# Bartok's Methods



## millionrainbows (Jun 23, 2012)

Been reading a book by Erno Lenvai, about Bartok and his use of The Golden Mean in his composition. Bartok used 2 main approaches in his method: a "harmonic" method based on Golden Ratios, and an "axis system" based on relations in the circle of fifths. I will explain the key differences in these 2 approaches as we go.

Any objections to the Golden Mean being discussed? I know in the past, elsewhere, this caused great disagreement.


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## ptr (Jan 22, 2013)

Please Millions, go on!

I read Lendvai's book 15 years ago when I took musicology at Uni. It'll be fun to see if Your thoughts on this will jog my blurred memory! 

/ptr


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## millionrainbows (Jun 23, 2012)

*Firstly,* you must realize that Golden Mean numbers are _irrational;_ that is, like *pi,* they cannot be expressed as whole numbers, but they *can* be expressed geometrically, and this is a key difference.

*Then* you must realize that our 12-note music system was invented by Pythagoras, as a compromise; the fifths, or 3:2's he was after are not possible to fit into an octave cycle, no matter how large; so the harmonic truth he sought was a compromise of arithmetic vs. ratio. Add to that the fact that we now use equal temperament, and the system becomes even more obscured.

Only after these concessions, imperfections, and compromises are acknowledged can we begin to make progress through the complexities, and realize what a beautiful system Bartok developed.

It built on tradition, more so than Schoenberg's cerebral and non-harmonic system did.

Bartok's Axis system had functional affinities with tonality, such as tonic, dominant, and subdominant; it had relative major and minor keys; it had overtone relations; it had leading notes; dominant and subdominant tensions were apposite, as in tonality; distance has a direct bearing on tonality.

In many ways, Bartok's Axis method is an inverse version of tonality; instead of going "outward," past the octave into new tonal centers, it goes "inward" into new tonal centers by using small-divisions of the octave, rather than fifths and fourths.


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## Blake (Nov 6, 2013)

You've inspired me to go listen to Bartok again. I'll have some quartets, please.


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## hpowders (Dec 23, 2013)

Takacs, Emerson, Hungarian, Julliard, Vegh or Tokyo?


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## KenOC (Mar 7, 2011)

hpowders said:


> Takacs, Emerson, Hungarian, Julliard, Vegh or Tokyo?


Takacs. Trust me on this!


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## Blake (Nov 6, 2013)

I have the Emerson.


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## hpowders (Dec 23, 2013)

I do too, but I think I could do better. They remind me a bit of a Pierre Boulez performance-all of the notes are clearly there and they play with virtuosity, but it sounds clinical. A bit more passion would be welcome.


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## Blake (Nov 6, 2013)

hpowders said:


> I do too, but I think I could do better. They remind me a bit of a Pierre Boulez performance-all of the notes are clearly there and they play with virtuosity, but it sounds clinical. A bit more passion would be welcome.


I love the cool precision of Boulez and Emerson. It's a welcome contrast when I'm not feeling like engaging so emotionally.


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## hpowders (Dec 23, 2013)

I'd prefer a bit more vibrato at times in the quartets.


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## Crudblud (Dec 29, 2011)

hpowders said:


> Takacs, Emerson, Hungarian, Julliard, Vegh or Tokyo?


I'm quite fond of Tátrai's recordings for Hungaroton.


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## EdwardBast (Nov 25, 2013)

millionrainbows said:


> Been reading a book by Erno Lenvai, about Bartok and his use of The Golden Mean in his composition. Bartok used 2 main approaches in his method: a "harmonic" method based on Golden Ratios, and an "axis system" based on relations in the circle of fifths. I will explain the key differences in these 2 approaches as we go.
> 
> Any objections to the Golden Mean being discussed? I know in the past, elsewhere, this caused great disagreement.


Quick question: Did Bartok claim to have used the golden mean in his compositions or is this Lenvai's interpretation based on the works?


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## hpowders (Dec 23, 2013)

Crudblud said:


> I'm quite fond of Tátrai's recordings for Hungaroton.


Me too, but alas, I don't possess their performances.


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## spradlig (Jul 25, 2012)

Did Bartok himself admit using the Golden Mean? It is easy to find the Golden Mean everywhere you look if you want to find it. Sorry I don't have a reference at hand, but the myth that the Golden Mean is prevalent in visual art and music has been thoroughly debunked.



millionrainbows said:


> Been reading a book by Erno Lenvai, about Bartok and his use of The Golden Mean in his composition. Bartok used 2 main approaches in his method: a "harmonic" method based on Golden Ratios, and an "axis system" based on relations in the circle of fifths. I will explain the key differences in these 2 approaches as we go.
> 
> Any objections to the Golden Mean being discussed? I know in the past, elsewhere, this caused great disagreement.


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## KenOC (Mar 7, 2011)

spradlig said:


> Did Bartok himself admit using the Golden Mean? It is easy to find the Golden Mean everywhere you look if you want to find it. Sorry I don't have a reference at hand, but the myth that the Golden Mean is prevalent in visual art and music has been thoroughly debunked.


I think some musicologists have time on their hands. Bartok is also supposed to have used Fibonacci numbers (in his Music for SP&C). And maybe he did. But there's no musical significance that I can see (or hear). In fact, there seems to be a relationship between the golden mean and Fibonacci numbers. 

http://en.wikipedia.org/wiki/Fibonacci_series


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## Blancrocher (Jul 6, 2013)

I just googled "Bartok + Fibonacci" and came up with a hilarious story about a shopping list, though it could be made up for all I know.


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## spradlig (Jul 25, 2012)

Yes, if you take the ratio of consecutive Fibonacci numbers (the bigger divided by the smaller), it converges to the Golden Mean as you go to infinity.

The problem with Fibonacci numbers is that _a lot_ of small numbers are Fibonacci numbers: the sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21,... so e.g. half the integers between 1 and 10 (inclusive) are Fibonacci numbers, and if you want to find them in random data, you'll find them.



KenOC said:


> I think some musicologists have time on their hands. Bartok is also supposed to have used Fibonacci numbers (in his Music for SP&C). And maybe he did. But there's no musical significance that I can see (or hear). In fact, there seems to be a relationship between the golden mean and Fibonacci numbers.
> 
> http://en.wikipedia.org/wiki/Fibonacci_series


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## spradlig (Jul 25, 2012)

@millionrainbows : you seem to know a lot about Bartok's Axis system, which I had not heard of. Does it involve the Golden Ratio? You did not mention that in your post.



millionrainbows said:


> *Firstly,* you must realize that Golden Mean numbers are _irrational;_ that is, like *pi,* they cannot be expressed as whole numbers, but they *can* be expressed geometrically, and this is a key difference.
> 
> *Then* you must realize that our 12-note music system was invented by Pythagoras, as a compromise; the fifths, or 3:2's he was after are not possible to fit into an octave cycle, no matter how large; so the harmonic truth he sought was a compromise of arithmetic vs. ratio. Add to that the fact that we now use equal temperament, and the system becomes even more obscured.
> 
> ...


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## Minona (Mar 25, 2013)

Yes, it has been debunked. There's a video on YouTube by one of the professors who promoted the idea in the 1970s but now admits it was a red herring.


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## ahammel (Oct 10, 2012)

Minona said:


> Yes, it has been debunked. There's a video on YouTube by one of the professors who promoted the idea in the 1970s but now admits it was a red herring.


"It" meaning the idea that the golden section shows up a lot in art in general, or in Bartók in particular?

The golden section theory of Bartók's harmony appears to be Lendvai's. I'm not sure that Bartók himself ever said anything about it. There's a pretty intentional-looking Fibonnaci sequence rhythm at the beginning of _Music for SP&C_, though. Non-retrogradeable as well. Messiaen would be proud.

The sequence of Fibonnaci intervals that Wiki gives from _Sonata for Two Pianos and Percussion_ looks less convincing, though. The intervals are cherry picked, and not in any particular order. Besides, 5 semi-tones make a perfect fourth, and 3 make a minor third: not exactly uncommon intervals in Western music.

So I'm skeptical, but I'd be interested to hear what Lendvai (or, even better, Bartók) has to say about the golden section in Bartók's music. Let's leave out the half-baked evolutionary psychology, though!


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## aleazk (Sep 30, 2011)

I always found all those claims related to the golden mean and the fibonacci numbers more akin to numerology than to actual sensible science.


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## Couac Addict (Oct 16, 2013)

What self respecting musician is faffing about with the Fibonacci sequence? Whatever happened to impressing chicks?


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## EdwardBast (Nov 25, 2013)

spradlig said:


> @millionrainbows : you seem to know a lot about Bartok's Axis system, which I had not heard of. Does it involve the Golden Ratio? You did not mention that in your post.


This is going back a ways in memory so don't count on its accuracy, but I believe the axis system ascribes tonic, dominant, and subdominant-like functions to octatonic collections within a given work (of Bartok). An octatonic scale or collection (jazz players sometimes call them diminished scales for obvious reasons) comprises alternating whole and half steps. Like diminished seventh chords (excluding enharmonic spellings and functions), there are only three different octatonic sets. So there would be three quasi-tonal axes based on or around the three octatonic scales. Presumably, once one identifies the tonic pitch, and by extension the tonic collection as well, the other two octatonic sets would be assigned the dominant and subdominant functions. I forget, however, which of the other two sets gets the dominant function and which the subdominant function after the tonic is determined, though my guess is the one containing the pitch a P5 above tonic would be the dominant collection.

Apologies if I have gotten something wrong here. Perhaps someone can verify, expand or correct this dim recollection. I don't own Lenvai's book.


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## Blancrocher (Jul 6, 2013)

spradlig said:


> Yes, if you take the ratio of consecutive Fibonacci numbers (the bigger divided by the smaller), it converges to the Golden Mean as you go to infinity.
> 
> The problem with Fibonacci numbers is that _a lot_ of small numbers are Fibonacci numbers: the sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21,... so e.g. half the integers between 1 and 10 (inclusive) are Fibonacci numbers, and if you want to find them in random data, you'll find them.


As with any structural analysis trying to prove something that isn't obvious, one would have to be careful in one's argument. As one moves along the Fibonacci Sequence towards higher adjacent numbers or multiples of low ones, one's chances of saying something credible improve: say, you have an 80-bar work and something very significant happens in the 50th (or 30th). The difficulty is in persuading people of that "very significant." Ideally, one would be able to show both that a composer is/was interested in golden ratios and that there is some discernible rationale for using it in the music.

Scholars seem to have shown this to be the case in Debussy's music, though I haven't looked into much on this topic aside from Wikipedia.

*p.s.* Wikipedia mentions that many scholars have disputed Lendvai's work, but doesn't provide sources. In any case, here is an essay by Lendvai putting forward some of his theories.


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## Blake (Nov 6, 2013)

Couac Addict said:


> What self respecting musician is faffing about with the Fibonacci sequence? Whatever happened to impressing chicks?


Hey man, there are chicks out there who aren't morons. :tiphat:


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## EdwardBast (Nov 25, 2013)

Since it fits with the general topic of this thread, I thought it might be fun to throw in a method we know Bartok actually used 

One thing that has always struck me about Bartok's writing is the pervasive invertibility of his themes, by which I mean that they often sound more or less equally satisfying and natural in their "straight up" and inverted forms. And Bartok commonly exploits both forms. One obvious example is the big brass fughetta in the first movement of the Concerto for Orchestra. In the exposition the theme is straight up, in the counter exposition it is inverted, and the passage ends with a perfectly symmetrical stretto combining the two forms. There are examples like this everywhere in Bartok's work. The Second Piano Concerto and Music for Strings, Percussion and Celeste, for example, are full of this kind of writing. 

What I wonder is: Did Bartok (have to) consciously tool his themes to function this way, or was it (or did it become over time) an automatic and subconscious capacity? That is, did his mind automatically run some kind of mirroring function while composing that influenced what kind of themes would occur to him? This sort of issue comes up with respect to any pervasively contrapuntal composer I imagine. For example, the mature J. S. Bach probably knew unconsciously what subjects would have interesting stretto possibilities without necessarily having to work out the stretto first (though I imagine he also must often have worked out the strettos first and worked back to the subject).


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## EdwardBast (Nov 25, 2013)

Blancrocher said:


> As with any structural analysis trying to prove something that isn't obvious, one would have to be careful in one's argument. As one moves along the Fibonacci Sequence towards higher adjacent numbers or multiples of low ones, one's chances of saying something credible improve: say, you have an 80-bar work and something very significant happens in the 50th (or 30th). The difficulty is in persuading people of that "very significant." Ideally, one would be able to show both that a composer is/was interested in golden ratios and that there is some discernible rationale for using it in the music.
> 
> Scholars seem to have shown this to be the case in Debussy's music, though I haven't looked into much on this topic aside from Wikipedia.


The problem with this theory is that in short to medium length works with any kind of overall "dramatic arc," more often than not, the climactic point is naturally going to occur at roughly two thirds of the way through. Think of climactic retransitions in movements by Beethoven, for example. Haven't actually done a statistical study, but I wouldn't be surprised if on average they cluster around a point that seems to be defined by the golden section.


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## Blancrocher (Jul 6, 2013)

EdwardBast said:


> The problem with this theory is that in short to medium length works with any kind of overall "dramatic arc," more often than not, the climactic point is naturally going to occur at roughly two thirds of the way through. Think of climactic retransitions in movements by Beethoven, for example. Haven't actually done a statistical study, but I wouldn't be surprised if on average they cluster around a point that seems to be defined by the golden section.


That seems right: this being the case, then, a composer had better not trust to 50/30 and had better use 55/34 if s/he wants to be understood!


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## millionrainbows (Jun 23, 2012)

Re: the quartets, watch out for the *Newton Classics 2-CD *reissue of the 1972-74* Guarneri *recordings. They are otherwise excellent, but there are two glitches which have shown up on the first disc alone, so far in my listening: a gap between cuts which should be continuous, and a huge pitch waver in another area, which sounds like an analog tape recorder starting up. A drag, because these are good performances. Does anyone recall the* Guarneri *releasing these on* RCA? *

Also, I noticed that the* Juilliard Quartet's *recording of the two* Charles Ives quartets *have also been released on *Newton;* These have NEVER been on CD until now. I think they were on* Nonesuch *as vinyl. I hope no glitches show up in these; I consider these the definitive versions (imprinted on me).


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## millionrainbows (Jun 23, 2012)

http://www.talkclassical.com/20772-xenakis-other-forum.html

I refer you to this thread, in reference to the wave of hostility to the Bartok/Golden Mean connection, and the Lendvai book. I am not going to waste time defending this; I'll simply present the ideas as they appeared in the book, and let those speak for themselves.


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## millionrainbows (Jun 23, 2012)

Bartok's "Axis" system has no relation to the Golden Mean, or to Fibonacci numbers; it is simply a way of establishing tonal centers and functions (tonic, dominant, subdominant), which is based on a symmetrical divisions of the octave. The beauty of Bartok's system is that, like tonality, it is based on harmonic principles, as well as structural principles of symmetry (like serialism is).
Howard Hanson explains the "projection" of intervals in complete form, so I refer you to his book; also, for those with less need for thorough understanding, go to "interval multiplication" in WIK. This is a basic idea of modern music, derived from Pythagoras' "stacking" of twelve 3:2's in order to divide and close the octave. Close, but no cigar: the octave (2:1) can never be divided by 3, 4, or any number except 1 itself (or multiples).
This must be "grokked" completely; that our 12-division is imperfect and impossible. Bartok's 3-5-8 proportion is only approximately correct, and is expressible only in irrational numbers (5:8.09061...), so this creates great confusion.

I cannot find an explicit statement by Bartok saying that he used Fibonacci numbers; but in Music for Strings, Percussion, and Celesta, the proportions speak for themselves: the first movement fugue (89 bars) are divided into sections of 55 and 34 bars by the peak. The section leading up to the climax (bar 55) show a division of 34 + 21. Those 21 bars are divided into 13 + 8.
_"We follow nature in composition," _wrote Bartok. He also called the sunflower his favorite plant, and kept fir-cones on his desk.

Bartok can also be quoted in At the Sources of Folk Music (1925):_ "Also, folk music is a phenomenon of nature. Its formations developed as spontaneously as other living natural organisms: the flowers, animals, etc."

_*Ok, I'd much rather delve into the particulars of Bartok's methods, rather than defend Erno Lendvai and the Golden Mean. Perhaps the connection will emerge more clearly after that.*


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## ptr (Jan 22, 2013)

It really does not matter if Bartók used any of these, they are influential tools in to understanding his music! As my Music Theory Professor used to say, analysis will help You to understand the musical structure, it won't help You to enjoy the music, that is a much more difficult pseudo scientific issue you should talk to your shrink about... 

/ptr


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## millionrainbows (Jun 23, 2012)

Regarding the Axis idea, which is only a small part of Bartok's system, we must imagine the circle of fifths; not a chromatic circle, since, appropriately, these Axis relations are based on harmonic principles as well.

On the circle, with C at the top as tonic, we can draw a straight diameter line to F#, which divides the octave into 6 + 6. Another diameter line, perpendicular to the first, connects A (clockwise from C) to Eb/D#. Note that these are tritone relations. These 4 points are "tonic" stations. These divide the octave into 4 parts. This corresponds to the "relative" majors and minors of classical music: C major and A minor, C minor and Eb major, etc.
An intimate relationship exists between the opposite poles: C/F# and A/Eb. These pairs are interchangeable, like the inversion of a tri-tone. 
These minor-third relations are further reinforced by the Romantic period's use of "upper relatives," as in C maj/a min/Eb maj.


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## Couac Addict (Oct 16, 2013)

Vesuvius said:


> Hey man, there are chicks out there who aren't morons. :tiphat:


What hope do I have with them?


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## Blake (Nov 6, 2013)

Couac Addict said:


> What hope do I have with them?


You'd have to start blowing your instrument correctly and get off the couac.


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## millionrainbows (Jun 23, 2012)

Regarding Bartok's axis system, and our 12-note octave in general, I'd like to discuss a crucial distinction, which I feel must be fully "grokked" before a more complete understanding can be approached.

This distinction is the geometric vs. the arithmetic. Geometric relations are often irrational, i.e., they cannot be expressed as whole numbers. Nonetheless, they represent very real relations. The arith-metic is expressed as rational whole numbers.

The natural overtone series is arithmetical; it is based on whole number ratios: 3:2 (fifth), 3:4 (fourth), 4:5 (major third), octave (2:1), etc.

Although Pythagoras based our 12-note chromatic scale on the "stacking" of fifths (3:2), he stopped stacking after 12 "cycles," apparently thinking that the "octave circle" had closed: F-B-E-A-D-G-C, in fourths. This was not a perfect closure, giving rise to the so-called "Pythagoran comma," the gap left when the last 3:2 exceeded the circular octave limit (by 2 cents). But in order to preserve the octave (we hear octaves as multiples of the same pitch, called pitch identity), the circle was closed, and the octave was preserved. In reality, the octave (1 and its multiples, 2,4,8, etc.) cannot be divided by any number other than itself and come out even, because it is a prime. So, any division of the octave other than by 1 or its multiples (going backwards, 1/2, 1/4, 1/8, etc.) is impossible, and will be an artificial division not based on "natural" ratios, as our present system is.

This inability to evenly divide an octave was not as much a problem for Pythagoras and pre-modern music-makers, because music did not modulate, and started from "1" at all times; therefore, Pythagoras' scale had uneven divisions, since Greek music was essentially melodic and did not change key.

Now, we have developed equal temperament, so our chromatic circle of "fifths" is evenly divided. Thus, we are faced with a geometric circle of "fifths" (or expressed as a 12-note chromatic circle). This is an imperfect circle from an acoustic, whole-number standpoint; it is essentially an artificial geometric construction.

Once you fully grasp this, we can proceed with Bartok's axis system, which was developed to encompass his "*chromatic" *scale, and his *"diatonic"* scales. His "chromatic" scale is geometric in nature, based on *Golden Section *(heretofore referred to as* GS*); the "diatonic" scale is based on acoustic principles, i.e. whole-number ratios such as 3:2, 4:5, etc.

Let's take the *chromatic *scale first. This scale is based on a symmetrical (geometric) division of the octave at the tri-tone. The tri-tone in itself is an arbitrary geometric division of the 12-note "circle." This is entirely arbitrary, based on the number 12. All the other chromatic relations are derived from further divisions of the 12-circle. This division is based on the GS proportion of *5:3:2,* which is part of the Fibonacci sequence. The divisions are (in chromatic steps):

model 1:5 --- alternating minor seconds and perfect 4ths; e.g. C-C#-F#-G-C-C#-F#-G. . .
model 1:3 --- alternating m2's and m3rds; e.g. C-C#-E-F-G#-A-C. . .
model 1:2 --- alternating m2's and Major 2's; e.g. C-C#-Eb-E-F#-G-A-Bb-C. . .

If you draw these onto a chromatic circle (clock-face), the geometric divisions will be much easier to see. All of them are part of the same subsets: 1:3 is a subset of 1:2, and 1:5 is a subset of both 1:3 and 1:2.

Model 1:2 is otherwise known as the diminished scale, which divides the 12 notes of the octave into 8 notes. Chordally, it creates 3 diminished-seventh chords of 4 notes each (3 X 4 = 12).

Now, we can proceed to see how Bartok used these ideas.


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## millionrainbows (Jun 23, 2012)

Bartok's use of the *pentatonic scale* is usually ascribed to his studies of folk music. Actually, the pentatonic is the prototype of our 3:2 based scale. If we begin on C and stack 3:2's (fifths), the first 5 pitches that appear are C-G-D-A-E. Rearranged into the octave, this yields the pentatonic scale C-D-E-G-A. This is really a part of the chromatic/geometric system, although it has acoustic features, if perfect 3:2's are used, and no key changes are demanded. Thus the universal, primal acoustic character of folk musics which use pentatonics.

If the pentatonic is also a part of the geometric/chromatic paradigm, then how, you may ask, is this related to the *GS* sequence? Seen on the chromatic circle, the pentatonic is 2+2+3+2+3+3 (C-D-E-G-A-C). Note that 3+2=5 (a Fibonacci number), and also, the Fibonacci number 8: the major third (C-E), when inverted, is a minor sixth, or a distance of 8 chromatic notes. So, we have the relations of 2+3, 5+3, and in its generation, 5+5+5+5...as fourths, the inversion of fifths. If you insist in going by fifths, 5X7=35. (a fifth is 7 half steps).

Going back to the "axis" system, let's see how the chromatic scale fits in to the three axes. There are 3 axes, or areas of tonality. If you draw these on to a circle of fifths, this is easier to see. The tonic axis is C-A-F#-Eb; the dominant axis is G-E-Db-Bb; the subdominant is F-D-B-Gb. This covers all 12 notes.

Going back to the "models," three different 1:2 models can be constructed:

tonic: C-C#-Eb-E-F#-G-A-Bb;
dominant: C#-D-E-F-G-Ab-Bb-B;
subdominant: D-Eb-F-F#-Ab-A-B-C.

These are the 3 different "diminished" or half-whole/whole-half scales. The symmetry is obvious, and each area has its own key-note and tri-tone counterpart (interchangeable tonics).


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## Minona (Mar 25, 2013)

Maybe I'm oversimplifying but... 1:1.618 isn't that far from 1:1.5, which is a perfect 5th.

Perhaps, the golden rectangle (for example) might be appealing because it's close to 1:1.5, that is one side is 1.5 x the size of the other, which I think is about right. I know I like music paper that size , but anyway... yes, couldn't the simple Perfect 5th or 12th explain the 'golden ratio' occurrences?


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## PetrB (Feb 28, 2012)

millionrainbows said:


> Regarding the Axis idea, which is only a small part of Bartok's system, we must imagine the circle of fifths; not a chromatic circle, since, appropriately, these Axis relations are based on harmonic principles as well.
> 
> On the circle, with C at the top as tonic, we can draw a straight diameter line to F#, which divides the octave into 6 + 6. Another diameter line, perpendicular to the first, connects A (clockwise from C) to Eb/D#. Note that these are tritone relations. These 4 points are "tonic" stations. These divide the octave into 4 parts. This corresponds to the "relative" majors and minors of classical music: C major and A minor, C minor and Eb major, etc.
> An intimate relationship exists between the opposite poles: C/F# and A/Eb. These pairs are interchangeable, like the inversion of a tri-tone.
> These minor-third relations are further reinforced by the Romantic period's use of "upper relatives," as in C maj/a min/Eb maj.


Bartok had _*a system?*_ How utterly tidy of him.


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## Eschbeg (Jul 25, 2012)

millionrainbows said:


> I cannot find an explicit statement by Bartok saying that he used Fibonacci numbers; but in Music for Strings, Percussion, and Celesta, the proportions speak for themselves: the first movement fugue (89 bars) are divided into sections of 55 and 34 bars by the peak. The section leading up to the climax (bar 55) show a division of 34 + 21. Those 21 bars are divided into 13 + 8.


Analysts who hear the Fibonacci sequence in this piece have never quite come to terms with how selectively the measures have to be counted in order to make the theory work. For example, contrary to common assumption, the first movement has 88 measures, not 89. In what I've read over the years I've seen two general methods for getting around this problem. The first way is to just pretend the movement really does have 89 measures. That's what Jonathan Kramer does, for example, in his book _Listen to the Music_, where he says something to the effect of "The movement has 89 measures: 88 actual bars plus the final silence," whatever that means. The second way is to count the eighth-note pick-up at the beginning of the movement as one of the measures. But if we go that route, a lot of the other Fibonacci moments in the movement get thrown off. The above-mentioned climax at m. 55, for example, is now at m. 56. (I'm charitably putting aside the more glaring problem that the true climax of the movement is pretty obviously the triple-forte 9/8 measure that _follows_ this measure, whichever way you count it.)

It is sometimes claimed that the voices of the fugue always enter in measure numbers falling within the Fibonacci sequence, but this too requires us to be pretty selective about how we define each entry. The second voice, for example, can be said to enter at m. 5, a Fibonacci number, _if_ we don't count the melody's pick-up. The third voice, by contrast, can be said to enter at m. 8, another Fibonacci number, only if we _do_ count that melody's pick-up.

When you compile all the inconsistencies, the Fibonacci series becomes a far less compelling analytical model for this piece than the axis system, which is not only consistent with the entry of fugal voices but is explicitly summarized, melodically, in the last three measures of the movement.


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## EdwardBast (Nov 25, 2013)

millionrainbows said:


> Going back to the "models," three different 1:2 models can be constructed:
> 
> tonic: C-C#-Eb-E-F#-G-A-Bb;
> dominant: C#-D-E-F-G-Ab-Bb-B;
> ...


The more common name for this configuration is the octatonic scale.


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## millionrainbows (Jun 23, 2012)

Well, Lendvai may be seeing things that would have been there already, but perhaps the Fibonacci sequence results whenever geometries are examined. In the case of our 12-note circle, this is an entirely fabricated geometric compromise. (Funny how Minona can see how close the Golden ratio is to a 3:2...



> Maybe I'm oversimplifying but... 1:1.618 isn't that far from 1:1.5, which is a perfect 5th.
> 
> Perhaps, the golden rectangle (for example) might be appealing because it's close to 1:1.5, that is one side is 1.5 x the size of the other, which I think is about right. I know I like music paper that size , but anyway... yes, couldn't the simple Perfect 5th or 12th explain the 'golden ratio' occurrences?


...yet Eschbeg is a non-believer because of an extra pick-up measure. For me, Lendvai makes an elegant case, especially in seeing the other aspects of Bartok's methods.

Most importantly, he grasps the "inner" nature of the chromatic axis, while recognizing the "outward" or expanding nature of the "diatonic" or acoustic scale of Bartok's, based on overtones, and sees how these invert yet complement each other.


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## Eschbeg (Jul 25, 2012)

millionrainbows said:


> ...yet Eschbeg is a non-believer because of an extra pick-up measure.


Nope. I'm a non-believer because the moments that emerge as structurally important according to the Fibonacci sequence don't align with the moments that are aurally important to the ear nor to what is notated in the score.


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## millionrainbows (Jun 23, 2012)

Eschbeg said:


> Nope. I'm a non-believer because the moments that emerge as structurally important according to the Fibonacci sequence don't align with the moments that are aurally important to the ear nor to what is notated in the score.


Not according to Lendvai. Remember the "peak" I mentioned in the earlier post #31 (above). Your refutations seem to all depend on pick-up measures.


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## Whistler Fred (Feb 6, 2014)

Simple soul that I am, I wonder if Bartok just had an instinct for good proportions in much of his music. Applying things like the Golden Section and the Fibonacci sequence may help explain why they are so satisfying. Or maybe not…


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## millionrainbows (Jun 23, 2012)

Whistler Fred said:


> Simple soul that I am, I wonder if Bartok just had an instinct for good proportions in much of his music. Applying things like the Golden Section and the Fibonacci sequence may help explain why they are so satisfying. Or maybe not…


What is meant by 'good proportions?' This could mean an acoustic truth, like small-number ratio intervals, or it could mean a geometric, numerical division of the octave. He has both.

But regardless of Eschbeg's statement that...

_"the moments that emerge as structurally important according to the Fibonacci sequence don't align with the moments that are aurally important to the ear nor to what is notated in the score,"

..._we must recognize that Bartok was not just 'composing by ear' when he wrote his music; if nothing else, please acknowledge the fact that Bartok was a musical thinker, and was using techniques of dividing the octave that were mathematical and geometric in nature, based on the number of chromatic notes (12) and its symmetries. This is common knowledge, and these sorts of ideas were a basic part of modern musical thinking.

I must conclude that Eschbeg's agenda here seems to be to "disavow' the use of mathematics in the precious 'aural' world of music, to 'save' it from the 'evils' of modernism.

A forelorn strategy, and way too late. If I have 'ruined' the fantasy of 'Bartok the artist' composing by ear and sheer artistry, with no systems or calculations, then so be it. This is common knowledge, and relates to all kinds of musical thinking, all through history.

Gee, I really dislike having to defend such basic 'givens' to those with 'noble agendas' of 'saving' the composers they like from being* 'mathematical thinkers'* instead of* 'sensitive, intuitive artists who do everything by instinct.'
*


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