# Symmetry, Bartok etc...



## smackhead999 (May 31, 2013)

I had an idea yesterday about something that I wanted to try, so I started researching it, assuming that it must have been used before if I had conceived it. Realized the idea was basically polymodal chromaticism. Interesting read.

Spent some time reading about Bartoks obsession with symmetry. The concept of polymodal chromaticism was pretty easy to grasp. While in the mood, I took another look at the pitch axis and its symmetry. Still not totally grasping the concept of this. I think the biggest problem with that is that a lot of people are writing about it on the internet, and dont really understand it themselves. I have searched on here and am still not comfortably wrapped around it. I will keep working on trying to grasp this one.

As far as the polymodal concept goes, that was fun last night. In about 5 minutes I whipped up a cool, progressive movement. I asked the old lady what she thought... and she said it sounded good but could not make out the back and forth between the modes, but that she knew it sounded different than what I usually play. So, I feel that I succeeded. It sounded right, so I guess I pulled it off okay, and it was very different than my comfort zone. I really liked it.

Most of the references to this polymodal counterpoint I see are assuming two melodic lines. Counterpoint movements. One ascending in lydian, one descending in phrygian. What I was doing was obviously one instrument, one melodic line at a time. So, I basically broke up the counterpoint into phrases. Ascending phrases in lydian, descending phrases in phrygian, tonal center of D. Occasionally I would break the rule and just use a chromatic step motion, particularly for the one note that was not in either mode, B/Ab (edit: me thinks I was failing on the authentic lydian spelling..oops. That explains why there was only 11 notes), which certainly had its own way of taking the focus I guess.

I also attempted some chord changes following the same rule, but was less impressed with that. I used a VI-i followed by a i-vi, which now realize would have sounded better had I used a chord from the lydian scale before moving to vi. This will take more work. Jumping to a phrygian chord movement following a lydian phrase was cool and vice versa, basically just bouncing back and forth, adhering to the ascending/descending concept.

I think I need to go back and read up on the counterpoint species again. Been a while.

Most of the examples for this all used the lydian/phrygian example, but I dont see why you couldnt use any mirrored modes that had a tonal center and some shared notes, I guess particularly the dominant. Also curious about linking these concepts together, such as transposing into a new tonal center with a dominant that is present in the previous mode with a different tonal center... basically utilizing say... 4 modes, with shared notes between which ever mode is being transitioned into and from. Obviously you couldnt have a shared dominant from two modes and then transpose to another tonal center and still have a common dominant, but, maybe a mediant... I dont know.


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## Osorio (May 31, 2013)

Care to post the things you came up with? Either sheet or some clips?

I'm really interested in this topic. My harmony studies are finally getting to the point where I can sort of understand what this would be about, but not enough that I feel comfortable in saying or debating anything about it. 
Eagerly waiting for SchecterWhore to descend upon this topic as he usually does. Feel like some very interesting contributions could spring from this.


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## smackhead999 (May 31, 2013)

I would have to transcribe it. I guess I could figure out how to record it and put it up. The old lady has guitar pro, and I can try that. Let me find some time.

My understanding of harmony is... well, could certainly use help too.

Interested in whether SW tells me Im right on or totally off base.


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## Osorio (May 31, 2013)

No rush, I just got curious. I believe it could enrich the discussion as well, since we will have something to look at or hear. Guitar pro is probably a good idea if you can transcribe it.

My harmony knowledge is FINALLY getting out of the "mediocre" zone. I'm, at last, diving into chromaticism and altered chords, stuff that I know almost nothing about. A book that has really helped me get to grasp with most stuff regarding harmony was a SW recommendation (that I pass on almost every time some brings up wanting to learn theory), which was "Tonal Harmony" by Kostka and Payne. I have a Counterpoint book here too, but I haven't looked to deeply into it yet. My knowledge of Harmony always seemed so lacking that I never got into it... It's kind hard to compose interesting counterpoint with non-sensical harmonic progressions. I look forward to getting the Harmony stuff done with and being able to dive into Counterpoint, because, as a concept, I really like the idea and would love to implement it more in the stuff I write.


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## smackhead999 (May 31, 2013)

My original idea I guess was more about using different modes to capture a particular "feel" in different phrases, alternating between phrases. This is kind of right on with Bartok's concept, but I realized that I was lacking the cohesiveness of some symmetrical structure. Rules so to speak. It sounds really crappy if you just go noodling through modes, but with a symmetrical set of rules applied to the alternations in modes, it really keeps things together. Maintaining that tonic and dominant keep both modes familiar, but the other notes being polar opposites allows each phrase to develop its own thing.

I really need to spend some time and hash out the modes and their mirror opposites so that I can expand rather than have a black and white, back and forth conversation with the composition.


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## Mr. Big Noodles (May 31, 2013)

< Bartók fanboi.

The first thing I would suggest is to get hold of the six volumes of the Mikrokosmos. He lays out his compositional style in a very logical manner, and if you follow along, you can learn more from analyzing those pieces than you can from what any words can describe. Next, we have to lay aside some preconceptions. You need a good solid understanding of harmony to understand Bartók, but that's not because he is going to use chord progressions like you're taught to think of them. His music very often exists in a world of expanded tonality, where you find vestigial remains of the old tonal system (as far as classical music goes), recontextualized to work in his polytonal/polymodal style. That means you need to know how Common Practice Period harmony works, so that you can see some weird thing that Mozart would never use (although his later works have some very strange moments) and say "That's like the thing that Mozart used". His harmony comes from a zoomed-in view on intervallic relationships. Counterpoint, too, is approached differently in his music versus the classical model: whereas doing species counterpoint will help you to understand Bach, Haydn, Mozart, Beethoven, and Brahms, it doesn't have quite as much application in Bartók. Still, it is an outcropping of those melodic styles (particularly of J.S. Bach, and later from Richard Strauss and Richard Wagner), so you can expect to find some similarities between his counterpoint and the old counterpoint. I actually find Bartók's counterpoint to be much more intuitive than counterpoint à la Gradus Ad Parnassum. He really did a lot to develop the traditional tonal language beyond its key-centric origins, but always by building on top of the old ways.

Bartók wrote the Mikrokosmos to initially teach his son the piano, so it is a pedagogical text in a way. The first book opens up with a bunch of "unison studies", meaning one melody played in octaves, which isn't the greatest thing in the world, but they do expose us to short compositions that use pentachords as a basis of melodic material, which is something that he expands upon greatly. Following those initial unison studies are pieces that are in the same vein, but working with repeated notes, different rhythms, alternating hands, and parallel motion. Really basic stuff. Where it kicks off is when he starts getting into contrary motion.

Béla Bartók - Mikrokosmos No.12








Looking at the first harmonic interval on this one, you're probably thinking "Woah, that's weird, it starts on a second." There is a compositional reason for that interval: it comes from the organization of the pitch material into tetrachords and pentachords, an approach I advocated for modal writing in a recent thread. Bartók wanted the unison D on the downbeat of the second measure, so he backtracked down the tetrachords of D dorian to approach the D from below in the soprano and from above in the bass, leaving us with a second when he gets to the end of the process. Contrary motion and reflection factor prominently into his tonal and contrapuntal style. Here is the next piece like that:

Mikrokosmos No.17

(Starting at 8:52)






This is where Bartók's polymodal/polytonal writing starts to become apparent. The same device is used here as in the previous study - contrary motion. Let's examine the soprano line. The first four measures are a C major pentachord, played ascending then descending. Measures 5-8 are a G major pentachord, done the same way. Measures 9 and 10 are the upper tetrachord of C major, ascending. Measures 11-12 are in C major, but using G as a pitch axis for a descending tetrachord. Then the last four measures are a recapitulation of the first four. In terms of harmony, there is a very clear link to Common Practice Period music: there is an exposition in the tonic key (1-4), a modulation to the dominant key (5-8), an episode on the dominant chord (9-12), and a recapitulation in the tonic key (13-16). It's a hypercompressed ABA form. The weird part is the left hand, which can be viewed two ways: it is the reflection of what the right hand is doing with the same pitch axes (1-4 is C, 5-8 is G, 9-16 is C), but in the key of the right hand's dominant, or it could be the parallel lydian mode of whatever's going on in the right hand. Either way, we get some polymodal stuff going on.

Mikrokosmos No.23







Everything that we've looked at so far is homorhythmic, meaning that the two voices line up on the same rhythm. What Bartók does to show us how his counterpoint is conceived is to displace one of the voices. This piece has imitation at the unison, and has the same melody in both voices, but displaced by two beats from each other. Obviously, they come together at cadences, but the idea is basically there. I don't know where the "inversion" part of "Imitation and Inversion" is coming from, maybe an erratum on the publisher's part, or possibly from Mr. B himself.

Mikrokosmos No.29







We have more imitation here, but the bass line is more free. Notice that the bass voice is the inverted version of the soprano voice. The first figure has intervallic significance when it is imitated: the soprano starts by neighboring that E with an upper major second, then there is a major trichord (all major seconds). When imitated in the bass, the E is neighbored by a lower major second, then the scalar part is also composed entirely of major seconds. The harmonic effect is one of polymodality, since the bass has the upper tetrachord of E minor. Some might be tempted to analyze that as a single pitch collection, and say that it's the "melodic major scale" (an awful term for a scale), or as Joe Satriani calls it, "aeolian dominant". Don't do this. Bartók was thinking intervals, trichords, tetrachords, pentachords, inversion, and imitation. You do yourself a disservice when analyzing music that is not scale-based with scales. And really, saying that something is "X scale" is usually ignorant of any harmonic underpinnings that are more important anyway.

The bass voice breaks off from the E pitch axis in measure 5, moving down to C while the soprano stays on E. You could argue for polytonality at this point, but the two pitch sets never overlap: we don't have G and G# happening to tell us whether the left hand is truly in E minor, and there's no superimposition of F and F# to suggest that both C major and E major are going on at the same time, so it's a bit fuzzy. The two voices are firmly confined from each other behind the wall of B. Take a look at what happens from measure 11 to the end. In 11, we have scales in contrary motion approaching an octave B, the dominant of E. From there, the soprano has a descending figure from E major's lower pentachord while the bass has an ascending figure from E minor's upper tetrachord, cadencing on the octave E. There is a clear I V I thing going on, for all you Schenkerians out there, but it is obscured by the polytonal and polymodal treatment of the pitch material.

Mikrokosmos No.70







Polymodal and polytonal inversion play out on a larger level, too. Take note of the key signatures in this piece. The two hands are stratified into two separate pitch collections. There are two ideas happening: a melody and an accompaniment figure, though not like this. The melody, first heard in the right hand, is modally ambiguous, consisting of the degrees 1 2 4 5 (if 1 is F#). That same melody migrates to the pitch center of A in the left hand in measure 10, where it is inverted (making the pitch material 1 &#9837;7 5 4, if 1 is A). The accompaniment is itself a mirror-inverted figure, also modally ambiguous as well as tonally ambiguous, but we can look at it in context and say that the accompaniment belongs to the same pitch center as the melody that goes with that hand. This is what's going on episodically:

1-9: Melody (prime form) in right hand, accompaniment in left hand.
10-16: Melody (inverted form) in left hand, accompaniment in right hand.
16-19: Inverted melody in C# in right hand, prime melody in A in left hand.
20-23: Accompaniment in different pitch areas in both hands. Uh, D&#8710;? Sometimes his polytonal explorations reveal a conglomerate pitch idea at the end, in this case a tertian sonority that is conceived of as two perfect fifths.

In terms of composition, it's pretty clear that he's presenting each idea in all of its configurations (melody vs. accompaniment, melody vs. melody, accompaniment vs. accompaniment, switching between prime and inverted, switching hands, etc.). He does this to make a longer form out of short ideas.

The lydian/phrygian polymodal chromaticism thing is hard to find an example of in the Mikrokosmos. He does a lot of descending phrygian tetrachords and ascending major tetrachords/trichords (I'm looking at 123 now, which has this happening in measure 16), but full scales are difficult to point to in his smaller works. Don't get me wrong, different ascending and descending pitches are very much what he's about (what my composition teacher calls "inverted chromaticism" in the context of Bartók), but major/minor or other scalar hybrids seem to be more commonplace. Nevertheless, I know of an instance in the Six Bulgarian Dances where you can find lydian/phrygian polymodal chromaticism:

Mikrokosmos No.148-153


Skip to 4:30.






The rest of the score: Page 2, page 3, page 4.

Béla Bartók on piano. His interpretation is different than pretty much everybody that has played this after him. In measure 17, there is an ascending A lydian figure, and in measure 18, there is a descending A phrygian (edit:locrian*; Missed it the first time around. Fairly close, but still.) figure. A quick glance tells me that it only happens once. Kind of underwhelming, huh? The fact that it took us this long to find an example of an ascending lydian scale followed by a descending phrygian scale in Bartók's music, and that it's crammed all the way in the back of the last volume of the Mikrokosmos (there are 153 pieces in total) might tell us a few things. 1.) We're looking in the wrong place. I am familiar with the Mikrokosmos, but he wrote a decent amount of student pieces for piano and other instruments, and there are many other pieces along the way that I have not taken my microscope to. 2.) It's a fancy term, but ultimately nothing to dwell on for too long. Bartók did not use complementary interlocking lydian and phrygian scales as a large part of his compositional language. Maybe the terminology is too literal and overstates the importance of the technique. 3.) Either that, or Bartók did not experiment with this technique until later in his life. We're seeing it in small form at the end of the last Mikrokosmos book, which was published in 1939, and he died in 1945. It might be worthwhile to leaf through the pages of some of his later and larger scores to find more developed usage of lydian/phrygian polymodal chromaticism.

* Look at the intervals. Ascending:


```
A-C# : M3
C#-D# : M2
D#-F# : m3
F#-G# : M2
G#-A : m2
```
Descending:


```
A-F : M3
F-E&#9837; : M2
E&#9837;- C : m3
C-B&#9837; : M2
B&#9837;-A : m2
```
Chromatic inversion. 

Music for Strings, Percussion, & Celesta - Mvt.2







This is the first page of the second movement of Music for Strings, Percussion, and Celesta (1937). In measure 5, there is an ascending scale going C D E F F#. That sort of meets our lydian requirement. The descent is E&#9837; D&#9837; C. That's about as phrygian as they come, and he is filling in the spaces on the descent. Still not the pure example I'm looking for, but if you scoured the movement, you might be able to find the lydian/phrygian thing more complete, as half of the music is ascending and descending scales.

I'll keep looking around. Bartók is my obsession, so please ask any questions you might have.


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## ElRay (Jun 1, 2013)

Big Bartok Fans here too. "Hungarian Folk Song" is one of my daughter's favorite pieces. I've heard one version of this on Classical guitar (Skip to 1:06)
​
but I haven't been able to find the arrangement. That's in my "To Do" file.

The IMSLP has PDF's of all six books: http://imslp.org/wiki/Mikrokosmos,_Sz.107_(Bartók,_Béla)

And Last.fm has about 96 of the pieces: http://www.last.fm/music/Béla+Bartók/BARTOK:+Mikrokosmos+(Complete)

Ray


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## smackhead999 (Jun 1, 2013)

Dude, that is a wicked long, and extremely in depth explanation. I would not have given that much effort.

However, I listened and read it all and it further reinforces the end that I have come to tonight. It follows:

What I have to keep reminding myself, is that in the method of Bartok, note labels, the existence of scales, their values and weight, and generally anything nominal about sound is pointless... save a method to communicate. Because what is important is the symmetrical collection around a focal point and the relationship between items in one collection, themselves, but also their symmetrical mates, opposite the focus.

Before your lengthy response, I started pondering how I kept seeing pentachords etc... coming up, that were second intervals apart, and what the deal was. I realized, I was still thinking in classical thirds harmony. A movement on one hand creating a pentachord is somewhat irrelevant unless you consider the relation with the other hand and what harmonies that makes. It doesn't really matter that one is Lydian and the other phrygian, but rather the relationship between the intervals. You could effectively make up some scales, as long as all 12 Intervals are covered and there is a center for focus.


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## smackhead999 (Jun 1, 2013)

Contemplating the various qualities of harmonies possible as well as building and releasing tension. I think my brain might explode.


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## Mr. Big Noodles (Jun 1, 2013)

ElRay said:


> Big Bartok Fans here too. "Hungarian Folk Song" is one of my daughter's favorite pieces.



Your daughter is going to be such a badass, dude.



> The IMSLP has PDF's of all six books: http://imslp.org/wiki/Mikrokosmos,_Sz.107_(Bartók,_Béla)


Not if you're in the US. Bartók's still under copyright. I know, because that's the only reason I have bookmarks for websites in languages I don't understand.



smackhead999 said:


> Dude, that is a wicked long, and extremely in depth explanation. I would not have given that much effort.



Well, sure, but this is me we're talking about. 

I remember that I encountered lydian/phrygian polymodal stuff in the fourth movement of Strings, Percussion, & Celesta, but I don't recall exactly where. However, you're right that it's not so much that there's a lydian scale and then a phrygian scale, but more than that, he's trying to find new ways to use chromaticism. Bartók's also paying attention to Schönberg - both of them wanted ways to include all twelve chromatic pitches in their music, so they invent systems that encourage a motivic approach to chromatic writing.

The cool thing about lydian and phrygian is that that pairing not only contains the complete chromatic set, but also preserves the tonic and dominant tones. C D E F# G A B C B&#9837; A&#9837; G F E&#9837; D&#9837; C. The more I study post-tonal music, the more I shy away from terms like "atonal". A lot of the big composers that are labelled as atonalists, or whatever, are really very tonal; they just use many little pockets of tonality rather than making big statements in a single key. Particularly with Bartók, everything is pentachords and motives that have their own tonal areas, but superimposed onto other pentachords and motives in other tonal areas. Anton Webern, the exemplar of twelve-tone technique, worked hard to make sure his music was dense with interlocking motivic statements. After getting past the stupid twelve-tone matrices, you realize that Symphony op.21 has some damn good writing.


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## ElRay (Jun 1, 2013)

SchecterWhore said:


> Your daughter is going to be such a badass, dude.


Danka. They both are. My Wife&I try hard. I'm working with the youngest to use the 3rd string B in one of her pieces so she can put vibrato on the 1/4 notes at the end of the phrases -- Then she be about as badass as a 6-1/2 year old playing a pink sparkly electric can be. 


SchecterWhore said:


> ElRay said:
> 
> 
> > The IMSLP has PDF's of all six books: Mikrokosmos, Sz.107 (Bartók, Béla) - IMSLP/Petrucci Music Library: Free Public Domain Sheet Music
> ...


I didn't catch that. I saw that the piece was nearly 40 years old and that is was a Polish engraving. I see it down at the bottom: Non-PD US 

Ray


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## Mr. Big Noodles (Jun 1, 2013)

ElRay said:


> Then she be about as badass as a 6-1/2 year old playing a pink sparkly electric can be.


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## smackhead999 (Jun 1, 2013)

Ok. Some observations on 151.

I dont have my notes in front of me at the moment, so I will go from memory without re-analyzing this piece.

I think the A Lydian and A Phrygian appear from the very beginning, although not developing fully until bar 10. However, I think the focus is C. It seems rather ambiguous from the beginning, without any particular qualities that identify a mode. Then the F# appears in bar 2 on the downward movement, suggesting the 6th if assuming A (could be Lydian). Then the F# steps down to F on the upward movement suggesting b6 (could be Phrygian), but also is a stepwise movement, opposite the upper melody. In bar 4, C# is brought out on the downward movement suggesting Lydian moving down on the melody, and inverted on the trichord. Then in bar 10 we have the A#, C#, and F# in the downward movement, but also the C, more or less completing both mode's development. This got me and the only thing I come up with is that at this point, both modes are occurring simultaneously. I think I hear a much stronger harmony coming in here, but Im not really sure.


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## smackhead999 (Jun 1, 2013)

I broke down the first 4 bars and measured the intervals between the notes, horizontally and vertically, and I really could not come up with any correlation between interval choices, harmonies, or tension resolve. It may take a little longer.

SW: What are your thoughts on the use of intervals, 2nd, 3rd, 4th, 5th harmonies related to this, and tension, resolve, and cadence?


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## ElRay (Jun 1, 2013)

SchecterWhore said:


> The cool thing about lydian and phrygian is that that pairing not only contains the complete chromatic set, but also preserves the tonic and dominant tones. C D E F# G A B C B&#9837; A&#9837; G F E&#9837; D&#9837; C.



So basically this is acting like a melodic minor, but you're playing lydian ascending and phrygian descending?

Ray


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## smackhead999 (Jun 1, 2013)

ElRay said:


> So basically this is acting like a melodic minor, but you're playing lydian ascending and phrygian descending?
> 
> Ray



I see where you get that from, because of the nature of the melodic minor being both minor and major in attributes. I think that is merely a coincidence in this case, but we will see what SW says about it.

I think the piece that I referenced for analysis, is all about the symmetry and what is happening with it. Starting very ambiguous, from nothing, slowly developing opposite qualities depending on the direction of the movement, and then once the qualities are fully developed, it seems that they come together and give a unique sound. From nothing, to opposing modes, to one conglomerate chord. But along the way, focusing on 2nd and 3rd intervals, and occasionally throwing in the 4th, etc... Not sure if that is his attempt to establish alternative (non tertian) harmony, or if its more about establishing "this is a lydian sounding collection, and this is a phrygian sounding collection, and now I bring them together."


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## smackhead999 (Jun 2, 2013)

I actually just came across this thread: 

http://www.sevenstring.org/forum/music-theory-lessons-techniques/212862-someone-help-explain-chromatics-me.html

I stumbled onto this concept of circle of 12ths, as he calls it, some time ago. It was beyond me when I discovered it. But as I was familiarizing myself with the polymodal chromaticism, I began to think of this. My original effort the other day of back and forths between lydian and phrygian were very reminiscent of the Blotted Science method. It really had that feel to it. Its very axial.


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## Ror (Jun 5, 2013)

smackhead999 said:


> I see where you get that from, because of the nature of the melodic minor being both minor and major in attributes. I think that is merely a coincidence in this case, but we will see what SW says about it.



I think what he actually meant was that it would function similarly to the melodic minor scale, in that you would use one set of tones when ascending melodically and a different one when descending. I don't think that's what SW was getting at, but that is nonetheless an interesting idea.

I forget where I encountered it, and I'm no expert so I don't know if this is a well-know method, but another method for creating a 12-tone piece is to separate the chromatic scale in to two hexachord groupings and then alternate the hexachords within the compostion, which I think could also be applied in a way that would imply polymodal compositon, but with a more serialistic approach. 

As a side note, I find all of this stuff very interesting, but to be honest thinking about this polytonal stuff really hurts my brain...


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## smackhead999 (Jun 5, 2013)

That 12 tone idea is pretty much what Ron Jazerombek (spelling?) Did with his circle of 12ths. Throws all of the notes up on a wheel in random order, then grabs systematic clusters and uses them as if they were chords. Its totally asymmetrical, but when used in a strict logic/rule based format, it becomes pretty cohesive.


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## Mr. Big Noodles (Jun 5, 2013)

ElRay said:


> So basically this is acting like a melodic minor, but you're playing lydian ascending and phrygian descending?
> 
> Ray





smackhead999 said:


> I see where you get that from, because of the nature of the melodic minor being both minor and major in attributes. I think that is merely a coincidence in this case, but we will see what SW says about it.



I don't think it's a bad observation. Melodic minor's ascending portion is meant to approach the tonic from below, and its descending portion is meant to approach the dominant from above. The lydian/phrygian thing is an approach to the dominant then the tonic from below, and then the dominant and tonic from above, so it's really the same idea. We can further abstract this to the tendency for "raised" notes to ascend and "lowered" notes to descend. Keep in mind that this is also a feature of non-Western modal music, and Bartók's language is partly derived from Eastern European folk idioms. I think he was fascinated by both the idea of following those harmonic tendencies as well as the organization of the chromatic scale that he got out of ascending/descending patterns.

Also, I've never enjoyed the explanation of melodic minor as a major scale with a &#9837;3, because while that covers the names of the notes that are played, it doesn't underline the harmonic and melodic reasons for the 6 and 7.



> I think the piece that I referenced for analysis, is all about the symmetry and what is happening with it. Starting very ambiguous, from nothing, slowly developing opposite qualities depending on the direction of the movement, and then once the qualities are fully developed, it seems that they come together and give a unique sound. From nothing, to opposing modes, to one conglomerate chord. But along the way, focusing on 2nd and 3rd intervals, and occasionally throwing in the 4th, etc... Not sure if that is his attempt to establish alternative (non tertian) harmony, or if its more about establishing "this is a lydian sounding collection, and this is a phrygian sounding collection, and now I bring them together."


I don't see it that way, and Bartók wasn't that kind of composer. I see C lydian hinted at on the first half of the first page, then a whole lot of modulations and superimpositions, but always with the same pitch set classes (for the most part). Third relationships seem to play a prominent part in the piece; the first modulation is to A, then the next is to F# - modulating by minor third each time. Then, the first time the melody is bitonally superimposed on itself (last system of the first page), it's at the interval of a major third. The return of the tonality of C (starting with the last measure of the first system on page 2) is marked by a different rhythm in the left hand and a different dynamic. Nothing special, standard variation fare. Where it gets back to forte, it's more messing around with thirds in a very explicit way. In fact, I hardly see anything from the minor/phrygian/locrian side of the table, unless I'm missing it, in which case I'd like to see it pointed out. Most everything seems to be lydian, ignoring the brief bouts of chromaticism that pepper the pitch material.



Ror said:


> I forget where I encountered it, and I'm no expert so I don't know if this is a well-know method, but another method for creating a 12-tone piece is to separate the chromatic scale in to two hexachord groupings and then alternate the hexachords within the compostion, which I think could also be applied in a way that would imply polymodal compositon, but with a more serialistic approach.



Schönberg and Webern were very much into interlocking hexachords (as well as other groupings), as was Josef Matthais Hauer. The last one made a system of composition around such a technique. I seem to remember a Berg piece in which a tone row was made from two whole tone scales, more or less. He wasn't as big of a serialist as his cohorts were, though. Maybe I'll remember the piece later.


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## ElRay (Jun 6, 2013)

ElRay said:


> ... but I haven't been able to find the arrangement ...



I found this: Amazon.com: Bela Bartok For Children : 25 Selected Pieces for Guitar.: Bela Bartok, arrangements Ferencz Brodsky: Books today. Now I just need to find a copy. I'm flying this weekend, I may try arranging it this weekend.

Ray


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