# Understanding Bartók



## millionrainbows (Jun 23, 2012)

I'm skimming through "Bartok: An Analysis of his Music" by Elliott Antokoletz, and there's an interesting chapter called 'Basic Principles of Symmetrical Pitch Construction."

It states, basically, that traditional Western music was based on an uneven division of the octave, namely the perfect fourth and fifth.

Look at all the intervals: all of them have complementary intervals which add up to an octave (min. 3rd/maj. 6th, etc.), and the smaller of these two complements generates a cycle which divides the octave symmetrically: one cycle of m2, two cycles of M2, three of m3, four cycles of M3, and six cycles of tritones; except the p4 and p5: this complementary interval does not generate a cycle which divides the octave symmetrically, but must extend through many octaves in order to reach its initial starting point again. Thus, there is only one cycle of perfect fourths, or perfect fifths.

In terms of pure set theory, the reason that perfect fourths and fifths behave this way is that 5 (a perfect fourth is five half steps) and 7 (a perfect fifth is seven half steps) are not divisors of 12...neither 5 nor 7 go into 12; 
until:

5 goes into 60, a multiple of 12 (circle of fourths, five octaves: C F Bb Eb Ab Db Gb B E A D G)

7 goes into 84, a multiple of 12 (circle of fifths, seven octaves: C G D A E B Gb Db Ab Eb Bb F)..."

The reason why this 'difference' of fourths and fifths was brought up is because the author of the Bartok book is saying that Bartok based his music on an even division of the octave, namely, the tritone. 

From a perspective of pure arithmetic, the octave can be seen as 'unity.' The octave, without regard to register, in terms of pitch identity and relation to a 'root,' can be called '1' or unity. On a number line, anything less than one, proceeding back to zero (infinity), is fractional. Anything larger than one proceeds forward, into the 'other' infinity of octaves.

Perhaps this is why the 4th & 5th are different; instead of dividing the octave fractionally, they are expansive by nature; they go 'outward' past one, past the octave, into other 'root' stations. Hence, the use of 4ths & 5ths to create root movement.

Every interval has its complement. All the intervals except perfect fourths & fifths have a smaller number which divides the octave (12) symmetrically; 

So each interval has 2 numbers which add up to an octave. 

The m2 has itself 1 and 11; 
M2 is 2 and 10; 
m3 is 3 and 9; 
M3 is 4 and 8;
p4 is 5 and 7;
tritone is 6 and 6;
p5 is 7 and 5;
m6 is 8 and 4;
M6 is 9 and 3;
m7 is 10 and 2;
and M7 is 11 and 1.

You can see the symmetry in this; and if we eliminate the redundancies, such as 10-2/2-10, we have 6 essential intervals.

Again, neither 5 nor 7 go into 12; until
5 goes into 60, a multiple of 12 (circle of fourths, five octaves: C F Bb Eb Ab Db Gb B E A D G)
7 goes into 84, a multiple of 12 (circle of fifths, seven octaves: C G D A E B Gb Db Ab Eb Bb F)"

The fourth and fifth, as pointed out, cannot be used as divisors of 12 (the octave); therefore, they can be seen as "expanding" in nature, as they generate cycles of 12 notes (outside the octave). Remember, 60 and 84 had to be used as the common denominators for 5 and 7. These large numbers can be seen as 'outside the octave' or as a 'greater referential point.' Hence, the reason the 4th and 5th are the basis of traditional Western music; this facilitates movement outside the octave, to a new reference point or new key.

This means that 'modern' music, like Bartok's, is 'inward-going' or 'introspective' if you like to indulge in metaphor (after all, this is art, not science). This is what Marshall McLuhan was getting at in his book "Through the Vanishing Point," in which he explains how our perspective on things is literally reversed in modern art, putting us at the other end of the 'vanishing point.' Like looking down the wrong end of a telescope, or rather a microscope, the 'inner' world now becomes our universe, heading towards the 'other infinity' towards zero; just like our number line, where anything less than one, proceeding back to zero (infinity), is fractional, and anything larger than one proceeds forward, into the 'other' infinity of octaves.

The Ernö Lendvai book deals a lot with the 'meta-concepts' of Bartók's methods. It generalizes to a great extent, and is not a very lengthy book, but it states the case elegantly, and it is a beautiful book. It divides Bartók's ideas into two main categories: the 'GS' approach, which has to do with the "Golden Section" and the Fibonacci series, and is also called his 'chromatic system'; and Bartók's 'diatonic system,' which is based on acoustic principles.

The beauty of all this is that the two approaches reflect each other in an inverse relationship.

In this quote by Ernö Lendvai, he reveals the most profound aspect of Bartók's system:

"A secret of Bartók's music, and perhaps the most profound, is that the 'closed' world of the GS (Golden Section) (1,2,3 and 6 being 'closed' or 'inward-directed' intervals, as opposed to 4ths and 5ths) is counterbalanced by the 'open' sphere of the acoustic system. The former always pre-supposes the presence of the complete system -- it is not accidental that we have always depicted chromatic formations in the closed circle of fifths. In the last, all relations are dependent on one tone since the natural sequence of overtones emerges from one single root: therefore it is open. Thus, the diatonic system has a fundamental 'root' note, and the chromatic system a 'central' note...Bartók's GS system always involves the concentric expansion or contraction of intervals..."

So we can see from this exposition of the intervals that modern music started moving away from traditional tonality by way of exploiting the INHERENT SYMMETRIES in the 12-note scale. 

In the bigger picture, what these small, recursive intervals do is allow the creation of pitch cells; these are aggregates of notes which expand around an axis of symmetry. Thus, localized areas of tonal centricity can be created on any note. 

An analogy would be, traditional tonality is like a tree which grows up in one direction from one 'rooted' spot; in the chromatic approach, tonality becomes radiant 'flowers' of pitch, centering on any possible note in the vertical spectrum.

Another aspect of Bartók's approach which has puzzled many is the fact that he still uses the fifth & fourth as generators of traditional tonality, sometimes mixing the two approaches.

All of these ideas were 'in the air' so to speak, around the turn of the century, and were not unique to Bartók; examples of symmetry began showing up as early as R. Strauss, in his 'Elektra' and 'Metamorphosen,' before he retreated back into conservative classicism. Debussy, as most of us know, used the whole-tone scale in his music, most notably the prelude 'Voiles' from Book I. The 6-note whole-tone scale itself is a symmetrical projection of the major second, and there are only two of them; Debussy exploits this characteristic to create 2 areas of contrasting tonality. Schoenberg was influenced by this idea as well.


Historically, it was the tritone (in both V7-I's and in diminished seventh chords) which was the first emergent symmetry which led to the expansion of tonality; this interval was the color tone in the V7-I progression, being the major third and flat-seven, which would then exchange places for the next cycle. This gave rise to new roots, moving chromatically instead of by fifths. This was tied-in (as mentioned above) with 'flat-nine' dominant altered chords, which are closely related to the diminished seventh. The use of 'flat-nine dominants' as true V chords appears as early as Beethoven and Bach. The vii degree of the major scale, a diminished triad, has always been treated as an incomplete dominant ninth with G as the 'imaginary' root, and resolved as a V7 chord would be (to C).
So, it can be seen from all this that 'tonality' underwent great changes around the dawn of the 20th century; and one should not confuse this expanded chromatic version of tonality with Schoenberg's 12-tone method, which just confuses the issue.

In closing, this quote by co-authors George Perle and Paul Lansky:

"Perhaps the most important influence of Schoenberg's method is not the 12-note idea itself, but along with it the individual concepts of permutation, inversional symmetry, invariance under transformation, etc.....Each of these ideas by itself, or in conjunction with many others, is focused upon with varying degrees...by...Bartók, Stravinsky, Berg, Webern, Varèse, etc...In this sense the development of the serial idea may be viewed not as a radical break with the past but as an especially brilliant coordination of musical ideas which had developed in the course of recent history. The symmetrical divisions of the octave so often found in Liszt and Wagner, for example, are not momentary abberations in tonal music which led to its ultimate destruction, but, rather, important musical ideas which, in defying integration into a given concept of a musical language, challenged the boundaries of that language."


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