# Bartok



## millionrainbows

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.

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; in an old post of mine from an Amazon thread, "Schoenberg's Op. 26 Wind Quintet", I pointed this out:
________________________________________

[The row is (first hexad) Eb-G-A-B-C#-C, which gives an augmented/whole-tone scale feel, with a "resolution" to C at the end, then (second hexad) Bb-D-E-F#-G#-F, which is very similar in its augmented/whole-tone scale structure, which only makes sense: there are only two whole-tone scales in the chromatic collection, each a chromatic half-step away from the other. I've heard Debussy use the two whole-tone scales in this manner, moving down a half-step to gain entry to the new key area. This is why Schoenberg used a "C" in the first hexad, and the "F" in the second; these are "gateways" into the chromatically adjacent scale area. Chromatic half-step relations like these can also be seen as "V-I" relations, when used as dual-identity "tri-tone substitutions" as explained following.
Another characteristic of whole-tone scales is their use (as in Thelonious Monk's idiosyncratic whole-tone run) as an altered dominant, or V chord. There is a tritone present, which creates a b7/3-3/b7 ambiguity, exploited by jazz players as "tri-tone substitution". The tritone (if viewed as b7-3 rather than I-b5) creates a constant harmonic movement, which is what chromatic jazzers, as well as German expressionists, are after.
So Schoenberg had several ideas in mind of the tonal implications when he chose this row.]
________________________________________________

Also, from this we can see that, 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.

Quoting Lendvai: "In fact, I am more critical of Schoenberg than I ever was before; his method treated dissonances like consonances, and renounced a tonal center. But dissonance is not the same as consonance; it has different acoustical and physiological effects. Therefore, dissonance ought not be treated as if it were identical with consonance. Plus, Schoenberg's renunciation of a tonal center does not follow from any previously stated proposition, and is merely an assertion of his dogmatic belief that the negation of tonality was 'historically inevitable.'"

However, that's a whole 'nother can of worms.

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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## Mahlerian

millionrainbows said:


> 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.
> 
> Quoting Lendvai: "In fact, I am more critical of Schoenberg than I ever was before; his method treated dissonances like consonances, and renounced a tonal center. But dissonance is not the same as consonance; it has different acoustical and physiological effects. Therefore, dissonance ought not be treated as if it were identical with consonance. Plus, Schoenberg's renunciation of a tonal center does not follow from any previously stated proposition, and is merely an assertion of his dogmatic belief that the negation of tonality was 'historically inevitable.'"
> 
> However, that's a whole 'nother can of worms.
> 
> 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."


You realize that these two quotes are in conflict with each other, right?

Schoenberg didn't renounce a tonal center in the sense that Lendvai means that Bartok has centers. He renounced a _key_, and that this is what he meant is clear from the passage that people cite in connection with the renunciation idea. Furthermore, he didn't treat all dissonances the same, just as he didn't treat all consonances the same. Of course a minor seventh has different properties from a perfect fifth. That is obvious.

Schoenberg, furthermore, did not believe in the negation of tonality, but in the expansion of tonal possibilities, just as Bartok did.


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## millionrainbows

Mahlerian said:


> Schoenberg didn't renounce a tonal center in the sense that Lendvai means that Bartok has centers. He renounced a _key_, and that this is what he meant is clear from the passage that people cite in connection with the renunciation idea.


Schoenberg was using tone rows, not scales, so the net result is atonal, regardless of key or no key. *Scales are an inherently tonal device, *since they are an 'index' of notes with no order (unlike ordered rows), and have a starting point, which also suggests tonality. Most usual scales cover an octave, which is also a suggestion of tonality. Tone row pitches do not suggest any particular octave, or even register, which is why 12-tone and atonal music has such wide, melody-destroying leaps.

By contrast, Bartok's devices are quasi-tonal by nature, in that they divide the octave in various ways (whole tone and diminished scales), and that these divisions have patterns which can be seen as starting points or reference points which govern smaller in-octave areas; 2, 3, 4, 6, etc. This is a form of 'micro-tonality.'

Schoenberg's tone rows have nothing to do with the inherently tonal notion of 'dividing the octave.' They are isolated series of notes unrelated to any notes except the one preceding and the one following ('related only to each other' by his definition).



> Furthermore, he didn't treat all dissonances the same, just as he didn't treat all consonances the same. Of course a minor seventh has different properties from a perfect fifth. That is obvious.


A *dissonance, or consonance* (inclusively known as *'sonance'*) *is an inherently tonal idea.* The notion of sonance is a matter of relative degree, and relates each interval to a reference of "1" or the key reference note. This is why sonance, as an interval, is often expressed as fractions or ratios. This is a _comparative_ concept.

Schoenberg's tone rows do not use sonance in reference to a '1' or key reference note, so they cannot be expressed as fractions in terms of consonance or dissonance; they stand alone as 'sonances,' _without comparison in terms of which is more or less consonant or dissonant.
_This is what is meant when they say that Schoenberg treated dissonance and consonance objectively, or as if there was no difference, since to compare the two is an inherently tonal idea.



> Schoenberg, furthermore, did not believe in the negation of tonality, but in the expansion of tonal possibilities, just as Bartok did.


When ordered rows are used as the basis of the music, then by my criteria, the music is 'atonal' or not tonal.


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## Mahlerian

millionrainbows said:


> Then ordered rows are used as the basis of the music, then by my criteria, the music is 'atonal' or not tonal.


Only if you show contempt for logic. You haven't proven anything, you've just stated that it's atonal, repeatedly. You've even ignored the fact that the paragraph you quoted says that there's no fundamental difference between Bartok's division of the octave and Schoenberg's serial methods.


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## millionrainbows

Mahlerian said:


> Only if you show contempt for logic. You haven't proven anything, you've just stated that it's atonal, repeatedly. You've even ignored the fact that...


Ad hominem, irrelevant.



> ...the paragraph you quoted says that there's no fundamental difference between Bartok's division of the octave and Schoenberg's serial methods.


No. It says there are commonalities in "modern" musical thinking such as _"the individual concepts of permutation, inversional symmetry, invariance under transformation, etc....." _These are things that pure serial thinking shares with advanced tonal thinking, a concept I have long championed.

Again, "dissonance" and "consonance" are comparative terms, and only work if there is a reference to "1" or a tonic note. That's why intervals are expressed as fractions: they are not absolute quantities, but are relationships. This is basic math.

It's a little misleading to say, in tonal lingo, that Schoenberg treated dissonance as if it were consonance, in the sense that these terms do not apply to his materials, which are tone-rows.

It would be more accurate to say that he did not even "treat" them at all. The idea of dissonance does not apply in the world of 12-tone and serialism. Only the intervals between notes of the row create sonance, and these are referenced only to each other.


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## Mahlerian

millionrainbows said:


> Ad hominem, irrelevant.


It's neither an ad hominem nor irrelevant. Your entire argument is just stating your conclusion repeatedly, or otherwise rewording it as a premise. That doesn't prove anything, unless you show contempt for logic.



millionrainbows said:


> No. It says there are commonalities in "modern" musical thinking such as _"the individual concepts of permutation, inversional symmetry, invariance under transformation, etc....." _These are things that pure serial thinking shares with advanced tonal thinking, a concept I have long championed.


It says "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."

I'd like to hear how quoting that supports the idea that there is some separation between tonality in Stravinsky, Debussy, and Bartok, and atonality in Schoenberg, Berg, and Webern. I have long been of the opinion that there is no way to make this idea square up. Either both groups are tonal or both are non-tonal, but you can't separate them on that basis.



millionrainbows said:


> Again, "dissonance" and "consonance" are comparative terms, and only work if there is a reference to "1" or a tonic note. That's why intervals are expressed as fractions: they are not absolute quantities, but are relationships. This is basic math.


You're confusing two different kinds of dissonance here. There's structural dissonance, which is why a D-flat major triad can be dissonant in the context of a C major piece, and there's dissonance as a function of the immediate harmony. In the latter sense, dissonances (seconds, fourths, sevenths, augmented/diminished intervals) are always dissonant, and consonances (thirds, fifths, sixths, octaves) always consonant, although context may make them sound more or less harsh.



millionrainbows said:


> It's a little misleading to say, in tonal lingo, that Schoenberg treated dissonance as if it were consonance, in the sense that these terms do not apply to his materials, which are tone-rows.
> 
> It would be more accurate to say that he did not even "treat" them at all. The idea of dissonance does not apply in the world of 12-tone and serialism. Only the intervals between notes of the row create sonance, and these are referenced only to each other.


You're treating the material of the work as if it were identical to the finished piece. Schoenberg's works are not made up of tone rows, tone rows are used to generate the works themselves. In the abstract, a tone row may not have consonance or dissonance, but in the context of a piece of music it can be used to create harmony and harmonic motion, and this is what composers such as Schoenberg did.


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## millionrainbows

Mahlerian said:


> It says "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."
> 
> I'd like to hear how quoting that supports the idea that there is some separation between tonality in Stravinsky, Debussy, and Bartok, and atonality in Schoenberg, Berg, and Webern. I have long been of the opinion that there is no way to make this idea square up. Either both groups are tonal or both are non-tonal, but you can't separate them on that basis.


Serialism is the clean break from tonality. Once you cross that line, you are in atonal territory. Of course, serialism shares many of the same devices as modern tonal music; _"the individual concepts of permutation, inversional symmetry, invariance under transformation, etc....."_



> You're treating the material of the work as if it were identical to the finished piece. Schoenberg's works are not made up of tone rows, tone rows are used to generate the works themselves. In the abstract, a tone row may not have consonance or dissonance, but in the context of a piece of music it can be used to create harmony and harmonic motion, and this is what composers such as Schoenberg did.


No, he used tone rows according to their internal structure, and if it created a harmonic effect, this is a result of the tone row, not "harmony and harmonic motion."


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## Mahlerian

millionrainbows said:


> Serialism is the clean break from tonality. Once you cross that line, you are in atonal territory. Of course, serialism shares many of the same devices as modern tonal music; _"the individual concepts of permutation, inversional symmetry, invariance under transformation, etc....."_


But there is no clean break from tonality in serialism. It's an expansion and crystallization of principles of harmonic/melodic coordination that had been in use for at least two decades, and the quote *you cited* says exactly that.



millionrainbows said:


> No, he used tone rows according to their internal structure, and if it created a harmonic effect, this is a result of the tone row, not "harmony and harmonic motion."


Again, you haven't shown that there is any contradiction between tone rows and harmony, you just keep saying it over and over.

Any use of a group of tones is harmonic, either by implication (horizontally) or direct statement (vertically). That's my position, and in response, your only recourse has been to say "tone rows are ordered sets." You could theoretically fabricate a row that produced the entirety of Bach's Air from the Orchestral Suite in D and call it an ordered set, and by god it would be one, stated once, and still wouldn't contradict either the principle of ordered sets or that of functional tonal harmony.


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## millionrainbows

Mahlerian said:


> But there is no clean break from tonality in serialism. It's an expansion and crystallization of principles of harmonic/melodic coordination that had been in use for at least two decades, and the quote *you cited* says exactly that.


No; you are confusing the commonalities of modern thinking which expanded tonality _shares_ with serial thinking; the clean break does occur, in the form of ordered rows.



> Again, you haven't shown that there is any contradiction between tone rows and harmony, you just keep saying it over and over.


I ask you to do the same; prove it by assertion.



> Any use of a group of tones is harmonic, either by implication (horizontally) or direct statement (vertically).


Sure, any set of tones has a harmonic presence, but this does not mean they are derived out of tonal considerations of consonance/dissonance, and the lack of harmonic context (harmonic goals derived from tonal function, harmonic goals established in musical space, voice-leading which enables the transition from one goal to the next) strengthens the argument that to be tonal, these conditions must be present and demonstrable.



> That's my position, and in response, your only recourse has been to say "tone rows are ordered sets." You could theoretically fabricate a row that produced the entirety of Bach's Air from the Orchestral Suite in D and call it an ordered set, and by god it would be one, stated once, and still wouldn't contradict either the principle of ordered sets or that of functional tonal harmony.


I've said more than that; namely, that the tonal language has pervasive stuctural elements which create tonality. These elements are: harmonic goals derived from tonal function, harmonic goals established in musical space, and voice-leading which enables the transition from one goal to the next.

!2-tone and serial music, by contrast, are motive-driven, and while atonal methods have much in common with tonality and advanced tonality (inversion, retrograde, etc.), they are revealed to be atonal by the way they are used: in a non-tonal way.

As an example, you could take Bach's Prelude No. 1 in C major from the WTC book one, and invert some of the phrases, retrograde them, etc, but unless this was done in a way that is consistent with tonal harmony, the result will not be tonal, nor will it sound tonal, and this will be immediately obvious to any listener.


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