As one listens to the piece it seems as though there are two major compositional formulas that begin to take shape. Each formula contains several components of theory and the use of compositional tools. The first, as the the pitch material descends, the note values increase allowing the effect of the music to slow down. The second, as the pitch material ascends, the note values decrease allowing the effect of the music to speed up. The climax on the first effect becomes distant, the dynamics decrease along with the canons, the strettos, the inversions, and the retrogrades become stretched which allows the transition of new themes. The climax on the second effect becomes closer, the dynamics increase bringing along a more intense use of the canon, the stretto, inversions, retrogrades, and intervals - a simply marvelous concept.
Measure 14 marks the beginning of the second theme including a third group of four notes. Using hemeolas, ametric time, and circular rhythm, Bartók brings in the new notes, G#, C#, D, G, while combining additional dynamics of pianissimo. Antokoletz identified 3 groups of four notes and put them into a formula of group, X, Y, and Z. Group X, represents the first group of four notes at the introduction of the piece, C, C#, D, D#. Group Y represents the second group at the exposition, Bb, C, D, E. And Group Z represents the last group, G#, C#, D, G. Antokoletz also refers to the notes as pitches in a twelve tone row between 0 and 11.
In an article written in The Journal of Musicology "The overall theory of utilizing three groups of four notes is a staple theory of Bartók and has become labeled Spacial and Psychoacoustic Factors in Atonal Prolongation."(11) Lendvai discovered, "No elements can be properly joined without the aid of a third one, for the two can only be united by the mediation of a link; but of all the links that one is the most beautiful which makes a complete whole of itself and of the elements united by it."(12)
When one analyzes Bartók they will find elements of the number three in many if not all of his later compositions. There is the arch form which is made up of three sections, there are numerous examples of hemeolas using three-over-four, the golden section featuring the .618 and its difference of .382 and even Bartók's axis theories shows three uses of tonic, dominant, and subdominant theories in the circle of fifths. On a side note, Lendvai points out Bartók's use of the circle of fifths using four note groups in opposite poles in example 4.
Lendvai explains, "chords based on the fundamental C, Eb, F#, A have a tonic function. Chords based on the fundamental F, B, D, Ab, have a subdominant function. Chords based on Db, G, E, Bb have a dominant function."(13)
Not everyone, including Bartók, however agrees with Erno Lendvai. Fred Lerdahl writes from his 1996 lecture regarding Spatial and Psychoacoustic Factors in Atonal Prolongation, " I see little interest in making a theory of atonal music that cannot find its place in a general theory of music."(14) Perhaps Bartók knew this idiom to be true because his pitch material and use of traditional forms reflect the stability of tonal musical theories, or perhaps Bartók just uses traditional compositional tools to bring familiarity to the piece so that there can be a common frame of reference. Halsey Stevens quotes Bartók from his book, The Life and Music of Belá Bartók, "I do not care to subscribe to any of the accepted contemporary musical tendencies...my ideal is a measures balance of these elements."(16)
Still More to Come...
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