post subject: Reply with quote  Mark this post and the followings unread

Golden Ratio and Fibonacci numbers
The golden mean ratio can be found in many compositions mainly because it is a "natural" way of dealing with divisions of time. One can find it in a lot of works by Mozart, Beethoven, Chopin, etc., etc. It is a question if it was used in a deliberate way or just intuitively (probably intuitively). On the other hand, composers like Debussy and Bartok have made a conscious attempt to use this ratio and the Fibonacci series of numbers which produces a similar effect (adjacent members of the series give ratios getting closer and closer to the golden mean ratio). Bartok intentionally writes melodies which contain only intervals whose sizes can be expressed in Fibonacci numbers of semitones. He also divides the formal sections of some of his pieces in ratios corresponding to the golden mean. Without going into much detail, Debussy also does this in some of his music and so does Xenakis (a composer who writes exclusively by using stochastic distributions, set theory, game theory, random walks, etc.) in his first major work, "Metastasis". The idea is not new, already in the Renaissance composers used it and built melodic lines around the Fibonacci sequence -just like Bartok's "Music for strings, percussion and celesta".

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PostPosted: Wed Dec 03, 2003 4:32 pm    Post subject: Reply with quote  Mark this post and the followings unread

What about the Moscovitz series? Idea

You take the difference of the previous two numbers.

0 1 1 0 -1 -1 0 1 1 0 -1 -1 0 ...

This is better in many respects. At least it's an oscillator. You can hear it. It's moving and inspiring.

You guys in Europe, please get the trademark on this for me.

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PostPosted: Wed Dec 03, 2003 4:35 pm    Post subject: Reply with quote  Mark this post and the followings unread

My understanding is that the great gothic cathedrals were architected using musical intervals for their varying dimensions, and that music performed in them was written to match these dimensions/intervals. Notre Dame (sp?) comes to mind, specifically.

I don't know, the historical elements of this discussion are a bit beyond my scope at this point in my life.

Cyx

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PostPosted: Wed Dec 03, 2003 4:36 pm    Post subject: Reply with quote  Mark this post and the followings unread

Then take the Multiplicative Moscovitz series.

0 1 0 0 0 0 0 0 0 0 0 0

That one is good if you like to be one with the universe.

For some reason, there is no Division series. They say it just can't be done. Crying or Very sad
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PostPosted: Wed Dec 03, 2003 4:39 pm    Post subject: Reply with quote  Mark this post and the followings unread

LAST BUT NOT LEAST

















Petals on flowers

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number: 

3 petals: lily, iris

5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)

8 petals: delphiniums

13 petals: ragwort, corn marigold, cineraria,

21 petals: aster, black-eyed susan, chicory

34 petals: plantain, pyrethrum

55, 89 petals: michaelmas daisies, the asteraceae family

Fibonacci Numbers in Nature

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PostPosted: Wed Dec 03, 2003 5:23 pm    Post subject: Reply with quote  Mark this post and the followings unread

Wow... great info on the Fibonacci numbers.

Actually, I have been using Fibonacci numbers myself... in my music. I like that approach better than staying with the silly old pop metrics. Sounds better too.

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PostPosted: Sat Jul 23, 2005 11:44 pm    Post subject: Re: Hidden Geometry in Music Reply with quote  Mark this post and the followings unread

elektro80 wrote:
Quote:
Hidden Geometry in Music of Bach and Schoenberg: Reflection, Rotation, Proportion
Jonathan Saggau

Abstract
Musicians often equate the composers Bach (1685-1750) and Schoenberg (1874-1951) with a nearly obsessive relationship to numbers and mathematics. We think often of Schoenberg as a father of the numerical music organization systems of atonality and dodecaphony while there is an oft cited encoding within Bach's works the numerical and musical representation of his surname. Analysts often limit these composers' creativity to numerical surface details in the case of Bach or the tabulations of 12-tone row forms in the case of Schoenberg; this paper illuminates elegant architectural structures so prevalent behind such musical edifices. It comprises an architectural, geometrical, and statistical walking tour of two beautiful constructions, namely Schoenberg's Op. 19 No. 2 for piano and the Courante from Bach's Suite for Solo Cello No. 2. Through graphic measurements taken through modeling each piece of music onto a pitch-time two-dimensional complex plane (after Cogan and Escot), the paper analyzes internal architectural and geometric proportions of these works illuminating a consistent use of arithmetic, geometric, harmonic, and golden mean proportions amid these composers' works. It also provides graphical illumination of various pitch and time bilaterally symmetrical structures within the Schoenberg. Statistical contour correlations and oppositions are also found between ordered pitch data sets obtained from equal-length sections of the Bach using Spearman, Pearson, and Kendall data correlation methods. Finally, the paper compares the statistical distribution of pitch classes within the Schoenberg underlining his use of statistically lowest total duration pitches as contextually unique information at architecturally significant moments.




Read the rest at http://www.sapaan.com/vol1/saggau.htm
This is insanely interesting stuff. It will keep you guys awake for days.



Just stumbled across this old thread...
And have to add something about hidden geometrie in music...
In the early 80´s i came along some aboriginal recordings...it was a very young trend at this time to use tribal soundrecordings and it was hard to get something from austaralia in germany, it was from ethnological science sources and not from your local recordshop... However years later when i had my first machines that was able to display the waveforms i found some hidden geometrie in the most ancient aboriginal style where they not even have didgeridous..just stone and voice..and this so called most primitiv music in the planet dont appears primitiv at all when beeing seen in an audio editor... triangels...squares...eliptoids...trapezoids... a wide varity of very precise geometrical envelopes one after another like a code... I never have seen vocalrecordings of any source looking that geometrical and intentional...100% precission in the volume articulation...
The most primitiv music on the globe... with lots of hidden geometry...
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PostPosted: Sun Jul 24, 2005 7:08 am    Post subject: Re: Hidden Geometry in Music Reply with quote  Mark this post and the followings unread

3phase wrote:

The most primitiv music on the globe... with lots of hidden geometry...

Interesting. Do you have any screen shots and/or recordings you can post?

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PostPosted: Sun Jul 24, 2005 12:48 pm    Post subject: Re: Hidden Geometry in Music Reply with quote  Mark this post and the followings unread

mosc wrote:
3phase wrote:

The most primitiv music on the globe... with lots of hidden geometry...

Interesting. Do you have any screen shots and/or recordings you can post?


not by hand..its maybe somwhere..but there was no screenshots..just the little display from a casio FZ1...but even the small display had a good waveform graphics. I might find the recordings at one point its a good idea to dokument that because i personally think that theese aborignal music is very underrated.. Its not primitiv at all . They just have another level of virtuosity. Its also interestinmg to apply echo to some of the stone percussions. What sounds like a drunken hitting an unrhytmic figure turns to samba when repeated...they just leave repetativ hits out... I dont think that this was by accident....
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PostPosted: Mon Jul 25, 2005 7:52 pm    Post subject: You think it kept YOU awake. Reply with quote  Mark this post and the followings unread

Shocked It REALLY kept me awake. Then again, I wrote it. Very Happy Please do check out the writings of my teachers, Pozzi Escot and Robert Cogan. Great stuff.

~Jonathan Saggau
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PostPosted: Mon Jul 25, 2005 8:20 pm    Post subject: Reply with quote  Mark this post and the followings unread

welcome jonmarimba. Good to have you here.

So, you are Jonathan Saggau? Shocked Cool

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PostPosted: Tue Jul 26, 2005 2:59 am    Post subject: Re: You think it kept YOU awake. Reply with quote  Mark this post and the followings unread

jonmarimba wrote:
Shocked It REALLY kept me awake. Then again, I wrote it. Very Happy Please do check out the writings of my teachers, Pozzi Escot and Robert Cogan. Great stuff.

~Jonathan Saggau


Shocked

You wrote that one?! !!!! Well done!

Great to see you here! Very Happy
I will Google for those names.. or do you have any direct links to some texts?

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PostPosted: Tue Jul 26, 2005 3:35 am    Post subject: Reply with quote  Mark this post and the followings unread

Jonathan's text references Hofstadter's GEB, I highly recomend that people also look at his analysis of some Chopin etudes in Methamagical themas. Hofstadter worked together with this guy who programed software to display musical scores as graphical peterns (I forgot his name and my copy is at Rob's). It's very interesting though it doesn't go into as much mathematical depth as Jonathan's text.
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PostPosted: Tue Jul 26, 2005 9:43 am    Post subject: Reply with quote  Mark this post and the followings unread

I'm not taking this off-topic, just continueing with Mathematics in Art.

A fascinating book I read is called "The Great Pyramid Speaks : An Adventure in Mathematical Archaeology" , by Joseph B. Gil

The author starts at the outside of the Pyramid, with it's four not-so-perfect sides. Once you find the door, and the *dimension*of the door, you realize that each side of the pyramid is ~365 door widths long. Each slightly off, but when added together, form the same "length" as our leep-year system, including the century and 4 century adjustments. Then you look at the door, which hieght to width ratio is Pi. The rest of the Pyramid's architecture reveals exponents, "e", and a staight-line visual guide to the north star (well, where it was 3,000 years ago), including an exact measurement of distance in *light years* to that star.

It was Christopher Walken as the angel Gabriel in "The Prophecy" who told the neighborhood kids..."Study your math...it's the key to the universe" (I think in europe this movie was called "God's Army").
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PostPosted: Tue Jul 26, 2005 10:57 am    Post subject: Reply with quote  Mark this post and the followings unread

If this it true, then the North Star pointer would date the structures at 3000 years old. There are people that say it is 10000 years old.
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PostPosted: Tue Jul 26, 2005 12:17 pm    Post subject: Reply with quote  Mark this post and the followings unread

jksuperstar wrote:

The author starts at the outside of the Pyramid, with it's four not-so-perfect sides. Once you find the door, and the *dimension*of the door, you realize that each side of the pyramid is ~365 door widths long. Each slightly off, but when added together, form the same "length" as our leep-year system, including the century and 4 century adjustments. Then you look at the door, which hieght to width ratio is Pi. The rest of the Pyramid's architecture reveals exponents, "e", and a staight-line visual guide to the north star (well, where it was 3,000 years ago), including an exact measurement of distance in *light years* to that star.


I'm down with Pi, e is a strech but I'll take it since it would've been quite usefull and they were clever. I'll take leap years too, even the exception once a century but 4 centuries is a long, long time.

Frankly I don't believe lightyears. That would require knowledge that light takes time to travel (how would they know?) and secondly a way of measuring it. I don't see how you'd do that in that time. Having e would be quite clever, knowing the speed of light would require atomic clocks and the like. Knowing the speed of sound would be a respectable feat in itself.

Quote:

It was Christopher Walken as the angel Gabriel in "The Prophecy" who told the neighborhood kids..."Study your math...it's the key to the universe" (I think in europe this movie was called "God's Army").


I'd say he's right, but within this context; would you turst a person often credited with keeping humanity out of paradise for the crime of seeking knowledge of good and evil?

;¬)

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