# BARTOK, part I



## millionrainbows (Jun 23, 2012)

Bartók can be hard to pin-down, because he used an arsenal of different techniques in composing. To some, his string quartets are the way in; to others, like me, the orchestral works such as "Music for Percussion..." that Sid James mentioned earlier are the key.

The "creepiness" that many listeners cite is due to the fact of Bartók's use of small intervals, as well as use of the diminished scale (symmetrical in nature & dividing the octave in half via the tritone).

Here is my long-winded explanation of Bartók, which you can take or leave, which I posted elsewhere: 

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.

This is about music, and the nature of tonality. The ideas I put forth about intervals, although fairly simple in one sense, are laying the groundwork for a larger, more all-encompassing understanding of tonality and chromaticism. I see it as a necessary reference to the ideas which will follow. The 'simple arithmetic' of intervals is necessary, in case some of these ideas about intervals & reciprocals might not be fully 'grokked' by some readers.

It's simple, and it's complicated, all at the same time; but after a thorough pondering and practical application (in composition) of Howard Hanson's ideas of interval projection, I decided it was time to tackle the ideas of another musical giant: Bartók, and what a revelation it has been! Especially the little book by Ernö Lendvai, which I highly recommend, that is, IF you are sufficiently prepared to read it. Some knowledge of intervals & reciprocals is necessary.
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.


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