Tonality and harmonic organization are questions that should be approached from two areas: that of "12-tone music" and that of its diametric opposite - "classical harmony". 12-tone music confronts us with a musical ideal that seeks to neutralize classical major-minor relationships, and be "intertonal" ¹⁾; consequently its harmonic building blocks will be tonally neutral or tonally indifferent equidistant chords (chords of equal pitch division) and their combinations, namely the circle of fifths or fourths:
 
 

Let us examine more closely in what manner the vocabulary of equidistant chords forms itself into the organic tonal system, which provides the harmonic unity of the Sonata for Two Pianos and Percussion, and of Bartók's music in general.

In the list given above, the only item which embraces the whole chromatic scale is the circle of fifths or fourths. One of the nearest natural harmonics, the perfect fifth, together with its inversion, the perfect fourth, is treated by Bartók - perhaps due to the influence of folk music - as the "queen" of intervals. Following these considerations, and the lessons to be drawn from analysis, the foundation of the present theory will be the "tonal compass rose" - the circle of fifths or fourths.

The augmented fourth (=diminished fifth) provides the constituents of the diminished seventh chord, while the whole-tone scale - lacking as it does possibilities for combination ²⁾ - proves to be impotent as a form-creating element; the main components of a system based upon the circle of fifths or fourths will therefore be primarily the augmented triad, and the diminished seventh chord. Let us see how!

A precondition of any system of tonality is a central point in relation to which others are dependent or subordinate. In the Sonata for Two Pianos and Percussion, the tonal system's central point c offers itself for the role of the "ruling" tonic note. If we set beside it "ministers" - on the right side a note with positive tension, on the left side a note with negative tension - then we have already simplified the question of the governement of the circle of fifths. Suitable for just this purpose is the equidistant tripartite division - ab-c-e - of the circle of fifths, the only question being what function we should apportion to the notes e and ab. If we look at the practice of European music, then e, the major third above the tonic, may with impunity be considered a dominant, and similarly ab, the major third below the tonic, a subdominant. Even in classical harmony we find these are lent a similar significance. ³⁾

Now we shall examine how the influence of the main functional notes - the tonic c, the dominant e, and the subdominant ab is distributed. If we proportionately divide the 12 tones of the chromatic scale between the 3 main functional notes, then each function acquires 4 "poles". The only arrangement of these poles that serves our ends, is to quarter by equidistant division the circle of fifths - thereby matching the "intertonal" musical ideal of 12-tone music - in which case we arrive at diminished seventh chords.

The following can therefore be affixed to one another:

this especially when we consider that the tonic character of c and a (degrees I and VI), the dominant character of g and e (degrees V and III) and the subdominant character of f and d (degrees IV and II) agree with the principles of classical harmony. On the other hand we have satisfied the requirements of 12-tone music by each of the above listed "tonally" joined notes c-a, g-e, f-d, being neutralized by its "counterpole" - slicing diagonally the circle of fifths - for example c is neutralized by f#.

This same result can be arrived at based upon another theoretical consideration as well. Taking note of the fifth-fourth attraction that binds the tonic to the dominant (V-I), the dominant to the subdominant (secondary dominant), and the subdominant to the tonic (I-IV), thus regarding Bartók's 12-tone musical functions based upon the circle of fifths or fourths, we see the following system:

If c is therefore regarded as the tonic, f (degree IV) as the subdominant and g (degree V) as the dominant, then a (as relative to the tonic) will have a tonic significance, d (as relative to the subdominant) a subdominant significance, and e (as relative to the dominant) a dominant significance. The circle of fifths segment f-c-g-d-a-e is, therefore, matched by the set of functions S-T-D-S-T-D. In this relationship a certain regularity is to be observed, as the S-T-D periodically recurs. In essence the above diagram does no more than extend this S-T-D attraction to the whole circle of fifths!

The clearly defined separate levels of the tonic, dominant and subdominant "axes" (to coin a name, lacking terminology) remain either practically intact, or are interrupted by an answer at the fifth during their participation in the overall formal structure.

Within each axis, viewed as a unified harmonic plane, the tonal tension produced may be of two kinds:

1. Tension between the opposing pole and counterpole. By counterpole is meant the note of the pole in the circle of fifths that at the same time splits the octave in half, e.g. c and f#. This pole-counterpole relationship should be regarded as one of the most fundamental formal principles of the Sonata. A basic tenet of the axis system theory is that the pole may be exchanged at any time for its counterpole without a change of function. (A cadential harmonic progression E-A-D-G-C-F in Bartók's system can, for example, be conceived as follows: E-A-Ab -C#-C-F, in which the d and g are replaced by ab and c#.)

2. Tension between the opposing main-branch and sub-branch. The "main-branches" - in the diagram (Figure 3) - are the lines joining the counterpoles of the main functional notes, that of the tonic axis being c-f#, that of the dominant e-bb, and of the subdominant ab-d. (For this reason the terms "Bartók dominant" and "Bartók subdominant" are used in this analysis to mean e or bb, and ab or d, respectively.) The "sub-branches" intersect the main-branches at right angles, that of the tonic axis being a-eb , that of the dominant g-c#, and of the subdominant f-b. ⁴⁾

It is essential that the individual axes be seen not as diminished sevenths, but as the synthesis of two tritone relationships, i.e. as a scheme wherein the "opposite" sides react to each other more sensitively than the "neighbouring" ones. The differentiation between the main-branch and the sub-branch became necessary not just systematically. We shall see how the "Bartók dominant and subdominant" are given a much more emphatic role than the classically understood dominant and subdominant. For example, in movement I a powerful appearance of the tonic c is almost always preceded by the tonality of bb. ⁵⁾

A dominant-tonic resolution, therefore, may be the following - in the tonality of c - based on the possibilities inherent in the axis system: G-C, E-C, Bb-C, the fourth and last being C#-C; that is to say, in the axis system a dominant of C is also C#. This resolution is reserved by Bartók for occasions when something unexpected takes place, such as a change of scene. The explanation for this is that after C#, cadentially we expect F#, whereupon the F# is surprisingly exchanged for its counterpole, C, the expected C#-F# being resolved instead by the "Bartók deceptive cadence", the progression C#-C, (cf. Music for Strings, Percussion and Celesta, movement IV, bs. 73-74, 98-99, 113-114, 243-244).

The constituents of the axis system, therefore, are:
 

Unquestionably, the instrumental combination of the work, with the neutral sound of the percussion, provided a very attractive opportunity to thoroughly put into effect a tonal conception of this nature.

The minor third intervals that lie between the poles of a particular axis can best be compared to the relative major-minor relationship in the classical harmonic system. Nonetheless, the concept of "tonic", "subdominant" or "dominant" axes does not accord with the similarly named concepts in classical harmony. Thus the dominant axis resolves onto the tonic not via a leading note, but upwards via a dorian leap of a major second (often to be observed with the progression bb-c), or downwards via a phrygian leap of a minor second (db-c). In contrast to the classical dominant-tonic relationship (i.e. T-S-D-T), the axis system often follows the plagal succession subdominant-tonic (i.e. T-D-S-T).

The extraordinary tension of Bartók's tonal system is very much connected to the fact that it arises as a struggle between two spheres of opposite attraction: major-minor tonality (the tonality of diatonicism) and the axis system (the tonality of chromaticism).

If we look back at the history and development of harmonic thinking, then we are bound to say that the birth of the axis system was a historical necessity, signifying the logical continuation of the development of Western music and to some extent its climax. At the beginning of harmonic thinking, a tonic, subdominant or dominant significance was clearly ascribed only to degrees I, IV and V of the scale. Classical harmony began to include primary and secondary triads, degree I being replaced by the relative degree VI, degree IV by the relative II, and degree V by the relative III. Romantic harmony went even further, making extensive use of the upper relatives, for example in the tonality of C:
 

From here it is but a short step to the system "closing up": the axis extends the employment of relatives to the whole system, since f# is a relative common to the tonics a and eb, b is a relative common to the subdominants d and ab, and c# is a relative common to the dominants e and bb. The functional principle remains unchanged by all this; it is simply that the number of juxtaposed planes has increased and become more diverse - thanks to the twelve tones. Bartók's harmonic system, therefore, is not a resumption or a new departure, but a climax and a fulfilment (indeed historically speaking, the axis theory has a certain retrospective aspect for musical research). Here a distinction must be drawn between Bartók's dodecaphony and Schönberg's "Zwölftonmusik". Schönberg demolishes and annihilates tonality, while Bartók, with heroic effort, amalgamates the principles of harmonic thinking into what is so far the most exquisite and perfect synthesis, one which matches the technical standards of the times.

Here we must also touch upon the acoustic interpretation of the axis system. A proviso of moving from the dominant to the tonic is that we move from a harmonic to the fundamental. According to this, the dominant of c will be not only g, but also e and bb; it was precisely these that went to make up earlier the dominant axis main-branch.

And because the
 

then e and bb will be the dominants of the tonic base note c,

c and f# will be the tonics of the subdominant base note ab,

and ab and d will be the subdominants of the dominant base note e:

If to this we add the role played by the harmonic at the fifth, then from these relationships the complete axis system can be demonstrated.

An undeniable prerequisite of an "axis-like" organization of the tonal system was the given nature of musical feeling in general, and the fact that Bartók's ideas derived from the best traditions, not just regarding their theoretical organization and execution, but - what is even more significant - their meaning and content as well. At this point the revolutionary endeavours of Liszt and Moussorgsky come to mind; the pieces by Liszt built from perfect equidistance - unfortunately little known - and the "axis system" experiments of Moussorgsky. Without exception, all of these, from the "Nuages gris", "Unstern", "Preludio funebre", death music ("R.Wagner. Venezia") and ghostly "Funeral gondolas" of Liszt (all late piano pieces) to the Mad Scene from Boris Godunov (pure axis system), reflect the negative side of life - dark, demonic and irrational experiences. These ideas, either consciously or unconsciously, also find expression in Bartók, in that for him, chromaticism (12-tone music, equidistance, the axis system) is synonymous with a demonic and irrational world, while diatonicism is associated with an optimistic and bright one - a subject returned to and further demonstrated in the chapter entitled "Meaning".

To summarize, therefore: the above tonal system of Bartók has been shown from four different perspectives: 12-tone music (symmetrical divisions of the system), classical harmony (functional relations), the historical development of music, and acoustics. That in every case these result in a congruity of means, shows that Bartók, when he created his musical material, penetrated to the roots of music, to its most inherent elements. Everything that was true he recognized as the truth, and out of these truths he created a most exquisite unity. Before dealing further with a demonstration of the axis system, we must turn our attention to a number of matters concerning Bartók's use of proportion.