Below you will find my transcription of Bartók's 44 Duos for Two Violins. Below the transcription you will find that I have fragmented the piece into five sections with an analysis for each section. This piece is an excellent example of how Bartók used the octatonic scale in his exploration of post-tonal music.
I have tried to explain this piece in the vernacular. My goal is to take these complex ideas (ideas which text books often explain with overly-decorated language) and boil them down to their basic principles so that I and other musicians can use them in their work.
We can call this section A. Notice the exact pitches that the two voices play. The top violin plays A, B, C and D. The bottom violin plays D#, E#, F# and G#. It is kind of interesting how Bartók has rationed off four notes to each violin part. It is also interesting to see how each four-note set is based on a minor scale (A, B, C, and D are the first four notes in an A minor scale; D#, E#, F# and G# are the first four notes in a D# minor scale). Also notice how those two four-note sets are a tritone away from one another (A-D#). This is an easy way to remember the makeup of the OCTATONIC SCALE (an eight-note scale that alternates whole and half steps; a scale which contains two four-note groups based on a minor scale, each group being a tritone apart). In this case the complete scale is D# E# F# G# A B C D.
Now notice the range of each melodic line. You may notice how each melody spans a perfect fourth and how each has a a clear nadir (or lowest point). The returns to these low points serve to ground the listener into two pitch centers, each a tritone apart (D# and A). There is something about the lowest note of a phrase that can give our ear a hint as to what the pitch center may be. I think it is worth noting the explorations that Bartók is making in his music: one foot in atonality with the other still looking for ways to give the music some sort of pitch centricity. Refer to the right hand columns throughout this analysis to see the pitch centers for each violin part. This is a very simple example of how composers explored points of stability in post-tonal music.
The B section. Bartók uses the same 'splitting of the octatonic scale' technique, this time with the G# octatonic scale: the first four notes (G#, A#, B and C#) are played by the bottom voice and the second set of four notes by the top voice (D, E, F and G). This is the primary means by which the B section is tied to the A section. However, listen for all of the ways that the composer constrasts this section: new tempo, no slurs but now accent and tenuto markings, new dynamic, the voices are now in canon. Notice the emphasis on the low notes again.
Note: I am declaring G# as the starting pitch for spelling out this octatonic scale, since that is the lowest note in this section. It may be useful to remember that the octatonic scale is a symmetrical scale. In other words, you can only transpose the scale so many times before it begins to repeat itself. Compare the G# octatonic scale to the B octatonic scale to see this principle. This concept is also taught in most diatonic harmony courses (e.g. there are only so many fully diminished seventh chords).
Return to the A section with the violins trading melodies and inverting them. As a composer who is trying to learn from other composers I really enjoy the simplicity and straightforwardness of this piece.
I think it is important to look at how this section does not repeat the nadir of the melody as much as the first A section did. This provides the listener with a little less stability in terms of a pitch center or a foundation from which the melody is based. This is expected at this point in the piece. Compare this moment to the instability of the development section of sonata form.
Now the second violin leads the canon in the final iteration of the B section. Here we see Bartók move the point of imitation (the point in time at which the imitation begins). Notice how in the first B section the imitation begins in the second measure. This time the imitation is more immediate, occuring at the one-half measure. I like to think of this principle as somehow related to the harmonic acceleration that Classical composers applied to their phrase structures (harmony often changes more quickly as Classical music approaches a cadence). Since this is the fourth section of the piece, maybe we can even think of it like the fourth measure of a Mozart phrase (where Mozart often applied harmonic acceleration). I think it is important to take these compositional tools and think about them in a broader sense.
Another interesting fact is that this fourth section shares the same pitch centers as the first section. Perhaps that is a signpost pointing back to the opening material.
As the allargando takes us into the last four measures, we return to the original tempo and the A section material is inverted once again back to its original form. You will notice something very peculiar about the 8 notes that Bartok has used here at the end. The bottom voice plays Ab, Bb, Cb and Db and the top voice Eb, F, Gb and Ab. Both voices are once again playing the first four notes of a minor scale, but they are no longer seperated by a tritone but instead a perfect fifth. You will see that the Ab is found in both of these four-note collections. The resulting scale is not an octatonic one; it is instead an Ab minor scale with a raised 6th (Ab Dorian). For what reason could Bartok have decided to do this? Perhaps we should look at the pitch class that the original A section and B sections were built upon. D# is the lower pitch center of the entire A section and G# is the lower pitch center of the B section. These notes are enharmonic with the Ab/Eb pitch centers of this ending! It is as if these last few measures contain a mixture of section A and B in terms of pitch centricity. I feel like an analysis of this piece through the lense I have been seing it from is like an adventure that comes full circle.
For more information on post-tonal analysis (and on this piece in particular) refer to Understanding Post Tonal Music by Miguel Roig-Francoli.